Research ArticleAn Accurate Predictor-Corrector-Type Nonstandard FiniteDifference Scheme for an SEIR Epidemic Model

Asma Farooqi12 Riaz Ahmad1 Rashada Farooqi3 Sayer O Alharbi4

Dumitru Baleanu 567 Muhammad Rafiq8 Ilyas Khan 4 and M O Ahmad9

1Faculty of Science Yibin University Yibin 644000 China2School of Energy and Power Engineering Xirsquoan Jiaotong University Xirsquoan 710049 China3Wah Medical College POF Hospital Wah Cantt 47040 Pakistan4Department of Mathematics College of Science Al-Zulfi Majmaah University Al-Majmaah 11952 Saudi Arabia5Department of Mathematics Cankaya University Ankara 06790 Turkey6Institute of Space Sciences Magurele 077125 Romania7Department of Medical Research China Medical University Hospital China Medical University Taichung 40447 Taiwan8Department of Mathematics Faculty of Sciences University of Central Punjab Lahore Pakistan9Department of Mathematics and Statistics University of Lahore Lahore 54590 Pakistan

Correspondence should be addressed to Dumitru Baleanu baleanumailcmuhorgtw and Ilyas Khan isaidmuedusa

Received 11 September 2020 Revised 23 September 2020 Accepted 22 November 2020 Published 15 December 2020

Academic Editor Hijaz Ahmad

Copyright copy 2020 Asma Farooqi et al-is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

-e present work deals with the construction development and analysis of a viable normalized predictor-corrector-typenonstandard finite difference scheme for the SEIR model concerning the transmission dynamics of measles -e proposednumerical scheme double refines the solution and gives realistic results even for large step sizes thus making it economical whenintegrating over long time periods Moreover it is dynamically consistent with a continuous system and unconditionallyconvergent and preserves the positive behavior of the state variables involved in the system Simulations are performed toguarantee the results and its effectiveness is compared with well-known numerical methods such as RungendashKutta (RK) and Eulermethod of a predictor-corrector type

1 Introduction

Mathematicians and biologists have been working for a longtime on the biological process of life science-ey succeededin evaluating some remarkable results from their work Animportant mark in mathematical biology is the mathe-matical modeling of infectious diseases [1] Measles is one ofsuch a highly infectious childhood disease caused by re-spiratory infection by a Morbilli virus measles virus ArthurRansom first observed the irregular cyclic behavior ofmeasles that is considered as a mainly conspicuous facet ofmeasles -e age structure of the population contact im-migration rate and the school seasons were known as acrucial phase for the swell of measles [2ndash4] WilliamHammer in 1906 published a discrete numerical model for

the transmission of measles epidemic Later assumption ofldquoMass Actionrdquo is applied to that model which is the basic ruleto the current theory of deterministic modeling of infectiousdiseases [5]

Society has a keen concern in knowing the major evo-lution for the spread of diseases Analytical results give asolution to these problems but for limited cases and causesmany problems -e homotopy perturbation method andvariational iteration method can be used for the solution ofthe epidemic models [6] However the first choice to solvethese laws of nature is the numerical method based on adifference scheme for good approximations [7ndash9] In gen-eral already developed numerical schemes such as EulerRungendashKutta and others at times stop working by gener-ating nonphysical results -ese unnecessary oscillations

HindawiJournal of MathematicsVolume 2020 Article ID 8830829 18 pageshttpsdoiorg10115520208830829

contrived chaos and false fixed points [10] Moreover somemethods are unsuccessful if we check them on larger stepsizes [11] To avoid such discrepancies the numericalschemes based on the ldquononstandard finite difference method(NSFD)rdquo are established -ese techniques were first de-veloped by R E Mickens [7 12 13] -e created numericalschemes preserve the essential properties such as dynamicalconsistency stability and equilibrium points [14ndash21] Re-searchers have developed competitive NSFD schemes forepidemic diseases Many of these NSFD schemes are con-sistent for small step sizes with the continuousmodel but forlarge step sizes the unwanted oscillations have been ob-served Piyanwong Jansen and Twizel have constructed apositive and unconditionally stable scheme for SIR and SEIRmodels respectively [22 23] Nevertheless the lack of ap-plication of conservation law in their developed schemesexplicitly caused impracticable and unrealistic solutionswhile Abraham and Gilberto have developed NSFD schemesof the SIR epidemic model to obtain the physically realisticsolutions for all step sizes where they apply the conservationlaw in addition to nonlocal approximation [24]

In this paper we have developed a normalized NSFDMof predictor-corrector- (PC-) type inspired by the previouswork discussed to double refine the numerical solution of anonlinear dynamics regarding the transmission of measlesTo keep the method explicit we will use the forward dif-ference approximations for the first derivative terms -enonlocal approximations are used to tackle the nonlinearterms with φ(h) as a nonclassical denominator function Byusing this idea the measles model will converge to equi-librium points even for the larger step sizes

2 The Mathematical Frame of Work

In this work the dynamics of the measles epidemic de-scribed by the SEIRmathematical model suggested by Jansenand E H Twizel is considered [23] In the SEIR model thetotal human population is categorized as susceptible ex-posed infectious and recovered subpopulations denoted byS E I and R respectively Consider the flow of the SEIRmodel for the measles epidemic as shown in Figure 1Here

S susceptible individualsE exposed individualsI infected individualR recovered individualμ birth rate and death rateβ the rate at which susceptible individuals are in-fected by those who are infectiousσ the rate at which exposed individuals becomeinfectedc the rate at which infected individuals recover

Here μ β σ and c are considered as positive parametersFurthermore we assume that (E0 I0 ne 0) Consider N is theconstant size of the population so that the number of re-covered individuals R R(t) at time t is defined byR N minus S minus E minus I A susceptible is a move to the exposed

model where the individual is infected but yet not infec-tious After sometimes the individual becomes infectiousand enters into the infected compartment and in this waythe disease spreads into the population

-e mathematical model is written as

dS

dt μN minus μS minus βIS tgt 0 S(0) S

0

dE

dt βIS minus μE minus σE tgt 0 E(0) E

0

dI

dt σE minus μI minus cI tgt 0 I(0) I

0

dR

dt cI minus μR tgt 0 R(0) R

0

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(1)

where

S + E + I + R N (2)

Equation (2) shows that the total size of the populationremains constant which is connected with the continuoussystem (1) -e following two points give the equilibriumpoints of (1)

-e disease-free equilibrium (DFE) point (N 0 0)

-e endemic equilibrium (EE) point (NR0 μNμ+

σ(1 minus 1R0) μβ(R0 minus 1))

Here R0 σβN(μ + σ)(μ + c) is the basic reproductivenumber associated with the measles model

For the sake of brevity we are not mentioning RK-4 andEuler-PC scheme

3 Numerical Modeling

To the construction of NSFD scheme the continuous system(1) is discretized using the forward difference approximationfor the first-order time derivatives -us if f(t) is differ-entiable the fprime(t) can be approximated by

df(t)

dt

f(t + h) minus f(t)

φ(h)+ O(h) as h⟶ 0 (3)

where φ(h) is a real-valued function satisfying the conditionφ(h)⟶ 0 as h⟶ 0 We have

φ(h) 1 minus eminus h

(4)

μN βSI μE

μS

γI

μI μR

I(t) R(t)

S(t) E(t)

σE

Figure 1 Flowchart of the SEIR Model for measles epidemic

2 Journal of Mathematics

-us the NSFD scheme for system (1) takes the form

Sn+1

minus Sn

φ(h) μN minus μS

n+1minus βS

n+1I

n

En+1

minus En

φ(h) βS

n+1I

nminus μE

n+1minus σE

n+1

In+1

minus In

φ(h) σE

n+1minus μI

n+1minus cI

n+1

Rn+1

minus Rn

φ(h) cI

n+1minus μR

n+1

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(5)

From equation (5) we have

(1 + φ(h)μ) Sn+1

+ En+1

+ In+1

+ Rn+1

1113960 1113961

φ(h)μN minus φ(h)μ Sn

+ En

+ In

+ Rn

( 1113857(6)

-us if Sn + En + In + Rn N for all nge 0 then Sn+1+

En+1 + In+1 + Rn+1 N for all nge 0-us the conservation law property is satisfied System

(5) is written as

Sn+1

S

n+ μφ(h)N

1 + φ(h) μ + βIn

( 1113857

En+1

E

n+ φ(h)βS

n+1I

n

1 + φ(h)(μ + σ)

In+1

I

n+ φ(h)σE

n+1

1 + φ(h)(μ + c)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(7)

Now the positivity of Sn+1 En+1 In+1 andRn+1 is guar-anteed if 0lt Sn lt 1 0ltEn lt 1 0lt In lt 1 and 0ltRn lt 1 for allnge 0 -us the constructed scheme has the followingproperties

-e conservation law is satisfied-e positivity and boundedness of the solution issatisfied for system (7) we have that if 0lt Sn lt 10ltEn lt 1 0lt In lt 1 and 0ltRn lt 1 then 0lt Sn+1 lt 10ltEn+1 lt 1 0lt In+1 lt 1 and 0ltRn+1 lt 1 for all nge 0

4 NSFD Predictor-Corrector Scheme

-is section is improved by the approach of a predictor-corrector-type NSFD scheme (7) to obtain the benefits ofboth methods For the development of this scheme firstlysystem (7) is taken as a predictor scheme ie

Sn+1p

Sn

+ μφ(h)N

1 + φ(h) μ + βIn

( 1113857

En+1p

En

+ φ(h)βSn+1p I

n

1 + φ(h)(μ + σ)

In+1p

In

+ φ(h)σEn+1p

1 + φ(h)(μ + c)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(8)

Now we evaluate system (1) at time t + φ(h) and in-troduce the term ϵminus 1Sn+1φ(h) (where ϵ is just like the ac-celerating factor and its range is (0lt ϵlt 1)) -us theexpression will be NSFDCL scheme that preserves conser-vation law represented as

Sn+1

minus Sn

φ(h) μN minus μS

n+1minus βS

n+1I

n+1minusϵminus 1

Sn+1

φ(h)+ϵminus 1

Sn+1

φ(h)

En+1

minus En

φ(h) βS

n+1I

nminus (μ + σ)E

n+1

In+1

minus In

φ(h) σE

n+1minus (μ + c)I

n

Rn+1

minus Rn

φ(h) cI

n+1minus μR

n+1

(9)

-us the corrector scheme is obtained as

Sn+1c

Sn

+ μφ(h)N + ϵminus 1S

n+1p

1 + ϵminus 1 + μφ(h) + βφ(h)In+1p

En+1c

En

+ φ(h)βSn+1c I

n+1p

1 + φ(h)(μ + σ)

In+1c

In

+ φ(h)σEn+1c

1 + φ(h)(μ + c)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(10)

where Rn+1 N minus Sn+1c minus En+1

c minus In+1c

5 Convergence Analysis

In this section the unconditional convergence of the nu-merical solution is presented by the proposed methodTaking the values as

F1(S E I) S + μNϕ(h)

1 + ϕ(h)[μ + βI]

F2(S E I) E + βϕ(h)IF1

1 + ϕ(h)[μ + σ]

F3(S E I) I + σϕ(h)F2

1 + ϕ(h)[μ + c]

G1(S E I) S + μNϕ(h) + ϵminus 1

F1

1 + ϵminus 1 + μϕ(h) + βϕ(l)F3

G2(S E I) E + βϕ(h)G1F3

1 + ϕ(h)[μ + σ]

G3(S E I) I + σϕ(h)G2

1 + ϕ(h)[μ + c]

(11)

where F1 F2 andF3 are given as in equation (8)

Journal of Mathematics 3

-e convergence and stability analysis of scheme (10) iscarried out by calculating the eigenvalues of the Jacobian ofthe linearized scheme and studying its behavior and

evaluated at the fixed points If (Slowast Elowast Ilowast) is the fixed pointof system (1) then the Jacobian matrix JG is given by

JG Slowast Elowast Ilowast

( 1113857

zG1 Slowast Elowast Ilowast

( 1113857

zS

zG1 Slowast Elowast Ilowast

( 1113857

zE

zG1 Slowast Elowast Ilowast

( 1113857

zI

zG2 Slowast Elowast Ilowast

( 1113857

zS

zG2 Slowast Elowast Ilowast

( 1113857

zE

zG2 Slowast Elowast Ilowast

( 1113857

zI

zG3 Slowast Elowast Ilowast

( 1113857

zS

zG3 Slowast Elowast Ilowast

( 1113857

zE

zG3 Slowast Elowast Ilowast

( 1113857

zI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(12)

where

zF1

zS

1[1 + μϕ(h) + βϕ(h)I]

zF1

zE

zF1

zI

minus (S + μNϕ(h))βϕ(h)

[1 + μϕ(h) + βϕ(h)I]2

zF2

zS

βϕ(h)IzF1zS

1 + ϕ(h)[μ + σ]

zF2

zE1 + βϕ(h)IzF1zE

1 + ϕ(h)[μ + σ]

zF2

zIβϕ(h) F1 + IzF1zI1113864 1113865

1 + ϕ(h)[μ + σ]

zF3

zE

σϕ(h)zF2zE

1 + ϕ(h)[μ + c]

zF3

zI1 + σϕ(h)zF2zS

1 + ϕ(h)[μ + c]

zG1

zS

1 + ϵminus 1zF1zS1113872 1113873 1 + ϵminus 1

+ μϕ(h) + βϕ(h)F31113960 1113961 minus S + μNϕ(h) + ϵminus 1F11113872 1113873βϕ(h)zF3zS

1 + ϵminus 1 + μϕ(h) + βϕ(l)F31113960 11139612

zG1

zEϵminus 1

zF1zE1113872 1113873 1 + ϵminus 1+ μϕ(h) + βϕ(h)F31113960 1113961 minus S + μNϕ(h) + ϵminus 1

F11113872 1113873βϕ(h)zF3zE

1 + ϵminus 1 + μϕ(h) + βϕ(h)F31113960 11139612

zG1

zIϵminus 1

zF1zI1113872 1113873 1 + ϵminus 1+ μϕ(h) + βϕ(h)F31113960 1113961 minus S + μNϕ(h) + ϵminus 1

F11113872 1113873βϕ(h)zF3zI

1 + ϵminus 1 + μϕ(h) + βϕ(h)F31113960 11139612

zG2

zS

βϕ(h)

1 + ϕ(h)[μ + σ]

zG1

zSF3 + G1

zF3

zS1113896 1113897

zG2

zE1 + βϕ(h) zG1zEF3 + G1zF3zE1113858 1113859

1 + ϕ(h)[μ + σ]

4 Journal of Mathematics

zG2

zI

βϕ(h)

1 + ϕ(h)[μ + σ]

zG1

zIF3 + G1

zF3

zI1113896 1113897

zG3

zS

σϕ(h)zG2zS

1 + ϕ(h)[μ + c]

zG3

zE

σϕ(h)zG2zE

1 + ϕ(h)[μ + c]

zG3

zI1 + σϕ(h)zG2zI

1 + ϕ(h)[μ + c]

(13)

51 Disease-Free Equilibrium (SoEo Io) (N 0 0) -evalues of functions and its derivatives at DFE point (N 0 0)

are as

F1(N 0 0) N F2(N 0 0) 0 F3(N 0 0) 0

G1(N 0 0) N G2(N 0 0) 0 G3(N 0 0) 0

zF1(N 0 0)

zS

11 + μϕ(h)

zF1(N 0 0)

zE 0

zF1(N 0 0)

zI

minus Nβϕ(h)

[1 + μϕ(h)]

zF2(N 0 0)

zS 0

zF2(N 0 0)

zE

11 + ϕ(h)[μ + σ]

zF2(N 0 0)

zI

βϕ(h)

1 + ϕ(h)[μ + σ]

zF3(N 0 0)

zS 0

zF3(N 0 0)

zE

σϕ(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]]

zF3(N 0 0)

zI1 +[σϕ(h)βNϕ(h)1 + ϕ(h)(μ + c)]

1 + ϕ(h)[μ + c]

zG1(N 0 0)

zS

11 + μϕ(h)

zG1(N 0 0)

zE

minus Nσϕ(h)βϕ(h)

1 + μϕ(h) + ϵminus 11113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

Journal of Mathematics 5

zG1(N 0 0)

zI

minus Nβϕ(h)

1 + ϵminus 1+ μϕ(h)1113960 1113961

ϵminus 1

1 + μϕ(h)+

1[1 + ϕ(h)(μ + c)

+σNβϕ2

(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113890 1113891

zG2(N 0 0)

zS 0

zG2(N 0 0)

zE

11 + ϕ(h)(μ + σ)

1 +Nβσϕ2(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113890 1113891

zG2(N 0 0)

zI

βϕ(h)

[1 + ϕ(h)(μ + σ)]

1 + ϕ(h)(μ + σ) + Nσβϕ2(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113896 1113897

zG3(N 0 0)

zS

zG3(N 0 0)

zE

σϕ(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1 +

σβNϕ2(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113896 1113897

zG3(N 0 0)

zI

1

[1 + ϕ(h)(μ + σ)]+

Nσβϕ2(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]2 +

N2β2σ2ϕ4(h)

[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + σ)]

2

(14)

For ease in the calculation we use the followingconventions

α μNϕ(h)gt 0 η1 μϕ(h)gt 0 η2

βϕ(h)gt 0 η3 σϕ(h)gt 0

θ 1 + ϵminus 1+ μϕ(h) 1 + ϵminus 1

+ η1 gt 1

δ1 1 + ϕ(h)[μ + σ]gt 1 δ2

(15)

It is clear that θgt η1 1 + η1 gt δ1 1 + η1 gt δ2 and δ2 θ minus ϵminus 1 + cϕ(h) If R0 σβN(μ + σ)(μ + c)lt 1 the Ja-cobian calculated at disease-free points (Slowast0 Elowast0 Ilowast0 )

(N 0 0) is given as

JG(N 0 0)

zG1(N 0 0)

zS

zG1(N 0 0)

zE

zG1(N 0 0)

zI

zG1(N 0 0)

zS

zG1(N 0 0)

zE

zG1(N 0 0)

zI

zG1(N 0 0)

zS

zG1(N 0 0)

zE

zG1(N 0 0)

zI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)

Since the partial derivatives using the conventions are

zG1(N 0 0)

zS

11 + η1

zG1(N 0 0)

zE

minus Nη2η3θδ1δ2

zG1(N 0 0)

zI

minus Nη2θϵminus 1

1 + η1+

1δ2

+Nη2η3δ1δ2

1113890 1113891

zG2(N 0 0)

zS 0

zG2(N 0 0)

zE

1δ1

+Nη2η3δ21δ2

1113890 1113891

zG2(N 0 0)

zI

Nη2δ1δ2

+N

2η22η3δ21δ2

1113890 1113891

zG3(N 0 0)

zS 0

zG3(N 0 0)

zE

η3δ1δ2

+Nη2η

23

δ21δ22

1113890 1113891

zG3(N 0 0)

zI

1δ2

+Nη2η3δ1δ

22

+N

2η22η23

δ21δ22

1113890 1113891

(17)

6 Journal of Mathematics

So

JG(N 0 0)

11 + η1

minus Nη2η3θδ1δ2

minus Nη2θϵminus 1

1 + η1+

1δ2

+Nη2η3δ1δ2

1113890 1113891

01δ1

+Nη2η3δ21δ2

1113890 1113891Nη2δ1δ2

+N

2η22η3δ21δ2

1113890 1113891

0η3δ1δ2

+Nη2η

23

δ21δ22

1113890 11138911δ2

+Nη2η3δ1δ

22

+N

2η22η23

δ21δ22

1113890 1113891

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)

To calculate the eigenvalues put det (J minus λI) 0

λ1 11 + η1 11 + μϕ(h)lt 1 And

λ2 minus λδ1δ2 + Nη2η3

δ21δ21113888 1113889 +

δ21δ2 + δ1Nη2η3 + N2η22η

23

δ21δ22

1113888 11138891113890 1113891

+δ1δ2 + Nη2η3

δ21δ21113888 1113889

δ21δ2 + δ1Nη2η3 + N2η22η

23

δ21δ22

1113888 1113889 minusNη2δ1 + N

2η22η3δ21δ2

1113888 1113889η3δ1δ2 + Nη2η

23

δ21δ22

1113888 11138891113890 1113891 0

λ2 minus Aλ + B 0

(19)

where

A δ1δ2 + Nη2η3δ21δ21113872 1113873 + δ21δ2 + δ1Nη2η3 + N

2η22η23δ

21δ

221113872 1113873 trace J

lowast

B δ1δ2 + Nη2η3

δ21δ21113888 1113889

δ21δ2 + δ1Nη2η3 + N2η22η

23

δ21δ22

1113888 1113889 minusNη2δ1 + N

2η22η3δ21δ2

1113888 1113889η3δ1δ2 + Nη2η

23

δ21δ22

1113888 11138891113890 1113891 det Jlowast

Jlowast

1δ1

+Nη2η3δ21δ2

1113890 1113891Nη2δ1δ2

+N

2η22η3δ21δ2

1113890 1113891

η3δ1δ2

+Nη2η

23

δ21δ22

1113890 11138911δ2

+Nη2η3δ1δ

22

+N

2η22η23

δ21δ22

1113890 1113891

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

To calculate the eigenvalues of Jlowast the following lemma isproved

Lemma 1 For the quadratic equation λ2 minus λA + B 0 bothroots satisfy |λi|lt 1 i 2 3 if and only if the followingconditions are satisfied [25]

(1) 1 minus A + Bgt 0

(2) 1 + A + Bgt 0

(3)Blt 1

(21)

Let us define A TraceJlowast B DetJlowast -erefore

Journal of Mathematics 7

A δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N

2η22η23

δ21δ22

gt 0

B 1

δ1δ2+

Nη2η3δ21δ

22lt 1

f(minus 1) 1 + A + Bgt 0

f(0) B 1

δ1δ2+

Nη2η3δ21δ

22lt

1δ1δ2

+Nη2η3δ1δ2

1 + Nη2η3

δ1δ2

1 + R0(μ + σ)(μ + c)ϕ2(h)

1 + ϕ2(h)(μ + σ)(μ + c) + ϕ(h)[(μ + σ)(μ + c)]

lt 1 sinceR0 lt 1( 1113857

rArrBlt 1

(22)

Now for

f(1) 1 minus A + B

δ21δ

22 + δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N

2η22η23 + Nη2η3 + δ1δ2

δ21δ22

gt 0

(23)

As all the conditions of the lemma hold for R0 lt 1therefore the absolute value of both eigenvalues of Jlowast is less

than one whenever R0 lt 1 -us the numerical scheme inequations (8) and (10) will converge unconditionally todisease-free equilibrium (N 0 0) for any value of time step h

whenever R0 lt 1 which is also computationally verified inFigure 2(a)

52 Endemic Equilibrium (SeEe Ie) (NR0 μN μ + σ(1 minus

1 R0) μβ(R0 minus 1)) -e value of functions and its deriv-atives at an endemic point is as

F1 Se Ee Ie( 1113857 N

R0

F2 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857

(μ + σ)

F3 Se Ee Ie( 1113857 μβ

R0 minus 1( 1113857

G1 Se Ee Ie( 1113857 N

R0

G2 Se Ee Ie( 1113857 μβ

R0 minus 1( 1113857

G3 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857

(μ + σ)

(24)

8 Journal of Mathematics

zF1 Se Ee Ie( 1113857

zS

11 + μϕ(h)R0

zF1 Se Ee Ie( 1113857

zE 0

zF1 Se Ee Ie( 1113857

zI

minus Nβϕ(h)

1 + μϕ(h)R0

zF2 Se Ee Ie( 1113857

zS

μϕ(h) R0 minus 1( 1113857

1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + σ)]

zF2 Se Ee Ie( 1113857

zE

1[1 + ϕ(h)(μ + σ)]

zF2 Se Ee Ie( 1113857

zI

minus Nβϕ(h) μϕ(l)R20 minus 2μϕ(h)R0 minus 11113960 1113961

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

zF3 Se Ee Ie( 1113857

zS

μσϕ2(h) R0 minus 1( 1113857

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

zF3 Se Ee Ie( 1113857

zE

σϕ(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

zF3 Se Ee Ie( 1113857

zI

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h) minus 11113960 1113961

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

(25)

zG1 Se Ee Ie( 1113857

zS

11 + μϕ(h)R01113858 1113859

minusNμβσϕ3(h) R0 minus 1( 1113857

R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

zG1 Se Ee Ie( 1113857

zE

minus Nβσϕ2(h)

R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

Eige

n va

lues

Jacobian at DFE

0492

0494

0496

0498

05

0502

0504

0506

0508

051

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size

(a)

098

0981

0982

0983

0984

0985

0986

0987

0988

0989Jacobian at EE

Eige

n va

lues

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size

(b)

Figure 2 Spectral radius of the Jacobian at equilibrium points

Journal of Mathematics 9

zG1 Se Ee Ie( 1113857

zI

minus Nβϕ(h)

1 + ϵminus 1 + μϕ(h)R01113960 11139611 + μϕ(h)R0

middot ϵminus 1+

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

⎡⎣ ⎤⎦

zG2 Se Ee Ie( 1113857

zS

μβϕ(h) R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890

minusNμβ2σϕ4(h)μ R0 minus 1( 1113857

2

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nμβσϕ3(h) R0 minus 1( 1113857

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

1113891

zG2 Se Ee Ie( 1113857

zE

1[1 + ϕ(h)(μ + σ)]

+Nβσϕ2(h)

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]

minusNμβ2σϕ3

(h) R0 minus 1( 1113857

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961

zG2 Se Ee Ie( 1113857

zI

minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961

+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967

βR20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nβϕ(h)

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus

N2β2σϕ3(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

zG3 Se Ee Ie( 1113857

zS

σϕ(h)

[1 + ϕ(h)(μ + c)]

μβϕ(h) R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890

minusNμβ2σϕ4(h)μ R0 minus 1( 1113857

2

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nμβσϕ3(h) R0 minus 1( 1113857

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

1113891

zG3 Se Ee Ie( 1113857

zE

σϕ(h)

[1 + ϕ(h)(μ + c)]

1[1 + ϕ(h)(μ + σ)]

minusNμβ2σϕ3(h) R0 minus 1( 1113857

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961

⎡⎢⎣

+Nβσϕ2

(h)

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]

1113891

zG3 Se Ee Ie( 1113857

zI

σϕ(h)

[1 + ϕ(h)(μ + c)]

minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961⎡⎢⎣

+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967

βR20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

⎤⎥⎦

10 Journal of Mathematics

Table 1 Comparison of NSFD scheme with Euler and RK-4 schemes

l 001 01 100 1000Euler PC Convergence Divergence Divergence DivergenceRK-4 Convergence Divergence Divergence DivergenceNSFDPC Convergence Convergence Convergence Convergence

times107

NSFD-PC

Disease-free equilibrium

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Disease-free equilibriumtimes104

NSFD-PC

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

0

05

1

15

2

25

3

35

4

45

5

Expo

sed

fract

ion

(b)

times104

NSFD-PC

Disease-free equilibrium

0

05

1

15

2

25

3

Infe

cted

frac

tion

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

times107

NSFD-PC

X 5675Y 2435e + 007

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

1

15

2

25

3

35

Susc

eptib

le fr

actio

n

(d)

Figure 3 Continued

Journal of Mathematics 11

+Nβϕ(h)

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus

N2β2σϕ3(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

(26)

-e Jacobian evaluated at an endemic point(Slowaste Elowaste Ilowaste ) (Se Ee Ie) (NR0 μNμ + σ(1 minus 1R0) μβ(R0 minus 1)) is evaluated as

JG Slowaste Elowaste Ilowaste( 1113857

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(27)

It is tedious to calculate the eigenvalues of the Jacobian ofendemic equilibrium analytically so we have plotted thelargest eigenvalue against each step size and Figure 2(b)shows that for all step sizes the spectral radius of the Ja-cobian of EE remains less than one if R0 gt 1 which impliesthat the numerical scheme in equations (8) and (10) isunconditionally convergent if R0 gt 1 for all step sizes

6 Numerical Results and Discussion

All the methods showed the convergence for small step sizesin Table 1 However for large values of step sizes only thenormalized NSFDPC converge to the correct disease-freepoint for β 01 times 10minus 5 as well as the correct endemic pointfor β 03 times 10minus 5 -is means that for the value of the basic

reproductive number (threshold parameter) R0 lt 1 thenDFE is stable ie solution to the system (S E I R) areconverging to it -is can be seen from Figures 3(a)ndash3(c) forh 001 Also for the value of the basic reproductivenumber (threshold parameter) R0 gt 1 then EE is stable iesolution to the system (S E I R) is converging to it-is canbe seen from Figures 3(d)ndash3(f ) for h 001

Figures 4(a)ndash4(c) show how the NSFDPC method ofDFE converges to the equilibrium point for different stepsizes Figures 4(d)ndash4(f) show how the NSFDPC method ofEE converges to the equilibrium point for different step sizes

For comparison with the well-known RK-4 method tosystem (1) using the parameter values given in Table 2 it isfound that in Figures 5(a)ndash5(f) the numerical solutionconverges for h 001 for both equilibrium points and

times105

NSFD-PC

X 562Y 1125e + 004

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

0

1

2

3

4

5

6

7

8Ex

pose

d fra

ctio

n

(e)

times105

NSFD-PC

X 5523Y 7023

Infe

cted

frac

tion

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

0

05

1

15

2

25

3

35

4

45

5

(f )

Figure 3 Time plots of susceptible exposed and infected population using the NSFDPC method with h 001

12 Journal of Mathematics

times107

NSFD-PC

Disease-free equilibrium

20 40 60 80 100 120 140 160 180 2000Time (years) h = 01

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

(a)

times107

NSFD-PC

Disease-free equilibrium

500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(b)

Disease-free equilibriumtimes107

times105

NSFD-PC

05 1 15 2 25 3 35 40Time (years) h = 1000

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(c)

X 5827Y 7022

Infe

cted

frac

tion

Endemic equilibriumtimes104

NSFD-PC

100 200 300 400 500 6000Time (years) h = 01

0

2

4

6

8

10

12

14

16

18

(d)

Endemic equilibriumtimes104

NSFD-PC

X 9600Y 7022

Infe

cted

frac

tion

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10

0

05

1

15

2

25

3

(e)

Endemic equilibriumtimes104

times105

NSFD-PC

X 861e + 005Y 7022

Infe

cted

frac

tion

1 2 3 4 5 6 7 8 9 100Time (years) h = 1000

0

05

1

15

2

25

3

(f )

Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000

Journal of Mathematics 13

Table 2 Parameters value of the measles model

Parameters N μ σ c

5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1

DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5

EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5

Disease-free equilibrium

RK4

times107

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Expo

sed

fract

ion

Disease-free equilibrium

RK4

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

times104

Infe

cted

frac

tion

Disease-free equilibrium

RK4

0

05

1

15

2

25

3

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

times107

X 5578Y 2435e + 007

Endemic equilibrium

RK4

1

15

2

25

3

35

Susc

eptib

le F

ract

ion

100 200 300 400 500 6000Time (years) h = 001

(d)

Figure 5 Continued

14 Journal of Mathematics

Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively

Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces

overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8

-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent

times105

X 5772Y 1125e + 004

Endemic equilibrium

RK4

Expo

sed

fract

ion

0

1

2

3

4

5

6

7

8

9

10

100 200 300 400 500 6000Time (years) h = 001

(e)

times105

X 7659Y 7022

RK4

Endemic equilibrium

Infe

cted

frac

tion

0

1

2

3

4

5

6

100 200 300 400 500 600 700 8000Time (years) h = 001

(f )

Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001

times1087 RK-4 scheme (disease-free equilibrium)

SusceptibleExposedInfected

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

Popu

latio

n fra

ctio

ns

(a)

RK-4 scheme (endemic equilibrium)

SusceptibleExposedInfected

times10145

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

Popu

latio

n fra

ctio

ns

(b)

Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01

Journal of Mathematics 15

Euler-PC

Disease-free equilibriumtimes107

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Euler-PC

Disease-free equilibrium

Expo

sed

fract

ion

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

Euler-PC

Disease-free equilibrium

Infe

cted

frac

tion

times104

0

05

1

15

2

25

3

35

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

Euler-PC

X 5733Y 2435e + 007

Endemic equilibriumtimes107

1

15

2

25

3

35Su

scep

tible

frac

tion

100 200 300 400 500 6000Time (years) h = 001

(d)

Euler-PC

X 5662Y 1118e + 004

Endemic equilibrium

Expo

sed

fract

ion

times105

0

2

4

6

8

10

12

100 200 300 400 500 6000Time (years) h = 001

(e)

Euler-PC

X 5427Y 7025

Endemic equilibrium

Infe

cted

frac

tion

times105

0

1

2

3

4

5

6

7

100 200 300 400 500 6000Time (years) h = 001

(f )

Figure 7 Time plots of population using the Euler-PC method with h 001

16 Journal of Mathematics

7 Conclusions

In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the

positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory

Euler-PC scheme (disease-free equilibrium)times1041

ndash3

ndash2

ndash1

0

1

2

3

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(a)

Euler-PC scheme (endemic equilibrium)times1053

ndash6

ndash4

ndash2

0

2

4

6

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(b)

Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01

NSFD-PCRK-4

times1092

Infe

cted

frac

tion

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(a)

NSFD-PCEuler-PC

times10213

Infe

cted

frac

tion

ndash4

ndash35

ndash3

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(b)

Figure 9 Comparative convergence graphs of the NSFDPC method

Journal of Mathematics 17

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6

References

[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018

[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012

[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019

[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018

[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007

[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020

[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994

[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000

[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007

[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973

[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003

[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002

[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005

[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007

[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005

[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003

[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001

[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009

[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005

[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008

[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008

[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003

[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002

[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010

[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001

18 Journal of Mathematics

-us the NSFD scheme for system (1) takes the form

Sn+1

minus Sn

φ(h) μN minus μS

n+1minus βS

n+1I

n

En+1

minus En

φ(h) βS

n+1I

nminus μE

n+1minus σE

n+1

In+1

minus In

φ(h) σE

n+1minus μI

n+1minus cI

n+1

Rn+1

minus Rn

φ(h) cI

n+1minus μR

n+1

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(5)

From equation (5) we have

(1 + φ(h)μ) Sn+1

+ En+1

+ In+1

+ Rn+1

1113960 1113961

φ(h)μN minus φ(h)μ Sn

+ En

+ In

+ Rn

( 1113857(6)

-us if Sn + En + In + Rn N for all nge 0 then Sn+1+

En+1 + In+1 + Rn+1 N for all nge 0-us the conservation law property is satisfied System

(5) is written as

Sn+1

S

n+ μφ(h)N

1 + φ(h) μ + βIn

( 1113857

En+1

E

n+ φ(h)βS

n+1I

n

1 + φ(h)(μ + σ)

In+1

I

n+ φ(h)σE

n+1

1 + φ(h)(μ + c)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(7)

Now the positivity of Sn+1 En+1 In+1 andRn+1 is guar-anteed if 0lt Sn lt 1 0ltEn lt 1 0lt In lt 1 and 0ltRn lt 1 for allnge 0 -us the constructed scheme has the followingproperties

-e conservation law is satisfied-e positivity and boundedness of the solution issatisfied for system (7) we have that if 0lt Sn lt 10ltEn lt 1 0lt In lt 1 and 0ltRn lt 1 then 0lt Sn+1 lt 10ltEn+1 lt 1 0lt In+1 lt 1 and 0ltRn+1 lt 1 for all nge 0

4 NSFD Predictor-Corrector Scheme

-is section is improved by the approach of a predictor-corrector-type NSFD scheme (7) to obtain the benefits ofboth methods For the development of this scheme firstlysystem (7) is taken as a predictor scheme ie

Sn+1p

Sn

+ μφ(h)N

1 + φ(h) μ + βIn

( 1113857

En+1p

En

+ φ(h)βSn+1p I

n

1 + φ(h)(μ + σ)

In+1p

In

+ φ(h)σEn+1p

1 + φ(h)(μ + c)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(8)

Now we evaluate system (1) at time t + φ(h) and in-troduce the term ϵminus 1Sn+1φ(h) (where ϵ is just like the ac-celerating factor and its range is (0lt ϵlt 1)) -us theexpression will be NSFDCL scheme that preserves conser-vation law represented as

Sn+1

minus Sn

φ(h) μN minus μS

n+1minus βS

n+1I

n+1minusϵminus 1

Sn+1

φ(h)+ϵminus 1

Sn+1

φ(h)

En+1

minus En

φ(h) βS

n+1I

nminus (μ + σ)E

n+1

In+1

minus In

φ(h) σE

n+1minus (μ + c)I

n

Rn+1

minus Rn

φ(h) cI

n+1minus μR

n+1

(9)

-us the corrector scheme is obtained as

Sn+1c

Sn

+ μφ(h)N + ϵminus 1S

n+1p

1 + ϵminus 1 + μφ(h) + βφ(h)In+1p

En+1c

En

+ φ(h)βSn+1c I

n+1p

1 + φ(h)(μ + σ)

In+1c

In

+ φ(h)σEn+1c

1 + φ(h)(μ + c)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(10)

where Rn+1 N minus Sn+1c minus En+1

c minus In+1c

5 Convergence Analysis

In this section the unconditional convergence of the nu-merical solution is presented by the proposed methodTaking the values as

F1(S E I) S + μNϕ(h)

1 + ϕ(h)[μ + βI]

F2(S E I) E + βϕ(h)IF1

1 + ϕ(h)[μ + σ]

F3(S E I) I + σϕ(h)F2

1 + ϕ(h)[μ + c]

G1(S E I) S + μNϕ(h) + ϵminus 1

F1

1 + ϵminus 1 + μϕ(h) + βϕ(l)F3

G2(S E I) E + βϕ(h)G1F3

1 + ϕ(h)[μ + σ]

G3(S E I) I + σϕ(h)G2

1 + ϕ(h)[μ + c]

(11)

where F1 F2 andF3 are given as in equation (8)

Journal of Mathematics 3

-e convergence and stability analysis of scheme (10) iscarried out by calculating the eigenvalues of the Jacobian ofthe linearized scheme and studying its behavior and

evaluated at the fixed points If (Slowast Elowast Ilowast) is the fixed pointof system (1) then the Jacobian matrix JG is given by

JG Slowast Elowast Ilowast

( 1113857

zG1 Slowast Elowast Ilowast

( 1113857

zS

zG1 Slowast Elowast Ilowast

( 1113857

zE

zG1 Slowast Elowast Ilowast

( 1113857

zI

zG2 Slowast Elowast Ilowast

( 1113857

zS

zG2 Slowast Elowast Ilowast

( 1113857

zE

zG2 Slowast Elowast Ilowast

( 1113857

zI

zG3 Slowast Elowast Ilowast

( 1113857

zS

zG3 Slowast Elowast Ilowast

( 1113857

zE

zG3 Slowast Elowast Ilowast

( 1113857

zI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(12)

where

zF1

zS

1[1 + μϕ(h) + βϕ(h)I]

zF1

zE

zF1

zI

minus (S + μNϕ(h))βϕ(h)

[1 + μϕ(h) + βϕ(h)I]2

zF2

zS

βϕ(h)IzF1zS

1 + ϕ(h)[μ + σ]

zF2

zE1 + βϕ(h)IzF1zE

1 + ϕ(h)[μ + σ]

zF2

zIβϕ(h) F1 + IzF1zI1113864 1113865

1 + ϕ(h)[μ + σ]

zF3

zE

σϕ(h)zF2zE

1 + ϕ(h)[μ + c]

zF3

zI1 + σϕ(h)zF2zS

1 + ϕ(h)[μ + c]

zG1

zS

1 + ϵminus 1zF1zS1113872 1113873 1 + ϵminus 1

+ μϕ(h) + βϕ(h)F31113960 1113961 minus S + μNϕ(h) + ϵminus 1F11113872 1113873βϕ(h)zF3zS

1 + ϵminus 1 + μϕ(h) + βϕ(l)F31113960 11139612

zG1

zEϵminus 1

zF1zE1113872 1113873 1 + ϵminus 1+ μϕ(h) + βϕ(h)F31113960 1113961 minus S + μNϕ(h) + ϵminus 1

F11113872 1113873βϕ(h)zF3zE

1 + ϵminus 1 + μϕ(h) + βϕ(h)F31113960 11139612

zG1

zIϵminus 1

zF1zI1113872 1113873 1 + ϵminus 1+ μϕ(h) + βϕ(h)F31113960 1113961 minus S + μNϕ(h) + ϵminus 1

F11113872 1113873βϕ(h)zF3zI

1 + ϵminus 1 + μϕ(h) + βϕ(h)F31113960 11139612

zG2

zS

βϕ(h)

1 + ϕ(h)[μ + σ]

zG1

zSF3 + G1

zF3

zS1113896 1113897

zG2

zE1 + βϕ(h) zG1zEF3 + G1zF3zE1113858 1113859

1 + ϕ(h)[μ + σ]

4 Journal of Mathematics

zG2

zI

βϕ(h)

1 + ϕ(h)[μ + σ]

zG1

zIF3 + G1

zF3

zI1113896 1113897

zG3

zS

σϕ(h)zG2zS

1 + ϕ(h)[μ + c]

zG3

zE

σϕ(h)zG2zE

1 + ϕ(h)[μ + c]

zG3

zI1 + σϕ(h)zG2zI

1 + ϕ(h)[μ + c]

(13)

51 Disease-Free Equilibrium (SoEo Io) (N 0 0) -evalues of functions and its derivatives at DFE point (N 0 0)

are as

F1(N 0 0) N F2(N 0 0) 0 F3(N 0 0) 0

G1(N 0 0) N G2(N 0 0) 0 G3(N 0 0) 0

zF1(N 0 0)

zS

11 + μϕ(h)

zF1(N 0 0)

zE 0

zF1(N 0 0)

zI

minus Nβϕ(h)

[1 + μϕ(h)]

zF2(N 0 0)

zS 0

zF2(N 0 0)

zE

11 + ϕ(h)[μ + σ]

zF2(N 0 0)

zI

βϕ(h)

1 + ϕ(h)[μ + σ]

zF3(N 0 0)

zS 0

zF3(N 0 0)

zE

σϕ(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]]

zF3(N 0 0)

zI1 +[σϕ(h)βNϕ(h)1 + ϕ(h)(μ + c)]

1 + ϕ(h)[μ + c]

zG1(N 0 0)

zS

11 + μϕ(h)

zG1(N 0 0)

zE

minus Nσϕ(h)βϕ(h)

1 + μϕ(h) + ϵminus 11113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

Journal of Mathematics 5

zG1(N 0 0)

zI

minus Nβϕ(h)

1 + ϵminus 1+ μϕ(h)1113960 1113961

ϵminus 1

1 + μϕ(h)+

1[1 + ϕ(h)(μ + c)

+σNβϕ2

(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113890 1113891

zG2(N 0 0)

zS 0

zG2(N 0 0)

zE

11 + ϕ(h)(μ + σ)

1 +Nβσϕ2(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113890 1113891

zG2(N 0 0)

zI

βϕ(h)

[1 + ϕ(h)(μ + σ)]

1 + ϕ(h)(μ + σ) + Nσβϕ2(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113896 1113897

zG3(N 0 0)

zS

zG3(N 0 0)

zE

σϕ(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1 +

σβNϕ2(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113896 1113897

zG3(N 0 0)

zI

1

[1 + ϕ(h)(μ + σ)]+

Nσβϕ2(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]2 +

N2β2σ2ϕ4(h)

[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + σ)]

2

(14)

For ease in the calculation we use the followingconventions

α μNϕ(h)gt 0 η1 μϕ(h)gt 0 η2

βϕ(h)gt 0 η3 σϕ(h)gt 0

θ 1 + ϵminus 1+ μϕ(h) 1 + ϵminus 1

+ η1 gt 1

δ1 1 + ϕ(h)[μ + σ]gt 1 δ2

(15)

It is clear that θgt η1 1 + η1 gt δ1 1 + η1 gt δ2 and δ2 θ minus ϵminus 1 + cϕ(h) If R0 σβN(μ + σ)(μ + c)lt 1 the Ja-cobian calculated at disease-free points (Slowast0 Elowast0 Ilowast0 )

(N 0 0) is given as

JG(N 0 0)

zG1(N 0 0)

zS

zG1(N 0 0)

zE

zG1(N 0 0)

zI

zG1(N 0 0)

zS

zG1(N 0 0)

zE

zG1(N 0 0)

zI

zG1(N 0 0)

zS

zG1(N 0 0)

zE

zG1(N 0 0)

zI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)

Since the partial derivatives using the conventions are

zG1(N 0 0)

zS

11 + η1

zG1(N 0 0)

zE

minus Nη2η3θδ1δ2

zG1(N 0 0)

zI

minus Nη2θϵminus 1

1 + η1+

1δ2

+Nη2η3δ1δ2

1113890 1113891

zG2(N 0 0)

zS 0

zG2(N 0 0)

zE

1δ1

+Nη2η3δ21δ2

1113890 1113891

zG2(N 0 0)

zI

Nη2δ1δ2

+N

2η22η3δ21δ2

1113890 1113891

zG3(N 0 0)

zS 0

zG3(N 0 0)

zE

η3δ1δ2

+Nη2η

23

δ21δ22

1113890 1113891

zG3(N 0 0)

zI

1δ2

+Nη2η3δ1δ

22

+N

2η22η23

δ21δ22

1113890 1113891

(17)

6 Journal of Mathematics

So

JG(N 0 0)

11 + η1

minus Nη2η3θδ1δ2

minus Nη2θϵminus 1

1 + η1+

1δ2

+Nη2η3δ1δ2

1113890 1113891

01δ1

+Nη2η3δ21δ2

1113890 1113891Nη2δ1δ2

+N

2η22η3δ21δ2

1113890 1113891

0η3δ1δ2

+Nη2η

23

δ21δ22

1113890 11138911δ2

+Nη2η3δ1δ

22

+N

2η22η23

δ21δ22

1113890 1113891

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)

To calculate the eigenvalues put det (J minus λI) 0

λ1 11 + η1 11 + μϕ(h)lt 1 And

λ2 minus λδ1δ2 + Nη2η3

δ21δ21113888 1113889 +

δ21δ2 + δ1Nη2η3 + N2η22η

23

δ21δ22

1113888 11138891113890 1113891

+δ1δ2 + Nη2η3

δ21δ21113888 1113889

δ21δ2 + δ1Nη2η3 + N2η22η

23

δ21δ22

1113888 1113889 minusNη2δ1 + N

2η22η3δ21δ2

1113888 1113889η3δ1δ2 + Nη2η

23

δ21δ22

1113888 11138891113890 1113891 0

λ2 minus Aλ + B 0

(19)

where

A δ1δ2 + Nη2η3δ21δ21113872 1113873 + δ21δ2 + δ1Nη2η3 + N

2η22η23δ

21δ

221113872 1113873 trace J

lowast

B δ1δ2 + Nη2η3

δ21δ21113888 1113889

δ21δ2 + δ1Nη2η3 + N2η22η

23

δ21δ22

1113888 1113889 minusNη2δ1 + N

2η22η3δ21δ2

1113888 1113889η3δ1δ2 + Nη2η

23

δ21δ22

1113888 11138891113890 1113891 det Jlowast

Jlowast

1δ1

+Nη2η3δ21δ2

1113890 1113891Nη2δ1δ2

+N

2η22η3δ21δ2

1113890 1113891

η3δ1δ2

+Nη2η

23

δ21δ22

1113890 11138911δ2

+Nη2η3δ1δ

22

+N

2η22η23

δ21δ22

1113890 1113891

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

To calculate the eigenvalues of Jlowast the following lemma isproved

Lemma 1 For the quadratic equation λ2 minus λA + B 0 bothroots satisfy |λi|lt 1 i 2 3 if and only if the followingconditions are satisfied [25]

(1) 1 minus A + Bgt 0

(2) 1 + A + Bgt 0

(3)Blt 1

(21)

Let us define A TraceJlowast B DetJlowast -erefore

Journal of Mathematics 7

A δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N

2η22η23

δ21δ22

gt 0

B 1

δ1δ2+

Nη2η3δ21δ

22lt 1

f(minus 1) 1 + A + Bgt 0

f(0) B 1

δ1δ2+

Nη2η3δ21δ

22lt

1δ1δ2

+Nη2η3δ1δ2

1 + Nη2η3

δ1δ2

1 + R0(μ + σ)(μ + c)ϕ2(h)

1 + ϕ2(h)(μ + σ)(μ + c) + ϕ(h)[(μ + σ)(μ + c)]

lt 1 sinceR0 lt 1( 1113857

rArrBlt 1

(22)

Now for

f(1) 1 minus A + B

δ21δ

22 + δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N

2η22η23 + Nη2η3 + δ1δ2

δ21δ22

gt 0

(23)

As all the conditions of the lemma hold for R0 lt 1therefore the absolute value of both eigenvalues of Jlowast is less

than one whenever R0 lt 1 -us the numerical scheme inequations (8) and (10) will converge unconditionally todisease-free equilibrium (N 0 0) for any value of time step h

whenever R0 lt 1 which is also computationally verified inFigure 2(a)

52 Endemic Equilibrium (SeEe Ie) (NR0 μN μ + σ(1 minus

1 R0) μβ(R0 minus 1)) -e value of functions and its deriv-atives at an endemic point is as

F1 Se Ee Ie( 1113857 N

R0

F2 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857

(μ + σ)

F3 Se Ee Ie( 1113857 μβ

R0 minus 1( 1113857

G1 Se Ee Ie( 1113857 N

R0

G2 Se Ee Ie( 1113857 μβ

R0 minus 1( 1113857

G3 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857

(μ + σ)

(24)

8 Journal of Mathematics

zF1 Se Ee Ie( 1113857

zS

11 + μϕ(h)R0

zF1 Se Ee Ie( 1113857

zE 0

zF1 Se Ee Ie( 1113857

zI

minus Nβϕ(h)

1 + μϕ(h)R0

zF2 Se Ee Ie( 1113857

zS

μϕ(h) R0 minus 1( 1113857

1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + σ)]

zF2 Se Ee Ie( 1113857

zE

1[1 + ϕ(h)(μ + σ)]

zF2 Se Ee Ie( 1113857

zI

minus Nβϕ(h) μϕ(l)R20 minus 2μϕ(h)R0 minus 11113960 1113961

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

zF3 Se Ee Ie( 1113857

zS

μσϕ2(h) R0 minus 1( 1113857

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

zF3 Se Ee Ie( 1113857

zE

σϕ(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

zF3 Se Ee Ie( 1113857

zI

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h) minus 11113960 1113961

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

(25)

zG1 Se Ee Ie( 1113857

zS

11 + μϕ(h)R01113858 1113859

minusNμβσϕ3(h) R0 minus 1( 1113857

R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

zG1 Se Ee Ie( 1113857

zE

minus Nβσϕ2(h)

R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

Eige

n va

lues

Jacobian at DFE

0492

0494

0496

0498

05

0502

0504

0506

0508

051

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size

(a)

098

0981

0982

0983

0984

0985

0986

0987

0988

0989Jacobian at EE

Eige

n va

lues

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size

(b)

Figure 2 Spectral radius of the Jacobian at equilibrium points

Journal of Mathematics 9

zG1 Se Ee Ie( 1113857

zI

minus Nβϕ(h)

1 + ϵminus 1 + μϕ(h)R01113960 11139611 + μϕ(h)R0

middot ϵminus 1+

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

⎡⎣ ⎤⎦

zG2 Se Ee Ie( 1113857

zS

μβϕ(h) R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890

minusNμβ2σϕ4(h)μ R0 minus 1( 1113857

2

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nμβσϕ3(h) R0 minus 1( 1113857

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

1113891

zG2 Se Ee Ie( 1113857

zE

1[1 + ϕ(h)(μ + σ)]

+Nβσϕ2(h)

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]

minusNμβ2σϕ3

(h) R0 minus 1( 1113857

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961

zG2 Se Ee Ie( 1113857

zI

minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961

+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967

βR20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nβϕ(h)

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus

N2β2σϕ3(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

zG3 Se Ee Ie( 1113857

zS

σϕ(h)

[1 + ϕ(h)(μ + c)]

μβϕ(h) R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890

minusNμβ2σϕ4(h)μ R0 minus 1( 1113857

2

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nμβσϕ3(h) R0 minus 1( 1113857

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

1113891

zG3 Se Ee Ie( 1113857

zE

σϕ(h)

[1 + ϕ(h)(μ + c)]

1[1 + ϕ(h)(μ + σ)]

minusNμβ2σϕ3(h) R0 minus 1( 1113857

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961

⎡⎢⎣

+Nβσϕ2

(h)

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]

1113891

zG3 Se Ee Ie( 1113857

zI

σϕ(h)

[1 + ϕ(h)(μ + c)]

minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961⎡⎢⎣

+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967

βR20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

⎤⎥⎦

10 Journal of Mathematics

Table 1 Comparison of NSFD scheme with Euler and RK-4 schemes

l 001 01 100 1000Euler PC Convergence Divergence Divergence DivergenceRK-4 Convergence Divergence Divergence DivergenceNSFDPC Convergence Convergence Convergence Convergence

times107

NSFD-PC

Disease-free equilibrium

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Disease-free equilibriumtimes104

NSFD-PC

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

0

05

1

15

2

25

3

35

4

45

5

Expo

sed

fract

ion

(b)

times104

NSFD-PC

Disease-free equilibrium

0

05

1

15

2

25

3

Infe

cted

frac

tion

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

times107

NSFD-PC

X 5675Y 2435e + 007

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

1

15

2

25

3

35

Susc

eptib

le fr

actio

n

(d)

Figure 3 Continued

Journal of Mathematics 11

+Nβϕ(h)

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus

N2β2σϕ3(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

(26)

-e Jacobian evaluated at an endemic point(Slowaste Elowaste Ilowaste ) (Se Ee Ie) (NR0 μNμ + σ(1 minus 1R0) μβ(R0 minus 1)) is evaluated as

JG Slowaste Elowaste Ilowaste( 1113857

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(27)

It is tedious to calculate the eigenvalues of the Jacobian ofendemic equilibrium analytically so we have plotted thelargest eigenvalue against each step size and Figure 2(b)shows that for all step sizes the spectral radius of the Ja-cobian of EE remains less than one if R0 gt 1 which impliesthat the numerical scheme in equations (8) and (10) isunconditionally convergent if R0 gt 1 for all step sizes

6 Numerical Results and Discussion

All the methods showed the convergence for small step sizesin Table 1 However for large values of step sizes only thenormalized NSFDPC converge to the correct disease-freepoint for β 01 times 10minus 5 as well as the correct endemic pointfor β 03 times 10minus 5 -is means that for the value of the basic

reproductive number (threshold parameter) R0 lt 1 thenDFE is stable ie solution to the system (S E I R) areconverging to it -is can be seen from Figures 3(a)ndash3(c) forh 001 Also for the value of the basic reproductivenumber (threshold parameter) R0 gt 1 then EE is stable iesolution to the system (S E I R) is converging to it-is canbe seen from Figures 3(d)ndash3(f ) for h 001

Figures 4(a)ndash4(c) show how the NSFDPC method ofDFE converges to the equilibrium point for different stepsizes Figures 4(d)ndash4(f) show how the NSFDPC method ofEE converges to the equilibrium point for different step sizes

For comparison with the well-known RK-4 method tosystem (1) using the parameter values given in Table 2 it isfound that in Figures 5(a)ndash5(f) the numerical solutionconverges for h 001 for both equilibrium points and

times105

NSFD-PC

X 562Y 1125e + 004

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

0

1

2

3

4

5

6

7

8Ex

pose

d fra

ctio

n

(e)

times105

NSFD-PC

X 5523Y 7023

Infe

cted

frac

tion

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

0

05

1

15

2

25

3

35

4

45

5

(f )

Figure 3 Time plots of susceptible exposed and infected population using the NSFDPC method with h 001

12 Journal of Mathematics

times107

NSFD-PC

Disease-free equilibrium

20 40 60 80 100 120 140 160 180 2000Time (years) h = 01

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

(a)

times107

NSFD-PC

Disease-free equilibrium

500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(b)

Disease-free equilibriumtimes107

times105

NSFD-PC

05 1 15 2 25 3 35 40Time (years) h = 1000

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(c)

X 5827Y 7022

Infe

cted

frac

tion

Endemic equilibriumtimes104

NSFD-PC

100 200 300 400 500 6000Time (years) h = 01

0

2

4

6

8

10

12

14

16

18

(d)

Endemic equilibriumtimes104

NSFD-PC

X 9600Y 7022

Infe

cted

frac

tion

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10

0

05

1

15

2

25

3

(e)

Endemic equilibriumtimes104

times105

NSFD-PC

X 861e + 005Y 7022

Infe

cted

frac

tion

1 2 3 4 5 6 7 8 9 100Time (years) h = 1000

0

05

1

15

2

25

3

(f )

Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000

Journal of Mathematics 13

Table 2 Parameters value of the measles model

Parameters N μ σ c

5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1

DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5

EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5

Disease-free equilibrium

RK4

times107

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Expo

sed

fract

ion

Disease-free equilibrium

RK4

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

times104

Infe

cted

frac

tion

Disease-free equilibrium

RK4

0

05

1

15

2

25

3

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

times107

X 5578Y 2435e + 007

Endemic equilibrium

RK4

1

15

2

25

3

35

Susc

eptib

le F

ract

ion

100 200 300 400 500 6000Time (years) h = 001

(d)

Figure 5 Continued

14 Journal of Mathematics

Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively

Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces

overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8

-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent

times105

X 5772Y 1125e + 004

Endemic equilibrium

RK4

Expo

sed

fract

ion

0

1

2

3

4

5

6

7

8

9

10

100 200 300 400 500 6000Time (years) h = 001

(e)

times105

X 7659Y 7022

RK4

Endemic equilibrium

Infe

cted

frac

tion

0

1

2

3

4

5

6

100 200 300 400 500 600 700 8000Time (years) h = 001

(f )

Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001

times1087 RK-4 scheme (disease-free equilibrium)

SusceptibleExposedInfected

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

Popu

latio

n fra

ctio

ns

(a)

RK-4 scheme (endemic equilibrium)

SusceptibleExposedInfected

times10145

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

Popu

latio

n fra

ctio

ns

(b)

Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01

Journal of Mathematics 15

Euler-PC

Disease-free equilibriumtimes107

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Euler-PC

Disease-free equilibrium

Expo

sed

fract

ion

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

Euler-PC

Disease-free equilibrium

Infe

cted

frac

tion

times104

0

05

1

15

2

25

3

35

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

Euler-PC

X 5733Y 2435e + 007

Endemic equilibriumtimes107

1

15

2

25

3

35Su

scep

tible

frac

tion

100 200 300 400 500 6000Time (years) h = 001

(d)

Euler-PC

X 5662Y 1118e + 004

Endemic equilibrium

Expo

sed

fract

ion

times105

0

2

4

6

8

10

12

100 200 300 400 500 6000Time (years) h = 001

(e)

Euler-PC

X 5427Y 7025

Endemic equilibrium

Infe

cted

frac

tion

times105

0

1

2

3

4

5

6

7

100 200 300 400 500 6000Time (years) h = 001

(f )

Figure 7 Time plots of population using the Euler-PC method with h 001

16 Journal of Mathematics

7 Conclusions

In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the

positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory

Euler-PC scheme (disease-free equilibrium)times1041

ndash3

ndash2

ndash1

0

1

2

3

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(a)

Euler-PC scheme (endemic equilibrium)times1053

ndash6

ndash4

ndash2

0

2

4

6

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(b)

Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01

NSFD-PCRK-4

times1092

Infe

cted

frac

tion

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(a)

NSFD-PCEuler-PC

times10213

Infe

cted

frac

tion

ndash4

ndash35

ndash3

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(b)

Figure 9 Comparative convergence graphs of the NSFDPC method

Journal of Mathematics 17

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6

References

[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018

[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012

[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019

[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018

[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007

[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020

[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994

[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000

[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007

[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973

[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003

[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002

[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005

[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007

[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005

[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003

[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001

[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009

[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005

[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008

[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008

[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003

[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002

[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010

[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001

18 Journal of Mathematics

-e convergence and stability analysis of scheme (10) iscarried out by calculating the eigenvalues of the Jacobian ofthe linearized scheme and studying its behavior and

evaluated at the fixed points If (Slowast Elowast Ilowast) is the fixed pointof system (1) then the Jacobian matrix JG is given by

JG Slowast Elowast Ilowast

( 1113857

zG1 Slowast Elowast Ilowast

( 1113857

zS

zG1 Slowast Elowast Ilowast

( 1113857

zE

zG1 Slowast Elowast Ilowast

( 1113857

zI

zG2 Slowast Elowast Ilowast

( 1113857

zS

zG2 Slowast Elowast Ilowast

( 1113857

zE

zG2 Slowast Elowast Ilowast

( 1113857

zI

zG3 Slowast Elowast Ilowast

( 1113857

zS

zG3 Slowast Elowast Ilowast

( 1113857

zE

zG3 Slowast Elowast Ilowast

( 1113857

zI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(12)

where

zF1

zS

1[1 + μϕ(h) + βϕ(h)I]

zF1

zE

zF1

zI

minus (S + μNϕ(h))βϕ(h)

[1 + μϕ(h) + βϕ(h)I]2

zF2

zS

βϕ(h)IzF1zS

1 + ϕ(h)[μ + σ]

zF2

zE1 + βϕ(h)IzF1zE

1 + ϕ(h)[μ + σ]

zF2

zIβϕ(h) F1 + IzF1zI1113864 1113865

1 + ϕ(h)[μ + σ]

zF3

zE

σϕ(h)zF2zE

1 + ϕ(h)[μ + c]

zF3

zI1 + σϕ(h)zF2zS

1 + ϕ(h)[μ + c]

zG1

zS

1 + ϵminus 1zF1zS1113872 1113873 1 + ϵminus 1

+ μϕ(h) + βϕ(h)F31113960 1113961 minus S + μNϕ(h) + ϵminus 1F11113872 1113873βϕ(h)zF3zS

1 + ϵminus 1 + μϕ(h) + βϕ(l)F31113960 11139612

zG1

zEϵminus 1

zF1zE1113872 1113873 1 + ϵminus 1+ μϕ(h) + βϕ(h)F31113960 1113961 minus S + μNϕ(h) + ϵminus 1

F11113872 1113873βϕ(h)zF3zE

1 + ϵminus 1 + μϕ(h) + βϕ(h)F31113960 11139612

zG1

zIϵminus 1

zF1zI1113872 1113873 1 + ϵminus 1+ μϕ(h) + βϕ(h)F31113960 1113961 minus S + μNϕ(h) + ϵminus 1

F11113872 1113873βϕ(h)zF3zI

1 + ϵminus 1 + μϕ(h) + βϕ(h)F31113960 11139612

zG2

zS

βϕ(h)

1 + ϕ(h)[μ + σ]

zG1

zSF3 + G1

zF3

zS1113896 1113897

zG2

zE1 + βϕ(h) zG1zEF3 + G1zF3zE1113858 1113859

1 + ϕ(h)[μ + σ]

4 Journal of Mathematics

zG2

zI

βϕ(h)

1 + ϕ(h)[μ + σ]

zG1

zIF3 + G1

zF3

zI1113896 1113897

zG3

zS

σϕ(h)zG2zS

1 + ϕ(h)[μ + c]

zG3

zE

σϕ(h)zG2zE

1 + ϕ(h)[μ + c]

zG3

zI1 + σϕ(h)zG2zI

1 + ϕ(h)[μ + c]

(13)

51 Disease-Free Equilibrium (SoEo Io) (N 0 0) -evalues of functions and its derivatives at DFE point (N 0 0)

are as

F1(N 0 0) N F2(N 0 0) 0 F3(N 0 0) 0

G1(N 0 0) N G2(N 0 0) 0 G3(N 0 0) 0

zF1(N 0 0)

zS

11 + μϕ(h)

zF1(N 0 0)

zE 0

zF1(N 0 0)

zI

minus Nβϕ(h)

[1 + μϕ(h)]

zF2(N 0 0)

zS 0

zF2(N 0 0)

zE

11 + ϕ(h)[μ + σ]

zF2(N 0 0)

zI

βϕ(h)

1 + ϕ(h)[μ + σ]

zF3(N 0 0)

zS 0

zF3(N 0 0)

zE

σϕ(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]]

zF3(N 0 0)

zI1 +[σϕ(h)βNϕ(h)1 + ϕ(h)(μ + c)]

1 + ϕ(h)[μ + c]

zG1(N 0 0)

zS

11 + μϕ(h)

zG1(N 0 0)

zE

minus Nσϕ(h)βϕ(h)

1 + μϕ(h) + ϵminus 11113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

Journal of Mathematics 5

zG1(N 0 0)

zI

minus Nβϕ(h)

1 + ϵminus 1+ μϕ(h)1113960 1113961

ϵminus 1

1 + μϕ(h)+

1[1 + ϕ(h)(μ + c)

+σNβϕ2

(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113890 1113891

zG2(N 0 0)

zS 0

zG2(N 0 0)

zE

11 + ϕ(h)(μ + σ)

1 +Nβσϕ2(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113890 1113891

zG2(N 0 0)

zI

βϕ(h)

[1 + ϕ(h)(μ + σ)]

1 + ϕ(h)(μ + σ) + Nσβϕ2(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113896 1113897

zG3(N 0 0)

zS

zG3(N 0 0)

zE

σϕ(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1 +

σβNϕ2(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113896 1113897

zG3(N 0 0)

zI

1

[1 + ϕ(h)(μ + σ)]+

Nσβϕ2(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]2 +

N2β2σ2ϕ4(h)

[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + σ)]

2

(14)

For ease in the calculation we use the followingconventions

α μNϕ(h)gt 0 η1 μϕ(h)gt 0 η2

βϕ(h)gt 0 η3 σϕ(h)gt 0

θ 1 + ϵminus 1+ μϕ(h) 1 + ϵminus 1

+ η1 gt 1

δ1 1 + ϕ(h)[μ + σ]gt 1 δ2

(15)

It is clear that θgt η1 1 + η1 gt δ1 1 + η1 gt δ2 and δ2 θ minus ϵminus 1 + cϕ(h) If R0 σβN(μ + σ)(μ + c)lt 1 the Ja-cobian calculated at disease-free points (Slowast0 Elowast0 Ilowast0 )

(N 0 0) is given as

JG(N 0 0)

zG1(N 0 0)

zS

zG1(N 0 0)

zE

zG1(N 0 0)

zI

zG1(N 0 0)

zS

zG1(N 0 0)

zE

zG1(N 0 0)

zI

zG1(N 0 0)

zS

zG1(N 0 0)

zE

zG1(N 0 0)

zI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)

Since the partial derivatives using the conventions are

zG1(N 0 0)

zS

11 + η1

zG1(N 0 0)

zE

minus Nη2η3θδ1δ2

zG1(N 0 0)

zI

minus Nη2θϵminus 1

1 + η1+

1δ2

+Nη2η3δ1δ2

1113890 1113891

zG2(N 0 0)

zS 0

zG2(N 0 0)

zE

1δ1

+Nη2η3δ21δ2

1113890 1113891

zG2(N 0 0)

zI

Nη2δ1δ2

+N

2η22η3δ21δ2

1113890 1113891

zG3(N 0 0)

zS 0

zG3(N 0 0)

zE

η3δ1δ2

+Nη2η

23

δ21δ22

1113890 1113891

zG3(N 0 0)

zI

1δ2

+Nη2η3δ1δ

22

+N

2η22η23

δ21δ22

1113890 1113891

(17)

6 Journal of Mathematics

So

JG(N 0 0)

11 + η1

minus Nη2η3θδ1δ2

minus Nη2θϵminus 1

1 + η1+

1δ2

+Nη2η3δ1δ2

1113890 1113891

01δ1

+Nη2η3δ21δ2

1113890 1113891Nη2δ1δ2

+N

2η22η3δ21δ2

1113890 1113891

0η3δ1δ2

+Nη2η

23

δ21δ22

1113890 11138911δ2

+Nη2η3δ1δ

22

+N

2η22η23

δ21δ22

1113890 1113891

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)

To calculate the eigenvalues put det (J minus λI) 0

λ1 11 + η1 11 + μϕ(h)lt 1 And

λ2 minus λδ1δ2 + Nη2η3

δ21δ21113888 1113889 +

δ21δ2 + δ1Nη2η3 + N2η22η

23

δ21δ22

1113888 11138891113890 1113891

+δ1δ2 + Nη2η3

δ21δ21113888 1113889

δ21δ2 + δ1Nη2η3 + N2η22η

23

δ21δ22

1113888 1113889 minusNη2δ1 + N

2η22η3δ21δ2

1113888 1113889η3δ1δ2 + Nη2η

23

δ21δ22

1113888 11138891113890 1113891 0

λ2 minus Aλ + B 0

(19)

where

A δ1δ2 + Nη2η3δ21δ21113872 1113873 + δ21δ2 + δ1Nη2η3 + N

2η22η23δ

21δ

221113872 1113873 trace J

lowast

B δ1δ2 + Nη2η3

δ21δ21113888 1113889

δ21δ2 + δ1Nη2η3 + N2η22η

23

δ21δ22

1113888 1113889 minusNη2δ1 + N

2η22η3δ21δ2

1113888 1113889η3δ1δ2 + Nη2η

23

δ21δ22

1113888 11138891113890 1113891 det Jlowast

Jlowast

1δ1

+Nη2η3δ21δ2

1113890 1113891Nη2δ1δ2

+N

2η22η3δ21δ2

1113890 1113891

η3δ1δ2

+Nη2η

23

δ21δ22

1113890 11138911δ2

+Nη2η3δ1δ

22

+N

2η22η23

δ21δ22

1113890 1113891

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

To calculate the eigenvalues of Jlowast the following lemma isproved

Lemma 1 For the quadratic equation λ2 minus λA + B 0 bothroots satisfy |λi|lt 1 i 2 3 if and only if the followingconditions are satisfied [25]

(1) 1 minus A + Bgt 0

(2) 1 + A + Bgt 0

(3)Blt 1

(21)

Let us define A TraceJlowast B DetJlowast -erefore

Journal of Mathematics 7

A δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N

2η22η23

δ21δ22

gt 0

B 1

δ1δ2+

Nη2η3δ21δ

22lt 1

f(minus 1) 1 + A + Bgt 0

f(0) B 1

δ1δ2+

Nη2η3δ21δ

22lt

1δ1δ2

+Nη2η3δ1δ2

1 + Nη2η3

δ1δ2

1 + R0(μ + σ)(μ + c)ϕ2(h)

1 + ϕ2(h)(μ + σ)(μ + c) + ϕ(h)[(μ + σ)(μ + c)]

lt 1 sinceR0 lt 1( 1113857

rArrBlt 1

(22)

Now for

f(1) 1 minus A + B

δ21δ

22 + δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N

2η22η23 + Nη2η3 + δ1δ2

δ21δ22

gt 0

(23)

As all the conditions of the lemma hold for R0 lt 1therefore the absolute value of both eigenvalues of Jlowast is less

than one whenever R0 lt 1 -us the numerical scheme inequations (8) and (10) will converge unconditionally todisease-free equilibrium (N 0 0) for any value of time step h

whenever R0 lt 1 which is also computationally verified inFigure 2(a)

52 Endemic Equilibrium (SeEe Ie) (NR0 μN μ + σ(1 minus

1 R0) μβ(R0 minus 1)) -e value of functions and its deriv-atives at an endemic point is as

F1 Se Ee Ie( 1113857 N

R0

F2 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857

(μ + σ)

F3 Se Ee Ie( 1113857 μβ

R0 minus 1( 1113857

G1 Se Ee Ie( 1113857 N

R0

G2 Se Ee Ie( 1113857 μβ

R0 minus 1( 1113857

G3 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857

(μ + σ)

(24)

8 Journal of Mathematics

zF1 Se Ee Ie( 1113857

zS

11 + μϕ(h)R0

zF1 Se Ee Ie( 1113857

zE 0

zF1 Se Ee Ie( 1113857

zI

minus Nβϕ(h)

1 + μϕ(h)R0

zF2 Se Ee Ie( 1113857

zS

μϕ(h) R0 minus 1( 1113857

1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + σ)]

zF2 Se Ee Ie( 1113857

zE

1[1 + ϕ(h)(μ + σ)]

zF2 Se Ee Ie( 1113857

zI

minus Nβϕ(h) μϕ(l)R20 minus 2μϕ(h)R0 minus 11113960 1113961

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

zF3 Se Ee Ie( 1113857

zS

μσϕ2(h) R0 minus 1( 1113857

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

zF3 Se Ee Ie( 1113857

zE

σϕ(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

zF3 Se Ee Ie( 1113857

zI

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h) minus 11113960 1113961

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

(25)

zG1 Se Ee Ie( 1113857

zS

11 + μϕ(h)R01113858 1113859

minusNμβσϕ3(h) R0 minus 1( 1113857

R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

zG1 Se Ee Ie( 1113857

zE

minus Nβσϕ2(h)

R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

Eige

n va

lues

Jacobian at DFE

0492

0494

0496

0498

05

0502

0504

0506

0508

051

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size

(a)

098

0981

0982

0983

0984

0985

0986

0987

0988

0989Jacobian at EE

Eige

n va

lues

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size

(b)

Figure 2 Spectral radius of the Jacobian at equilibrium points

Journal of Mathematics 9

zG1 Se Ee Ie( 1113857

zI

minus Nβϕ(h)

1 + ϵminus 1 + μϕ(h)R01113960 11139611 + μϕ(h)R0

middot ϵminus 1+

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

⎡⎣ ⎤⎦

zG2 Se Ee Ie( 1113857

zS

μβϕ(h) R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890

minusNμβ2σϕ4(h)μ R0 minus 1( 1113857

2

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nμβσϕ3(h) R0 minus 1( 1113857

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

1113891

zG2 Se Ee Ie( 1113857

zE

1[1 + ϕ(h)(μ + σ)]

+Nβσϕ2(h)

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]

minusNμβ2σϕ3

(h) R0 minus 1( 1113857

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961

zG2 Se Ee Ie( 1113857

zI

minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961

+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967

βR20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nβϕ(h)

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus

N2β2σϕ3(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

zG3 Se Ee Ie( 1113857

zS

σϕ(h)

[1 + ϕ(h)(μ + c)]

μβϕ(h) R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890

minusNμβ2σϕ4(h)μ R0 minus 1( 1113857

2

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nμβσϕ3(h) R0 minus 1( 1113857

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

1113891

zG3 Se Ee Ie( 1113857

zE

σϕ(h)

[1 + ϕ(h)(μ + c)]

1[1 + ϕ(h)(μ + σ)]

minusNμβ2σϕ3(h) R0 minus 1( 1113857

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961

⎡⎢⎣

+Nβσϕ2

(h)

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]

1113891

zG3 Se Ee Ie( 1113857

zI

σϕ(h)

[1 + ϕ(h)(μ + c)]

minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961⎡⎢⎣

+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967

βR20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

⎤⎥⎦

10 Journal of Mathematics

Table 1 Comparison of NSFD scheme with Euler and RK-4 schemes

l 001 01 100 1000Euler PC Convergence Divergence Divergence DivergenceRK-4 Convergence Divergence Divergence DivergenceNSFDPC Convergence Convergence Convergence Convergence

times107

NSFD-PC

Disease-free equilibrium

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Disease-free equilibriumtimes104

NSFD-PC

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

0

05

1

15

2

25

3

35

4

45

5

Expo

sed

fract

ion

(b)

times104

NSFD-PC

Disease-free equilibrium

0

05

1

15

2

25

3

Infe

cted

frac

tion

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

times107

NSFD-PC

X 5675Y 2435e + 007

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

1

15

2

25

3

35

Susc

eptib

le fr

actio

n

(d)

Figure 3 Continued

Journal of Mathematics 11

+Nβϕ(h)

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus

N2β2σϕ3(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

(26)

-e Jacobian evaluated at an endemic point(Slowaste Elowaste Ilowaste ) (Se Ee Ie) (NR0 μNμ + σ(1 minus 1R0) μβ(R0 minus 1)) is evaluated as

JG Slowaste Elowaste Ilowaste( 1113857

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(27)

It is tedious to calculate the eigenvalues of the Jacobian ofendemic equilibrium analytically so we have plotted thelargest eigenvalue against each step size and Figure 2(b)shows that for all step sizes the spectral radius of the Ja-cobian of EE remains less than one if R0 gt 1 which impliesthat the numerical scheme in equations (8) and (10) isunconditionally convergent if R0 gt 1 for all step sizes

6 Numerical Results and Discussion

All the methods showed the convergence for small step sizesin Table 1 However for large values of step sizes only thenormalized NSFDPC converge to the correct disease-freepoint for β 01 times 10minus 5 as well as the correct endemic pointfor β 03 times 10minus 5 -is means that for the value of the basic

reproductive number (threshold parameter) R0 lt 1 thenDFE is stable ie solution to the system (S E I R) areconverging to it -is can be seen from Figures 3(a)ndash3(c) forh 001 Also for the value of the basic reproductivenumber (threshold parameter) R0 gt 1 then EE is stable iesolution to the system (S E I R) is converging to it-is canbe seen from Figures 3(d)ndash3(f ) for h 001

Figures 4(a)ndash4(c) show how the NSFDPC method ofDFE converges to the equilibrium point for different stepsizes Figures 4(d)ndash4(f) show how the NSFDPC method ofEE converges to the equilibrium point for different step sizes

For comparison with the well-known RK-4 method tosystem (1) using the parameter values given in Table 2 it isfound that in Figures 5(a)ndash5(f) the numerical solutionconverges for h 001 for both equilibrium points and

times105

NSFD-PC

X 562Y 1125e + 004

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

0

1

2

3

4

5

6

7

8Ex

pose

d fra

ctio

n

(e)

times105

NSFD-PC

X 5523Y 7023

Infe

cted

frac

tion

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

0

05

1

15

2

25

3

35

4

45

5

(f )

Figure 3 Time plots of susceptible exposed and infected population using the NSFDPC method with h 001

12 Journal of Mathematics

times107

NSFD-PC

Disease-free equilibrium

20 40 60 80 100 120 140 160 180 2000Time (years) h = 01

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

(a)

times107

NSFD-PC

Disease-free equilibrium

500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(b)

Disease-free equilibriumtimes107

times105

NSFD-PC

05 1 15 2 25 3 35 40Time (years) h = 1000

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(c)

X 5827Y 7022

Infe

cted

frac

tion

Endemic equilibriumtimes104

NSFD-PC

100 200 300 400 500 6000Time (years) h = 01

0

2

4

6

8

10

12

14

16

18

(d)

Endemic equilibriumtimes104

NSFD-PC

X 9600Y 7022

Infe

cted

frac

tion

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10

0

05

1

15

2

25

3

(e)

Endemic equilibriumtimes104

times105

NSFD-PC

X 861e + 005Y 7022

Infe

cted

frac

tion

1 2 3 4 5 6 7 8 9 100Time (years) h = 1000

0

05

1

15

2

25

3

(f )

Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000

Journal of Mathematics 13

Table 2 Parameters value of the measles model

Parameters N μ σ c

5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1

DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5

EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5

Disease-free equilibrium

RK4

times107

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Expo

sed

fract

ion

Disease-free equilibrium

RK4

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

times104

Infe

cted

frac

tion

Disease-free equilibrium

RK4

0

05

1

15

2

25

3

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

times107

X 5578Y 2435e + 007

Endemic equilibrium

RK4

1

15

2

25

3

35

Susc

eptib

le F

ract

ion

100 200 300 400 500 6000Time (years) h = 001

(d)

Figure 5 Continued

14 Journal of Mathematics

Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively

Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces

overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8

-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent

times105

X 5772Y 1125e + 004

Endemic equilibrium

RK4

Expo

sed

fract

ion

0

1

2

3

4

5

6

7

8

9

10

100 200 300 400 500 6000Time (years) h = 001

(e)

times105

X 7659Y 7022

RK4

Endemic equilibrium

Infe

cted

frac

tion

0

1

2

3

4

5

6

100 200 300 400 500 600 700 8000Time (years) h = 001

(f )

Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001

times1087 RK-4 scheme (disease-free equilibrium)

SusceptibleExposedInfected

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

Popu

latio

n fra

ctio

ns

(a)

RK-4 scheme (endemic equilibrium)

SusceptibleExposedInfected

times10145

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

Popu

latio

n fra

ctio

ns

(b)

Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01

Journal of Mathematics 15

Euler-PC

Disease-free equilibriumtimes107

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Euler-PC

Disease-free equilibrium

Expo

sed

fract

ion

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

Euler-PC

Disease-free equilibrium

Infe

cted

frac

tion

times104

0

05

1

15

2

25

3

35

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

Euler-PC

X 5733Y 2435e + 007

Endemic equilibriumtimes107

1

15

2

25

3

35Su

scep

tible

frac

tion

100 200 300 400 500 6000Time (years) h = 001

(d)

Euler-PC

X 5662Y 1118e + 004

Endemic equilibrium

Expo

sed

fract

ion

times105

0

2

4

6

8

10

12

100 200 300 400 500 6000Time (years) h = 001

(e)

Euler-PC

X 5427Y 7025

Endemic equilibrium

Infe

cted

frac

tion

times105

0

1

2

3

4

5

6

7

100 200 300 400 500 6000Time (years) h = 001

(f )

Figure 7 Time plots of population using the Euler-PC method with h 001

16 Journal of Mathematics

7 Conclusions

In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the

positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory

Euler-PC scheme (disease-free equilibrium)times1041

ndash3

ndash2

ndash1

0

1

2

3

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(a)

Euler-PC scheme (endemic equilibrium)times1053

ndash6

ndash4

ndash2

0

2

4

6

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(b)

Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01

NSFD-PCRK-4

times1092

Infe

cted

frac

tion

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(a)

NSFD-PCEuler-PC

times10213

Infe

cted

frac

tion

ndash4

ndash35

ndash3

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(b)

Figure 9 Comparative convergence graphs of the NSFDPC method

Journal of Mathematics 17

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6

References

[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018

[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012

[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019

[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018

[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007

[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020

[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994

[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000

[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007

[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973

[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003

[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002

[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005

[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007

[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005

[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003

[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001

[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009

[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005

[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008

[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008

[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003

[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002

[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010

[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001

18 Journal of Mathematics

zG2

zI

βϕ(h)

1 + ϕ(h)[μ + σ]

zG1

zIF3 + G1

zF3

zI1113896 1113897

zG3

zS

σϕ(h)zG2zS

1 + ϕ(h)[μ + c]

zG3

zE

σϕ(h)zG2zE

1 + ϕ(h)[μ + c]

zG3

zI1 + σϕ(h)zG2zI

1 + ϕ(h)[μ + c]

(13)

51 Disease-Free Equilibrium (SoEo Io) (N 0 0) -evalues of functions and its derivatives at DFE point (N 0 0)

are as

F1(N 0 0) N F2(N 0 0) 0 F3(N 0 0) 0

G1(N 0 0) N G2(N 0 0) 0 G3(N 0 0) 0

zF1(N 0 0)

zS

11 + μϕ(h)

zF1(N 0 0)

zE 0

zF1(N 0 0)

zI

minus Nβϕ(h)

[1 + μϕ(h)]

zF2(N 0 0)

zS 0

zF2(N 0 0)

zE

11 + ϕ(h)[μ + σ]

zF2(N 0 0)

zI

βϕ(h)

1 + ϕ(h)[μ + σ]

zF3(N 0 0)

zS 0

zF3(N 0 0)

zE

σϕ(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]]

zF3(N 0 0)

zI1 +[σϕ(h)βNϕ(h)1 + ϕ(h)(μ + c)]

1 + ϕ(h)[μ + c]

zG1(N 0 0)

zS

11 + μϕ(h)

zG1(N 0 0)

zE

minus Nσϕ(h)βϕ(h)

1 + μϕ(h) + ϵminus 11113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

Journal of Mathematics 5

zG1(N 0 0)

zI

minus Nβϕ(h)

1 + ϵminus 1+ μϕ(h)1113960 1113961

ϵminus 1

1 + μϕ(h)+

1[1 + ϕ(h)(μ + c)

+σNβϕ2

(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113890 1113891

zG2(N 0 0)

zS 0

zG2(N 0 0)

zE

11 + ϕ(h)(μ + σ)

1 +Nβσϕ2(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113890 1113891

zG2(N 0 0)

zI

βϕ(h)

[1 + ϕ(h)(μ + σ)]

1 + ϕ(h)(μ + σ) + Nσβϕ2(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113896 1113897

zG3(N 0 0)

zS

zG3(N 0 0)

zE

σϕ(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1 +

σβNϕ2(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113896 1113897

zG3(N 0 0)

zI

1

[1 + ϕ(h)(μ + σ)]+

Nσβϕ2(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]2 +

N2β2σ2ϕ4(h)

[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + σ)]

2

(14)

For ease in the calculation we use the followingconventions

α μNϕ(h)gt 0 η1 μϕ(h)gt 0 η2

βϕ(h)gt 0 η3 σϕ(h)gt 0

θ 1 + ϵminus 1+ μϕ(h) 1 + ϵminus 1

+ η1 gt 1

δ1 1 + ϕ(h)[μ + σ]gt 1 δ2

(15)

It is clear that θgt η1 1 + η1 gt δ1 1 + η1 gt δ2 and δ2 θ minus ϵminus 1 + cϕ(h) If R0 σβN(μ + σ)(μ + c)lt 1 the Ja-cobian calculated at disease-free points (Slowast0 Elowast0 Ilowast0 )

(N 0 0) is given as

JG(N 0 0)

zG1(N 0 0)

zS

zG1(N 0 0)

zE

zG1(N 0 0)

zI

zG1(N 0 0)

zS

zG1(N 0 0)

zE

zG1(N 0 0)

zI

zG1(N 0 0)

zS

zG1(N 0 0)

zE

zG1(N 0 0)

zI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)

Since the partial derivatives using the conventions are

zG1(N 0 0)

zS

11 + η1

zG1(N 0 0)

zE

minus Nη2η3θδ1δ2

zG1(N 0 0)

zI

minus Nη2θϵminus 1

1 + η1+

1δ2

+Nη2η3δ1δ2

1113890 1113891

zG2(N 0 0)

zS 0

zG2(N 0 0)

zE

1δ1

+Nη2η3δ21δ2

1113890 1113891

zG2(N 0 0)

zI

Nη2δ1δ2

+N

2η22η3δ21δ2

1113890 1113891

zG3(N 0 0)

zS 0

zG3(N 0 0)

zE

η3δ1δ2

+Nη2η

23

δ21δ22

1113890 1113891

zG3(N 0 0)

zI

1δ2

+Nη2η3δ1δ

22

+N

2η22η23

δ21δ22

1113890 1113891

(17)

6 Journal of Mathematics

So

JG(N 0 0)

11 + η1

minus Nη2η3θδ1δ2

minus Nη2θϵminus 1

1 + η1+

1δ2

+Nη2η3δ1δ2

1113890 1113891

01δ1

+Nη2η3δ21δ2

1113890 1113891Nη2δ1δ2

+N

2η22η3δ21δ2

1113890 1113891

0η3δ1δ2

+Nη2η

23

δ21δ22

1113890 11138911δ2

+Nη2η3δ1δ

22

+N

2η22η23

δ21δ22

1113890 1113891

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)

To calculate the eigenvalues put det (J minus λI) 0

λ1 11 + η1 11 + μϕ(h)lt 1 And

λ2 minus λδ1δ2 + Nη2η3

δ21δ21113888 1113889 +

δ21δ2 + δ1Nη2η3 + N2η22η

23

δ21δ22

1113888 11138891113890 1113891

+δ1δ2 + Nη2η3

δ21δ21113888 1113889

δ21δ2 + δ1Nη2η3 + N2η22η

23

δ21δ22

1113888 1113889 minusNη2δ1 + N

2η22η3δ21δ2

1113888 1113889η3δ1δ2 + Nη2η

23

δ21δ22

1113888 11138891113890 1113891 0

λ2 minus Aλ + B 0

(19)

where

A δ1δ2 + Nη2η3δ21δ21113872 1113873 + δ21δ2 + δ1Nη2η3 + N

2η22η23δ

21δ

221113872 1113873 trace J

lowast

B δ1δ2 + Nη2η3

δ21δ21113888 1113889

δ21δ2 + δ1Nη2η3 + N2η22η

23

δ21δ22

1113888 1113889 minusNη2δ1 + N

2η22η3δ21δ2

1113888 1113889η3δ1δ2 + Nη2η

23

δ21δ22

1113888 11138891113890 1113891 det Jlowast

Jlowast

1δ1

+Nη2η3δ21δ2

1113890 1113891Nη2δ1δ2

+N

2η22η3δ21δ2

1113890 1113891

η3δ1δ2

+Nη2η

23

δ21δ22

1113890 11138911δ2

+Nη2η3δ1δ

22

+N

2η22η23

δ21δ22

1113890 1113891

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

To calculate the eigenvalues of Jlowast the following lemma isproved

Lemma 1 For the quadratic equation λ2 minus λA + B 0 bothroots satisfy |λi|lt 1 i 2 3 if and only if the followingconditions are satisfied [25]

(1) 1 minus A + Bgt 0

(2) 1 + A + Bgt 0

(3)Blt 1

(21)

Let us define A TraceJlowast B DetJlowast -erefore

Journal of Mathematics 7

A δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N

2η22η23

δ21δ22

gt 0

B 1

δ1δ2+

Nη2η3δ21δ

22lt 1

f(minus 1) 1 + A + Bgt 0

f(0) B 1

δ1δ2+

Nη2η3δ21δ

22lt

1δ1δ2

+Nη2η3δ1δ2

1 + Nη2η3

δ1δ2

1 + R0(μ + σ)(μ + c)ϕ2(h)

1 + ϕ2(h)(μ + σ)(μ + c) + ϕ(h)[(μ + σ)(μ + c)]

lt 1 sinceR0 lt 1( 1113857

rArrBlt 1

(22)

Now for

f(1) 1 minus A + B

δ21δ

22 + δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N

2η22η23 + Nη2η3 + δ1δ2

δ21δ22

gt 0

(23)

As all the conditions of the lemma hold for R0 lt 1therefore the absolute value of both eigenvalues of Jlowast is less

than one whenever R0 lt 1 -us the numerical scheme inequations (8) and (10) will converge unconditionally todisease-free equilibrium (N 0 0) for any value of time step h

whenever R0 lt 1 which is also computationally verified inFigure 2(a)

52 Endemic Equilibrium (SeEe Ie) (NR0 μN μ + σ(1 minus

1 R0) μβ(R0 minus 1)) -e value of functions and its deriv-atives at an endemic point is as

F1 Se Ee Ie( 1113857 N

R0

F2 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857

(μ + σ)

F3 Se Ee Ie( 1113857 μβ

R0 minus 1( 1113857

G1 Se Ee Ie( 1113857 N

R0

G2 Se Ee Ie( 1113857 μβ

R0 minus 1( 1113857

G3 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857

(μ + σ)

(24)

8 Journal of Mathematics

zF1 Se Ee Ie( 1113857

zS

11 + μϕ(h)R0

zF1 Se Ee Ie( 1113857

zE 0

zF1 Se Ee Ie( 1113857

zI

minus Nβϕ(h)

1 + μϕ(h)R0

zF2 Se Ee Ie( 1113857

zS

μϕ(h) R0 minus 1( 1113857

1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + σ)]

zF2 Se Ee Ie( 1113857

zE

1[1 + ϕ(h)(μ + σ)]

zF2 Se Ee Ie( 1113857

zI

minus Nβϕ(h) μϕ(l)R20 minus 2μϕ(h)R0 minus 11113960 1113961

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

zF3 Se Ee Ie( 1113857

zS

μσϕ2(h) R0 minus 1( 1113857

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

zF3 Se Ee Ie( 1113857

zE

σϕ(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

zF3 Se Ee Ie( 1113857

zI

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h) minus 11113960 1113961

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

(25)

zG1 Se Ee Ie( 1113857

zS

11 + μϕ(h)R01113858 1113859

minusNμβσϕ3(h) R0 minus 1( 1113857

R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

zG1 Se Ee Ie( 1113857

zE

minus Nβσϕ2(h)

R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

Eige

n va

lues

Jacobian at DFE

0492

0494

0496

0498

05

0502

0504

0506

0508

051

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size

(a)

098

0981

0982

0983

0984

0985

0986

0987

0988

0989Jacobian at EE

Eige

n va

lues

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size

(b)

Figure 2 Spectral radius of the Jacobian at equilibrium points

Journal of Mathematics 9

zG1 Se Ee Ie( 1113857

zI

minus Nβϕ(h)

1 + ϵminus 1 + μϕ(h)R01113960 11139611 + μϕ(h)R0

middot ϵminus 1+

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

⎡⎣ ⎤⎦

zG2 Se Ee Ie( 1113857

zS

μβϕ(h) R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890

minusNμβ2σϕ4(h)μ R0 minus 1( 1113857

2

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nμβσϕ3(h) R0 minus 1( 1113857

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

1113891

zG2 Se Ee Ie( 1113857

zE

1[1 + ϕ(h)(μ + σ)]

+Nβσϕ2(h)

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]

minusNμβ2σϕ3

(h) R0 minus 1( 1113857

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961

zG2 Se Ee Ie( 1113857

zI

minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961

+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967

βR20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nβϕ(h)

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus

N2β2σϕ3(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

zG3 Se Ee Ie( 1113857

zS

σϕ(h)

[1 + ϕ(h)(μ + c)]

μβϕ(h) R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890

minusNμβ2σϕ4(h)μ R0 minus 1( 1113857

2

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nμβσϕ3(h) R0 minus 1( 1113857

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

1113891

zG3 Se Ee Ie( 1113857

zE

σϕ(h)

[1 + ϕ(h)(μ + c)]

1[1 + ϕ(h)(μ + σ)]

minusNμβ2σϕ3(h) R0 minus 1( 1113857

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961

⎡⎢⎣

+Nβσϕ2

(h)

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]

1113891

zG3 Se Ee Ie( 1113857

zI

σϕ(h)

[1 + ϕ(h)(μ + c)]

minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961⎡⎢⎣

+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967

βR20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

⎤⎥⎦

10 Journal of Mathematics

Table 1 Comparison of NSFD scheme with Euler and RK-4 schemes

l 001 01 100 1000Euler PC Convergence Divergence Divergence DivergenceRK-4 Convergence Divergence Divergence DivergenceNSFDPC Convergence Convergence Convergence Convergence

times107

NSFD-PC

Disease-free equilibrium

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Disease-free equilibriumtimes104

NSFD-PC

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

0

05

1

15

2

25

3

35

4

45

5

Expo

sed

fract

ion

(b)

times104

NSFD-PC

Disease-free equilibrium

0

05

1

15

2

25

3

Infe

cted

frac

tion

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

times107

NSFD-PC

X 5675Y 2435e + 007

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

1

15

2

25

3

35

Susc

eptib

le fr

actio

n

(d)

Figure 3 Continued

Journal of Mathematics 11

+Nβϕ(h)

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus

N2β2σϕ3(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

(26)

-e Jacobian evaluated at an endemic point(Slowaste Elowaste Ilowaste ) (Se Ee Ie) (NR0 μNμ + σ(1 minus 1R0) μβ(R0 minus 1)) is evaluated as

JG Slowaste Elowaste Ilowaste( 1113857

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(27)

It is tedious to calculate the eigenvalues of the Jacobian ofendemic equilibrium analytically so we have plotted thelargest eigenvalue against each step size and Figure 2(b)shows that for all step sizes the spectral radius of the Ja-cobian of EE remains less than one if R0 gt 1 which impliesthat the numerical scheme in equations (8) and (10) isunconditionally convergent if R0 gt 1 for all step sizes

6 Numerical Results and Discussion

All the methods showed the convergence for small step sizesin Table 1 However for large values of step sizes only thenormalized NSFDPC converge to the correct disease-freepoint for β 01 times 10minus 5 as well as the correct endemic pointfor β 03 times 10minus 5 -is means that for the value of the basic

reproductive number (threshold parameter) R0 lt 1 thenDFE is stable ie solution to the system (S E I R) areconverging to it -is can be seen from Figures 3(a)ndash3(c) forh 001 Also for the value of the basic reproductivenumber (threshold parameter) R0 gt 1 then EE is stable iesolution to the system (S E I R) is converging to it-is canbe seen from Figures 3(d)ndash3(f ) for h 001

Figures 4(a)ndash4(c) show how the NSFDPC method ofDFE converges to the equilibrium point for different stepsizes Figures 4(d)ndash4(f) show how the NSFDPC method ofEE converges to the equilibrium point for different step sizes

For comparison with the well-known RK-4 method tosystem (1) using the parameter values given in Table 2 it isfound that in Figures 5(a)ndash5(f) the numerical solutionconverges for h 001 for both equilibrium points and

times105

NSFD-PC

X 562Y 1125e + 004

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

0

1

2

3

4

5

6

7

8Ex

pose

d fra

ctio

n

(e)

times105

NSFD-PC

X 5523Y 7023

Infe

cted

frac

tion

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

0

05

1

15

2

25

3

35

4

45

5

(f )

Figure 3 Time plots of susceptible exposed and infected population using the NSFDPC method with h 001

12 Journal of Mathematics

times107

NSFD-PC

Disease-free equilibrium

20 40 60 80 100 120 140 160 180 2000Time (years) h = 01

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

(a)

times107

NSFD-PC

Disease-free equilibrium

500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(b)

Disease-free equilibriumtimes107

times105

NSFD-PC

05 1 15 2 25 3 35 40Time (years) h = 1000

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(c)

X 5827Y 7022

Infe

cted

frac

tion

Endemic equilibriumtimes104

NSFD-PC

100 200 300 400 500 6000Time (years) h = 01

0

2

4

6

8

10

12

14

16

18

(d)

Endemic equilibriumtimes104

NSFD-PC

X 9600Y 7022

Infe

cted

frac

tion

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10

0

05

1

15

2

25

3

(e)

Endemic equilibriumtimes104

times105

NSFD-PC

X 861e + 005Y 7022

Infe

cted

frac

tion

1 2 3 4 5 6 7 8 9 100Time (years) h = 1000

0

05

1

15

2

25

3

(f )

Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000

Journal of Mathematics 13

Table 2 Parameters value of the measles model

Parameters N μ σ c

5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1

DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5

EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5

Disease-free equilibrium

RK4

times107

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Expo

sed

fract

ion

Disease-free equilibrium

RK4

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

times104

Infe

cted

frac

tion

Disease-free equilibrium

RK4

0

05

1

15

2

25

3

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

times107

X 5578Y 2435e + 007

Endemic equilibrium

RK4

1

15

2

25

3

35

Susc

eptib

le F

ract

ion

100 200 300 400 500 6000Time (years) h = 001

(d)

Figure 5 Continued

14 Journal of Mathematics

Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively

Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces

overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8

-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent

times105

X 5772Y 1125e + 004

Endemic equilibrium

RK4

Expo

sed

fract

ion

0

1

2

3

4

5

6

7

8

9

10

100 200 300 400 500 6000Time (years) h = 001

(e)

times105

X 7659Y 7022

RK4

Endemic equilibrium

Infe

cted

frac

tion

0

1

2

3

4

5

6

100 200 300 400 500 600 700 8000Time (years) h = 001

(f )

Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001

times1087 RK-4 scheme (disease-free equilibrium)

SusceptibleExposedInfected

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

Popu

latio

n fra

ctio

ns

(a)

RK-4 scheme (endemic equilibrium)

SusceptibleExposedInfected

times10145

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

Popu

latio

n fra

ctio

ns

(b)

Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01

Journal of Mathematics 15

Euler-PC

Disease-free equilibriumtimes107

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Euler-PC

Disease-free equilibrium

Expo

sed

fract

ion

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

Euler-PC

Disease-free equilibrium

Infe

cted

frac

tion

times104

0

05

1

15

2

25

3

35

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

Euler-PC

X 5733Y 2435e + 007

Endemic equilibriumtimes107

1

15

2

25

3

35Su

scep

tible

frac

tion

100 200 300 400 500 6000Time (years) h = 001

(d)

Euler-PC

X 5662Y 1118e + 004

Endemic equilibrium

Expo

sed

fract

ion

times105

0

2

4

6

8

10

12

100 200 300 400 500 6000Time (years) h = 001

(e)

Euler-PC

X 5427Y 7025

Endemic equilibrium

Infe

cted

frac

tion

times105

0

1

2

3

4

5

6

7

100 200 300 400 500 6000Time (years) h = 001

(f )

Figure 7 Time plots of population using the Euler-PC method with h 001

16 Journal of Mathematics

7 Conclusions

In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the

positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory

Euler-PC scheme (disease-free equilibrium)times1041

ndash3

ndash2

ndash1

0

1

2

3

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(a)

Euler-PC scheme (endemic equilibrium)times1053

ndash6

ndash4

ndash2

0

2

4

6

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(b)

Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01

NSFD-PCRK-4

times1092

Infe

cted

frac

tion

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(a)

NSFD-PCEuler-PC

times10213

Infe

cted

frac

tion

ndash4

ndash35

ndash3

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(b)

Figure 9 Comparative convergence graphs of the NSFDPC method

Journal of Mathematics 17

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6

References

[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018

[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012

[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019

[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018

[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007

[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020

[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994

[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000

[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007

[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973

[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003

[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002

[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005

[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007

[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005

[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003

[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001

[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009

[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005

[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008

[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008

[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003

[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002

[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010

[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001

18 Journal of Mathematics

zG1(N 0 0)

zI

minus Nβϕ(h)

1 + ϵminus 1+ μϕ(h)1113960 1113961

ϵminus 1

1 + μϕ(h)+

1[1 + ϕ(h)(μ + c)

+σNβϕ2

(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113890 1113891

zG2(N 0 0)

zS 0

zG2(N 0 0)

zE

11 + ϕ(h)(μ + σ)

1 +Nβσϕ2(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113890 1113891

zG2(N 0 0)

zI

βϕ(h)

[1 + ϕ(h)(μ + σ)]

1 + ϕ(h)(μ + σ) + Nσβϕ2(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113896 1113897

zG3(N 0 0)

zS

zG3(N 0 0)

zE

σϕ(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1 +

σβNϕ2(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113896 1113897

zG3(N 0 0)

zI

1

[1 + ϕ(h)(μ + σ)]+

Nσβϕ2(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]2 +

N2β2σ2ϕ4(h)

[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + σ)]

2

(14)

For ease in the calculation we use the followingconventions

α μNϕ(h)gt 0 η1 μϕ(h)gt 0 η2

βϕ(h)gt 0 η3 σϕ(h)gt 0

θ 1 + ϵminus 1+ μϕ(h) 1 + ϵminus 1

+ η1 gt 1

δ1 1 + ϕ(h)[μ + σ]gt 1 δ2

(15)

It is clear that θgt η1 1 + η1 gt δ1 1 + η1 gt δ2 and δ2 θ minus ϵminus 1 + cϕ(h) If R0 σβN(μ + σ)(μ + c)lt 1 the Ja-cobian calculated at disease-free points (Slowast0 Elowast0 Ilowast0 )

(N 0 0) is given as

JG(N 0 0)

zG1(N 0 0)

zS

zG1(N 0 0)

zE

zG1(N 0 0)

zI

zG1(N 0 0)

zS

zG1(N 0 0)

zE

zG1(N 0 0)

zI

zG1(N 0 0)

zS

zG1(N 0 0)

zE

zG1(N 0 0)

zI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)

Since the partial derivatives using the conventions are

zG1(N 0 0)

zS

11 + η1

zG1(N 0 0)

zE

minus Nη2η3θδ1δ2

zG1(N 0 0)

zI

minus Nη2θϵminus 1

1 + η1+

1δ2

+Nη2η3δ1δ2

1113890 1113891

zG2(N 0 0)

zS 0

zG2(N 0 0)

zE

1δ1

+Nη2η3δ21δ2

1113890 1113891

zG2(N 0 0)

zI

Nη2δ1δ2

+N

2η22η3δ21δ2

1113890 1113891

zG3(N 0 0)

zS 0

zG3(N 0 0)

zE

η3δ1δ2

+Nη2η

23

δ21δ22

1113890 1113891

zG3(N 0 0)

zI

1δ2

+Nη2η3δ1δ

22

+N

2η22η23

δ21δ22

1113890 1113891

(17)

6 Journal of Mathematics

So

JG(N 0 0)

11 + η1

minus Nη2η3θδ1δ2

minus Nη2θϵminus 1

1 + η1+

1δ2

+Nη2η3δ1δ2

1113890 1113891

01δ1

+Nη2η3δ21δ2

1113890 1113891Nη2δ1δ2

+N

2η22η3δ21δ2

1113890 1113891

0η3δ1δ2

+Nη2η

23

δ21δ22

1113890 11138911δ2

+Nη2η3δ1δ

22

+N

2η22η23

δ21δ22

1113890 1113891

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)

To calculate the eigenvalues put det (J minus λI) 0

λ1 11 + η1 11 + μϕ(h)lt 1 And

λ2 minus λδ1δ2 + Nη2η3

δ21δ21113888 1113889 +

δ21δ2 + δ1Nη2η3 + N2η22η

23

δ21δ22

1113888 11138891113890 1113891

+δ1δ2 + Nη2η3

δ21δ21113888 1113889

δ21δ2 + δ1Nη2η3 + N2η22η

23

δ21δ22

1113888 1113889 minusNη2δ1 + N

2η22η3δ21δ2

1113888 1113889η3δ1δ2 + Nη2η

23

δ21δ22

1113888 11138891113890 1113891 0

λ2 minus Aλ + B 0

(19)

where

A δ1δ2 + Nη2η3δ21δ21113872 1113873 + δ21δ2 + δ1Nη2η3 + N

2η22η23δ

21δ

221113872 1113873 trace J

lowast

B δ1δ2 + Nη2η3

δ21δ21113888 1113889

δ21δ2 + δ1Nη2η3 + N2η22η

23

δ21δ22

1113888 1113889 minusNη2δ1 + N

2η22η3δ21δ2

1113888 1113889η3δ1δ2 + Nη2η

23

δ21δ22

1113888 11138891113890 1113891 det Jlowast

Jlowast

1δ1

+Nη2η3δ21δ2

1113890 1113891Nη2δ1δ2

+N

2η22η3δ21δ2

1113890 1113891

η3δ1δ2

+Nη2η

23

δ21δ22

1113890 11138911δ2

+Nη2η3δ1δ

22

+N

2η22η23

δ21δ22

1113890 1113891

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

To calculate the eigenvalues of Jlowast the following lemma isproved

Lemma 1 For the quadratic equation λ2 minus λA + B 0 bothroots satisfy |λi|lt 1 i 2 3 if and only if the followingconditions are satisfied [25]

(1) 1 minus A + Bgt 0

(2) 1 + A + Bgt 0

(3)Blt 1

(21)

Let us define A TraceJlowast B DetJlowast -erefore

Journal of Mathematics 7

A δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N

2η22η23

δ21δ22

gt 0

B 1

δ1δ2+

Nη2η3δ21δ

22lt 1

f(minus 1) 1 + A + Bgt 0

f(0) B 1

δ1δ2+

Nη2η3δ21δ

22lt

1δ1δ2

+Nη2η3δ1δ2

1 + Nη2η3

δ1δ2

1 + R0(μ + σ)(μ + c)ϕ2(h)

1 + ϕ2(h)(μ + σ)(μ + c) + ϕ(h)[(μ + σ)(μ + c)]

lt 1 sinceR0 lt 1( 1113857

rArrBlt 1

(22)

Now for

f(1) 1 minus A + B

δ21δ

22 + δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N

2η22η23 + Nη2η3 + δ1δ2

δ21δ22

gt 0

(23)

As all the conditions of the lemma hold for R0 lt 1therefore the absolute value of both eigenvalues of Jlowast is less

than one whenever R0 lt 1 -us the numerical scheme inequations (8) and (10) will converge unconditionally todisease-free equilibrium (N 0 0) for any value of time step h

whenever R0 lt 1 which is also computationally verified inFigure 2(a)

52 Endemic Equilibrium (SeEe Ie) (NR0 μN μ + σ(1 minus

1 R0) μβ(R0 minus 1)) -e value of functions and its deriv-atives at an endemic point is as

F1 Se Ee Ie( 1113857 N

R0

F2 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857

(μ + σ)

F3 Se Ee Ie( 1113857 μβ

R0 minus 1( 1113857

G1 Se Ee Ie( 1113857 N

R0

G2 Se Ee Ie( 1113857 μβ

R0 minus 1( 1113857

G3 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857

(μ + σ)

(24)

8 Journal of Mathematics

zF1 Se Ee Ie( 1113857

zS

11 + μϕ(h)R0

zF1 Se Ee Ie( 1113857

zE 0

zF1 Se Ee Ie( 1113857

zI

minus Nβϕ(h)

1 + μϕ(h)R0

zF2 Se Ee Ie( 1113857

zS

μϕ(h) R0 minus 1( 1113857

1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + σ)]

zF2 Se Ee Ie( 1113857

zE

1[1 + ϕ(h)(μ + σ)]

zF2 Se Ee Ie( 1113857

zI

minus Nβϕ(h) μϕ(l)R20 minus 2μϕ(h)R0 minus 11113960 1113961

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

zF3 Se Ee Ie( 1113857

zS

μσϕ2(h) R0 minus 1( 1113857

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

zF3 Se Ee Ie( 1113857

zE

σϕ(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

zF3 Se Ee Ie( 1113857

zI

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h) minus 11113960 1113961

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

(25)

zG1 Se Ee Ie( 1113857

zS

11 + μϕ(h)R01113858 1113859

minusNμβσϕ3(h) R0 minus 1( 1113857

R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

zG1 Se Ee Ie( 1113857

zE

minus Nβσϕ2(h)

R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

Eige

n va

lues

Jacobian at DFE

0492

0494

0496

0498

05

0502

0504

0506

0508

051

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size

(a)

098

0981

0982

0983

0984

0985

0986

0987

0988

0989Jacobian at EE

Eige

n va

lues

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size

(b)

Figure 2 Spectral radius of the Jacobian at equilibrium points

Journal of Mathematics 9

zG1 Se Ee Ie( 1113857

zI

minus Nβϕ(h)

1 + ϵminus 1 + μϕ(h)R01113960 11139611 + μϕ(h)R0

middot ϵminus 1+

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

⎡⎣ ⎤⎦

zG2 Se Ee Ie( 1113857

zS

μβϕ(h) R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890

minusNμβ2σϕ4(h)μ R0 minus 1( 1113857

2

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nμβσϕ3(h) R0 minus 1( 1113857

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

1113891

zG2 Se Ee Ie( 1113857

zE

1[1 + ϕ(h)(μ + σ)]

+Nβσϕ2(h)

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]

minusNμβ2σϕ3

(h) R0 minus 1( 1113857

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961

zG2 Se Ee Ie( 1113857

zI

minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961

+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967

βR20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nβϕ(h)

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus

N2β2σϕ3(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

zG3 Se Ee Ie( 1113857

zS

σϕ(h)

[1 + ϕ(h)(μ + c)]

μβϕ(h) R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890

minusNμβ2σϕ4(h)μ R0 minus 1( 1113857

2

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nμβσϕ3(h) R0 minus 1( 1113857

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

1113891

zG3 Se Ee Ie( 1113857

zE

σϕ(h)

[1 + ϕ(h)(μ + c)]

1[1 + ϕ(h)(μ + σ)]

minusNμβ2σϕ3(h) R0 minus 1( 1113857

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961

⎡⎢⎣

+Nβσϕ2

(h)

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]

1113891

zG3 Se Ee Ie( 1113857

zI

σϕ(h)

[1 + ϕ(h)(μ + c)]

minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961⎡⎢⎣

+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967

βR20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

⎤⎥⎦

10 Journal of Mathematics

Table 1 Comparison of NSFD scheme with Euler and RK-4 schemes

l 001 01 100 1000Euler PC Convergence Divergence Divergence DivergenceRK-4 Convergence Divergence Divergence DivergenceNSFDPC Convergence Convergence Convergence Convergence

times107

NSFD-PC

Disease-free equilibrium

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Disease-free equilibriumtimes104

NSFD-PC

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

0

05

1

15

2

25

3

35

4

45

5

Expo

sed

fract

ion

(b)

times104

NSFD-PC

Disease-free equilibrium

0

05

1

15

2

25

3

Infe

cted

frac

tion

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

times107

NSFD-PC

X 5675Y 2435e + 007

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

1

15

2

25

3

35

Susc

eptib

le fr

actio

n

(d)

Figure 3 Continued

Journal of Mathematics 11

+Nβϕ(h)

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus

N2β2σϕ3(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

(26)

-e Jacobian evaluated at an endemic point(Slowaste Elowaste Ilowaste ) (Se Ee Ie) (NR0 μNμ + σ(1 minus 1R0) μβ(R0 minus 1)) is evaluated as

JG Slowaste Elowaste Ilowaste( 1113857

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(27)

It is tedious to calculate the eigenvalues of the Jacobian ofendemic equilibrium analytically so we have plotted thelargest eigenvalue against each step size and Figure 2(b)shows that for all step sizes the spectral radius of the Ja-cobian of EE remains less than one if R0 gt 1 which impliesthat the numerical scheme in equations (8) and (10) isunconditionally convergent if R0 gt 1 for all step sizes

6 Numerical Results and Discussion

All the methods showed the convergence for small step sizesin Table 1 However for large values of step sizes only thenormalized NSFDPC converge to the correct disease-freepoint for β 01 times 10minus 5 as well as the correct endemic pointfor β 03 times 10minus 5 -is means that for the value of the basic

reproductive number (threshold parameter) R0 lt 1 thenDFE is stable ie solution to the system (S E I R) areconverging to it -is can be seen from Figures 3(a)ndash3(c) forh 001 Also for the value of the basic reproductivenumber (threshold parameter) R0 gt 1 then EE is stable iesolution to the system (S E I R) is converging to it-is canbe seen from Figures 3(d)ndash3(f ) for h 001

Figures 4(a)ndash4(c) show how the NSFDPC method ofDFE converges to the equilibrium point for different stepsizes Figures 4(d)ndash4(f) show how the NSFDPC method ofEE converges to the equilibrium point for different step sizes

For comparison with the well-known RK-4 method tosystem (1) using the parameter values given in Table 2 it isfound that in Figures 5(a)ndash5(f) the numerical solutionconverges for h 001 for both equilibrium points and

times105

NSFD-PC

X 562Y 1125e + 004

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

0

1

2

3

4

5

6

7

8Ex

pose

d fra

ctio

n

(e)

times105

NSFD-PC

X 5523Y 7023

Infe

cted

frac

tion

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

0

05

1

15

2

25

3

35

4

45

5

(f )

Figure 3 Time plots of susceptible exposed and infected population using the NSFDPC method with h 001

12 Journal of Mathematics

times107

NSFD-PC

Disease-free equilibrium

20 40 60 80 100 120 140 160 180 2000Time (years) h = 01

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

(a)

times107

NSFD-PC

Disease-free equilibrium

500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(b)

Disease-free equilibriumtimes107

times105

NSFD-PC

05 1 15 2 25 3 35 40Time (years) h = 1000

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(c)

X 5827Y 7022

Infe

cted

frac

tion

Endemic equilibriumtimes104

NSFD-PC

100 200 300 400 500 6000Time (years) h = 01

0

2

4

6

8

10

12

14

16

18

(d)

Endemic equilibriumtimes104

NSFD-PC

X 9600Y 7022

Infe

cted

frac

tion

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10

0

05

1

15

2

25

3

(e)

Endemic equilibriumtimes104

times105

NSFD-PC

X 861e + 005Y 7022

Infe

cted

frac

tion

1 2 3 4 5 6 7 8 9 100Time (years) h = 1000

0

05

1

15

2

25

3

(f )

Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000

Journal of Mathematics 13

Table 2 Parameters value of the measles model

Parameters N μ σ c

5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1

DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5

EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5

Disease-free equilibrium

RK4

times107

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Expo

sed

fract

ion

Disease-free equilibrium

RK4

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

times104

Infe

cted

frac

tion

Disease-free equilibrium

RK4

0

05

1

15

2

25

3

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

times107

X 5578Y 2435e + 007

Endemic equilibrium

RK4

1

15

2

25

3

35

Susc

eptib

le F

ract

ion

100 200 300 400 500 6000Time (years) h = 001

(d)

Figure 5 Continued

14 Journal of Mathematics

Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively

Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces

overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8

-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent

times105

X 5772Y 1125e + 004

Endemic equilibrium

RK4

Expo

sed

fract

ion

0

1

2

3

4

5

6

7

8

9

10

100 200 300 400 500 6000Time (years) h = 001

(e)

times105

X 7659Y 7022

RK4

Endemic equilibrium

Infe

cted

frac

tion

0

1

2

3

4

5

6

100 200 300 400 500 600 700 8000Time (years) h = 001

(f )

Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001

times1087 RK-4 scheme (disease-free equilibrium)

SusceptibleExposedInfected

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

Popu

latio

n fra

ctio

ns

(a)

RK-4 scheme (endemic equilibrium)

SusceptibleExposedInfected

times10145

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

Popu

latio

n fra

ctio

ns

(b)

Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01

Journal of Mathematics 15

Euler-PC

Disease-free equilibriumtimes107

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Euler-PC

Disease-free equilibrium

Expo

sed

fract

ion

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

Euler-PC

Disease-free equilibrium

Infe

cted

frac

tion

times104

0

05

1

15

2

25

3

35

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

Euler-PC

X 5733Y 2435e + 007

Endemic equilibriumtimes107

1

15

2

25

3

35Su

scep

tible

frac

tion

100 200 300 400 500 6000Time (years) h = 001

(d)

Euler-PC

X 5662Y 1118e + 004

Endemic equilibrium

Expo

sed

fract

ion

times105

0

2

4

6

8

10

12

100 200 300 400 500 6000Time (years) h = 001

(e)

Euler-PC

X 5427Y 7025

Endemic equilibrium

Infe

cted

frac

tion

times105

0

1

2

3

4

5

6

7

100 200 300 400 500 6000Time (years) h = 001

(f )

Figure 7 Time plots of population using the Euler-PC method with h 001

16 Journal of Mathematics

7 Conclusions

In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the

positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory

Euler-PC scheme (disease-free equilibrium)times1041

ndash3

ndash2

ndash1

0

1

2

3

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(a)

Euler-PC scheme (endemic equilibrium)times1053

ndash6

ndash4

ndash2

0

2

4

6

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(b)

Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01

NSFD-PCRK-4

times1092

Infe

cted

frac

tion

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(a)

NSFD-PCEuler-PC

times10213

Infe

cted

frac

tion

ndash4

ndash35

ndash3

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(b)

Figure 9 Comparative convergence graphs of the NSFDPC method

Journal of Mathematics 17

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6

References

[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018

[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012

[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019

[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018

[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007

[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020

[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994

[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000

[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007

[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973

[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003

[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002

[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005

[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007

[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005

[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003

[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001

[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009

[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005

[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008

[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008

[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003

[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002

[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010

[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001

18 Journal of Mathematics

So

JG(N 0 0)

11 + η1

minus Nη2η3θδ1δ2

minus Nη2θϵminus 1

1 + η1+

1δ2

+Nη2η3δ1δ2

1113890 1113891

01δ1

+Nη2η3δ21δ2

1113890 1113891Nη2δ1δ2

+N

2η22η3δ21δ2

1113890 1113891

0η3δ1δ2

+Nη2η

23

δ21δ22

1113890 11138911δ2

+Nη2η3δ1δ

22

+N

2η22η23

δ21δ22

1113890 1113891

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)

To calculate the eigenvalues put det (J minus λI) 0

λ1 11 + η1 11 + μϕ(h)lt 1 And

λ2 minus λδ1δ2 + Nη2η3

δ21δ21113888 1113889 +

δ21δ2 + δ1Nη2η3 + N2η22η

23

δ21δ22

1113888 11138891113890 1113891

+δ1δ2 + Nη2η3

δ21δ21113888 1113889

δ21δ2 + δ1Nη2η3 + N2η22η

23

δ21δ22

1113888 1113889 minusNη2δ1 + N

2η22η3δ21δ2

1113888 1113889η3δ1δ2 + Nη2η

23

δ21δ22

1113888 11138891113890 1113891 0

λ2 minus Aλ + B 0

(19)

where

A δ1δ2 + Nη2η3δ21δ21113872 1113873 + δ21δ2 + δ1Nη2η3 + N

2η22η23δ

21δ

221113872 1113873 trace J

lowast

B δ1δ2 + Nη2η3

δ21δ21113888 1113889

δ21δ2 + δ1Nη2η3 + N2η22η

23

δ21δ22

1113888 1113889 minusNη2δ1 + N

2η22η3δ21δ2

1113888 1113889η3δ1δ2 + Nη2η

23

δ21δ22

1113888 11138891113890 1113891 det Jlowast

Jlowast

1δ1

+Nη2η3δ21δ2

1113890 1113891Nη2δ1δ2

+N

2η22η3δ21δ2

1113890 1113891

η3δ1δ2

+Nη2η

23

δ21δ22

1113890 11138911δ2

+Nη2η3δ1δ

22

+N

2η22η23

δ21δ22

1113890 1113891

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

To calculate the eigenvalues of Jlowast the following lemma isproved

Lemma 1 For the quadratic equation λ2 minus λA + B 0 bothroots satisfy |λi|lt 1 i 2 3 if and only if the followingconditions are satisfied [25]

(1) 1 minus A + Bgt 0

(2) 1 + A + Bgt 0

(3)Blt 1

(21)

Let us define A TraceJlowast B DetJlowast -erefore

Journal of Mathematics 7

A δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N

2η22η23

δ21δ22

gt 0

B 1

δ1δ2+

Nη2η3δ21δ

22lt 1

f(minus 1) 1 + A + Bgt 0

f(0) B 1

δ1δ2+

Nη2η3δ21δ

22lt

1δ1δ2

+Nη2η3δ1δ2

1 + Nη2η3

δ1δ2

1 + R0(μ + σ)(μ + c)ϕ2(h)

1 + ϕ2(h)(μ + σ)(μ + c) + ϕ(h)[(μ + σ)(μ + c)]

lt 1 sinceR0 lt 1( 1113857

rArrBlt 1

(22)

Now for

f(1) 1 minus A + B

δ21δ

22 + δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N

2η22η23 + Nη2η3 + δ1δ2

δ21δ22

gt 0

(23)

As all the conditions of the lemma hold for R0 lt 1therefore the absolute value of both eigenvalues of Jlowast is less

than one whenever R0 lt 1 -us the numerical scheme inequations (8) and (10) will converge unconditionally todisease-free equilibrium (N 0 0) for any value of time step h

whenever R0 lt 1 which is also computationally verified inFigure 2(a)

52 Endemic Equilibrium (SeEe Ie) (NR0 μN μ + σ(1 minus

1 R0) μβ(R0 minus 1)) -e value of functions and its deriv-atives at an endemic point is as

F1 Se Ee Ie( 1113857 N

R0

F2 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857

(μ + σ)

F3 Se Ee Ie( 1113857 μβ

R0 minus 1( 1113857

G1 Se Ee Ie( 1113857 N

R0

G2 Se Ee Ie( 1113857 μβ

R0 minus 1( 1113857

G3 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857

(μ + σ)

(24)

8 Journal of Mathematics

zF1 Se Ee Ie( 1113857

zS

11 + μϕ(h)R0

zF1 Se Ee Ie( 1113857

zE 0

zF1 Se Ee Ie( 1113857

zI

minus Nβϕ(h)

1 + μϕ(h)R0

zF2 Se Ee Ie( 1113857

zS

μϕ(h) R0 minus 1( 1113857

1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + σ)]

zF2 Se Ee Ie( 1113857

zE

1[1 + ϕ(h)(μ + σ)]

zF2 Se Ee Ie( 1113857

zI

minus Nβϕ(h) μϕ(l)R20 minus 2μϕ(h)R0 minus 11113960 1113961

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

zF3 Se Ee Ie( 1113857

zS

μσϕ2(h) R0 minus 1( 1113857

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

zF3 Se Ee Ie( 1113857

zE

σϕ(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

zF3 Se Ee Ie( 1113857

zI

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h) minus 11113960 1113961

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

(25)

zG1 Se Ee Ie( 1113857

zS

11 + μϕ(h)R01113858 1113859

minusNμβσϕ3(h) R0 minus 1( 1113857

R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

zG1 Se Ee Ie( 1113857

zE

minus Nβσϕ2(h)

R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

Eige

n va

lues

Jacobian at DFE

0492

0494

0496

0498

05

0502

0504

0506

0508

051

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size

(a)

098

0981

0982

0983

0984

0985

0986

0987

0988

0989Jacobian at EE

Eige

n va

lues

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size

(b)

Figure 2 Spectral radius of the Jacobian at equilibrium points

Journal of Mathematics 9

zG1 Se Ee Ie( 1113857

zI

minus Nβϕ(h)

1 + ϵminus 1 + μϕ(h)R01113960 11139611 + μϕ(h)R0

middot ϵminus 1+

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

⎡⎣ ⎤⎦

zG2 Se Ee Ie( 1113857

zS

μβϕ(h) R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890

minusNμβ2σϕ4(h)μ R0 minus 1( 1113857

2

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nμβσϕ3(h) R0 minus 1( 1113857

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

1113891

zG2 Se Ee Ie( 1113857

zE

1[1 + ϕ(h)(μ + σ)]

+Nβσϕ2(h)

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]

minusNμβ2σϕ3

(h) R0 minus 1( 1113857

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961

zG2 Se Ee Ie( 1113857

zI

minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961

+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967

βR20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nβϕ(h)

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus

N2β2σϕ3(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

zG3 Se Ee Ie( 1113857

zS

σϕ(h)

[1 + ϕ(h)(μ + c)]

μβϕ(h) R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890

minusNμβ2σϕ4(h)μ R0 minus 1( 1113857

2

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nμβσϕ3(h) R0 minus 1( 1113857

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

1113891

zG3 Se Ee Ie( 1113857

zE

σϕ(h)

[1 + ϕ(h)(μ + c)]

1[1 + ϕ(h)(μ + σ)]

minusNμβ2σϕ3(h) R0 minus 1( 1113857

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961

⎡⎢⎣

+Nβσϕ2

(h)

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]

1113891

zG3 Se Ee Ie( 1113857

zI

σϕ(h)

[1 + ϕ(h)(μ + c)]

minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961⎡⎢⎣

+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967

βR20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

⎤⎥⎦

10 Journal of Mathematics

Table 1 Comparison of NSFD scheme with Euler and RK-4 schemes

l 001 01 100 1000Euler PC Convergence Divergence Divergence DivergenceRK-4 Convergence Divergence Divergence DivergenceNSFDPC Convergence Convergence Convergence Convergence

times107

NSFD-PC

Disease-free equilibrium

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Disease-free equilibriumtimes104

NSFD-PC

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

0

05

1

15

2

25

3

35

4

45

5

Expo

sed

fract

ion

(b)

times104

NSFD-PC

Disease-free equilibrium

0

05

1

15

2

25

3

Infe

cted

frac

tion

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

times107

NSFD-PC

X 5675Y 2435e + 007

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

1

15

2

25

3

35

Susc

eptib

le fr

actio

n

(d)

Figure 3 Continued

Journal of Mathematics 11

+Nβϕ(h)

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus

N2β2σϕ3(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

(26)

-e Jacobian evaluated at an endemic point(Slowaste Elowaste Ilowaste ) (Se Ee Ie) (NR0 μNμ + σ(1 minus 1R0) μβ(R0 minus 1)) is evaluated as

JG Slowaste Elowaste Ilowaste( 1113857

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(27)

It is tedious to calculate the eigenvalues of the Jacobian ofendemic equilibrium analytically so we have plotted thelargest eigenvalue against each step size and Figure 2(b)shows that for all step sizes the spectral radius of the Ja-cobian of EE remains less than one if R0 gt 1 which impliesthat the numerical scheme in equations (8) and (10) isunconditionally convergent if R0 gt 1 for all step sizes

6 Numerical Results and Discussion

All the methods showed the convergence for small step sizesin Table 1 However for large values of step sizes only thenormalized NSFDPC converge to the correct disease-freepoint for β 01 times 10minus 5 as well as the correct endemic pointfor β 03 times 10minus 5 -is means that for the value of the basic

reproductive number (threshold parameter) R0 lt 1 thenDFE is stable ie solution to the system (S E I R) areconverging to it -is can be seen from Figures 3(a)ndash3(c) forh 001 Also for the value of the basic reproductivenumber (threshold parameter) R0 gt 1 then EE is stable iesolution to the system (S E I R) is converging to it-is canbe seen from Figures 3(d)ndash3(f ) for h 001

Figures 4(a)ndash4(c) show how the NSFDPC method ofDFE converges to the equilibrium point for different stepsizes Figures 4(d)ndash4(f) show how the NSFDPC method ofEE converges to the equilibrium point for different step sizes

For comparison with the well-known RK-4 method tosystem (1) using the parameter values given in Table 2 it isfound that in Figures 5(a)ndash5(f) the numerical solutionconverges for h 001 for both equilibrium points and

times105

NSFD-PC

X 562Y 1125e + 004

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

0

1

2

3

4

5

6

7

8Ex

pose

d fra

ctio

n

(e)

times105

NSFD-PC

X 5523Y 7023

Infe

cted

frac

tion

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

0

05

1

15

2

25

3

35

4

45

5

(f )

Figure 3 Time plots of susceptible exposed and infected population using the NSFDPC method with h 001

12 Journal of Mathematics

times107

NSFD-PC

Disease-free equilibrium

20 40 60 80 100 120 140 160 180 2000Time (years) h = 01

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

(a)

times107

NSFD-PC

Disease-free equilibrium

500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(b)

Disease-free equilibriumtimes107

times105

NSFD-PC

05 1 15 2 25 3 35 40Time (years) h = 1000

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(c)

X 5827Y 7022

Infe

cted

frac

tion

Endemic equilibriumtimes104

NSFD-PC

100 200 300 400 500 6000Time (years) h = 01

0

2

4

6

8

10

12

14

16

18

(d)

Endemic equilibriumtimes104

NSFD-PC

X 9600Y 7022

Infe

cted

frac

tion

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10

0

05

1

15

2

25

3

(e)

Endemic equilibriumtimes104

times105

NSFD-PC

X 861e + 005Y 7022

Infe

cted

frac

tion

1 2 3 4 5 6 7 8 9 100Time (years) h = 1000

0

05

1

15

2

25

3

(f )

Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000

Journal of Mathematics 13

Table 2 Parameters value of the measles model

Parameters N μ σ c

5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1

DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5

EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5

Disease-free equilibrium

RK4

times107

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Expo

sed

fract

ion

Disease-free equilibrium

RK4

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

times104

Infe

cted

frac

tion

Disease-free equilibrium

RK4

0

05

1

15

2

25

3

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

times107

X 5578Y 2435e + 007

Endemic equilibrium

RK4

1

15

2

25

3

35

Susc

eptib

le F

ract

ion

100 200 300 400 500 6000Time (years) h = 001

(d)

Figure 5 Continued

14 Journal of Mathematics

Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively

Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces

overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8

-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent

times105

X 5772Y 1125e + 004

Endemic equilibrium

RK4

Expo

sed

fract

ion

0

1

2

3

4

5

6

7

8

9

10

100 200 300 400 500 6000Time (years) h = 001

(e)

times105

X 7659Y 7022

RK4

Endemic equilibrium

Infe

cted

frac

tion

0

1

2

3

4

5

6

100 200 300 400 500 600 700 8000Time (years) h = 001

(f )

Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001

times1087 RK-4 scheme (disease-free equilibrium)

SusceptibleExposedInfected

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

Popu

latio

n fra

ctio

ns

(a)

RK-4 scheme (endemic equilibrium)

SusceptibleExposedInfected

times10145

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

Popu

latio

n fra

ctio

ns

(b)

Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01

Journal of Mathematics 15

Euler-PC

Disease-free equilibriumtimes107

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Euler-PC

Disease-free equilibrium

Expo

sed

fract

ion

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

Euler-PC

Disease-free equilibrium

Infe

cted

frac

tion

times104

0

05

1

15

2

25

3

35

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

Euler-PC

X 5733Y 2435e + 007

Endemic equilibriumtimes107

1

15

2

25

3

35Su

scep

tible

frac

tion

100 200 300 400 500 6000Time (years) h = 001

(d)

Euler-PC

X 5662Y 1118e + 004

Endemic equilibrium

Expo

sed

fract

ion

times105

0

2

4

6

8

10

12

100 200 300 400 500 6000Time (years) h = 001

(e)

Euler-PC

X 5427Y 7025

Endemic equilibrium

Infe

cted

frac

tion

times105

0

1

2

3

4

5

6

7

100 200 300 400 500 6000Time (years) h = 001

(f )

Figure 7 Time plots of population using the Euler-PC method with h 001

16 Journal of Mathematics

7 Conclusions

In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the

positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory

Euler-PC scheme (disease-free equilibrium)times1041

ndash3

ndash2

ndash1

0

1

2

3

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(a)

Euler-PC scheme (endemic equilibrium)times1053

ndash6

ndash4

ndash2

0

2

4

6

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(b)

Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01

NSFD-PCRK-4

times1092

Infe

cted

frac

tion

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(a)

NSFD-PCEuler-PC

times10213

Infe

cted

frac

tion

ndash4

ndash35

ndash3

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(b)

Figure 9 Comparative convergence graphs of the NSFDPC method

Journal of Mathematics 17

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6

References

[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018

[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012

[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019

[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018

[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007

[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020

[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994

[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000

[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007

[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973

[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003

[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002

[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005

[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007

[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005

[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003

[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001

[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009

[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005

[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008

[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008

[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003

[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002

[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010

[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001

18 Journal of Mathematics

A δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N

2η22η23

δ21δ22

gt 0

B 1

δ1δ2+

Nη2η3δ21δ

22lt 1

f(minus 1) 1 + A + Bgt 0

f(0) B 1

δ1δ2+

Nη2η3δ21δ

22lt

1δ1δ2

+Nη2η3δ1δ2

1 + Nη2η3

δ1δ2

1 + R0(μ + σ)(μ + c)ϕ2(h)

1 + ϕ2(h)(μ + σ)(μ + c) + ϕ(h)[(μ + σ)(μ + c)]

lt 1 sinceR0 lt 1( 1113857

rArrBlt 1

(22)

Now for

f(1) 1 minus A + B

δ21δ

22 + δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N

2η22η23 + Nη2η3 + δ1δ2

δ21δ22

gt 0

(23)

As all the conditions of the lemma hold for R0 lt 1therefore the absolute value of both eigenvalues of Jlowast is less

than one whenever R0 lt 1 -us the numerical scheme inequations (8) and (10) will converge unconditionally todisease-free equilibrium (N 0 0) for any value of time step h

whenever R0 lt 1 which is also computationally verified inFigure 2(a)

52 Endemic Equilibrium (SeEe Ie) (NR0 μN μ + σ(1 minus

1 R0) μβ(R0 minus 1)) -e value of functions and its deriv-atives at an endemic point is as

F1 Se Ee Ie( 1113857 N

R0

F2 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857

(μ + σ)

F3 Se Ee Ie( 1113857 μβ

R0 minus 1( 1113857

G1 Se Ee Ie( 1113857 N

R0

G2 Se Ee Ie( 1113857 μβ

R0 minus 1( 1113857

G3 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857

(μ + σ)

(24)

8 Journal of Mathematics

zF1 Se Ee Ie( 1113857

zS

11 + μϕ(h)R0

zF1 Se Ee Ie( 1113857

zE 0

zF1 Se Ee Ie( 1113857

zI

minus Nβϕ(h)

1 + μϕ(h)R0

zF2 Se Ee Ie( 1113857

zS

μϕ(h) R0 minus 1( 1113857

1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + σ)]

zF2 Se Ee Ie( 1113857

zE

1[1 + ϕ(h)(μ + σ)]

zF2 Se Ee Ie( 1113857

zI

minus Nβϕ(h) μϕ(l)R20 minus 2μϕ(h)R0 minus 11113960 1113961

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

zF3 Se Ee Ie( 1113857

zS

μσϕ2(h) R0 minus 1( 1113857

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

zF3 Se Ee Ie( 1113857

zE

σϕ(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

zF3 Se Ee Ie( 1113857

zI

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h) minus 11113960 1113961

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

(25)

zG1 Se Ee Ie( 1113857

zS

11 + μϕ(h)R01113858 1113859

minusNμβσϕ3(h) R0 minus 1( 1113857

R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

zG1 Se Ee Ie( 1113857

zE

minus Nβσϕ2(h)

R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

Eige

n va

lues

Jacobian at DFE

0492

0494

0496

0498

05

0502

0504

0506

0508

051

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size

(a)

098

0981

0982

0983

0984

0985

0986

0987

0988

0989Jacobian at EE

Eige

n va

lues

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size

(b)

Figure 2 Spectral radius of the Jacobian at equilibrium points

Journal of Mathematics 9

zG1 Se Ee Ie( 1113857

zI

minus Nβϕ(h)

1 + ϵminus 1 + μϕ(h)R01113960 11139611 + μϕ(h)R0

middot ϵminus 1+

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

⎡⎣ ⎤⎦

zG2 Se Ee Ie( 1113857

zS

μβϕ(h) R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890

minusNμβ2σϕ4(h)μ R0 minus 1( 1113857

2

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nμβσϕ3(h) R0 minus 1( 1113857

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

1113891

zG2 Se Ee Ie( 1113857

zE

1[1 + ϕ(h)(μ + σ)]

+Nβσϕ2(h)

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]

minusNμβ2σϕ3

(h) R0 minus 1( 1113857

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961

zG2 Se Ee Ie( 1113857

zI

minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961

+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967

βR20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nβϕ(h)

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus

N2β2σϕ3(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

zG3 Se Ee Ie( 1113857

zS

σϕ(h)

[1 + ϕ(h)(μ + c)]

μβϕ(h) R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890

minusNμβ2σϕ4(h)μ R0 minus 1( 1113857

2

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nμβσϕ3(h) R0 minus 1( 1113857

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

1113891

zG3 Se Ee Ie( 1113857

zE

σϕ(h)

[1 + ϕ(h)(μ + c)]

1[1 + ϕ(h)(μ + σ)]

minusNμβ2σϕ3(h) R0 minus 1( 1113857

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961

⎡⎢⎣

+Nβσϕ2

(h)

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]

1113891

zG3 Se Ee Ie( 1113857

zI

σϕ(h)

[1 + ϕ(h)(μ + c)]

minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961⎡⎢⎣

+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967

βR20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

⎤⎥⎦

10 Journal of Mathematics

Table 1 Comparison of NSFD scheme with Euler and RK-4 schemes

l 001 01 100 1000Euler PC Convergence Divergence Divergence DivergenceRK-4 Convergence Divergence Divergence DivergenceNSFDPC Convergence Convergence Convergence Convergence

times107

NSFD-PC

Disease-free equilibrium

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Disease-free equilibriumtimes104

NSFD-PC

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

0

05

1

15

2

25

3

35

4

45

5

Expo

sed

fract

ion

(b)

times104

NSFD-PC

Disease-free equilibrium

0

05

1

15

2

25

3

Infe

cted

frac

tion

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

times107

NSFD-PC

X 5675Y 2435e + 007

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

1

15

2

25

3

35

Susc

eptib

le fr

actio

n

(d)

Figure 3 Continued

Journal of Mathematics 11

+Nβϕ(h)

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus

N2β2σϕ3(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

(26)

-e Jacobian evaluated at an endemic point(Slowaste Elowaste Ilowaste ) (Se Ee Ie) (NR0 μNμ + σ(1 minus 1R0) μβ(R0 minus 1)) is evaluated as

JG Slowaste Elowaste Ilowaste( 1113857

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(27)

It is tedious to calculate the eigenvalues of the Jacobian ofendemic equilibrium analytically so we have plotted thelargest eigenvalue against each step size and Figure 2(b)shows that for all step sizes the spectral radius of the Ja-cobian of EE remains less than one if R0 gt 1 which impliesthat the numerical scheme in equations (8) and (10) isunconditionally convergent if R0 gt 1 for all step sizes

6 Numerical Results and Discussion

All the methods showed the convergence for small step sizesin Table 1 However for large values of step sizes only thenormalized NSFDPC converge to the correct disease-freepoint for β 01 times 10minus 5 as well as the correct endemic pointfor β 03 times 10minus 5 -is means that for the value of the basic

reproductive number (threshold parameter) R0 lt 1 thenDFE is stable ie solution to the system (S E I R) areconverging to it -is can be seen from Figures 3(a)ndash3(c) forh 001 Also for the value of the basic reproductivenumber (threshold parameter) R0 gt 1 then EE is stable iesolution to the system (S E I R) is converging to it-is canbe seen from Figures 3(d)ndash3(f ) for h 001

Figures 4(a)ndash4(c) show how the NSFDPC method ofDFE converges to the equilibrium point for different stepsizes Figures 4(d)ndash4(f) show how the NSFDPC method ofEE converges to the equilibrium point for different step sizes

For comparison with the well-known RK-4 method tosystem (1) using the parameter values given in Table 2 it isfound that in Figures 5(a)ndash5(f) the numerical solutionconverges for h 001 for both equilibrium points and

times105

NSFD-PC

X 562Y 1125e + 004

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

0

1

2

3

4

5

6

7

8Ex

pose

d fra

ctio

n

(e)

times105

NSFD-PC

X 5523Y 7023

Infe

cted

frac

tion

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

0

05

1

15

2

25

3

35

4

45

5

(f )

Figure 3 Time plots of susceptible exposed and infected population using the NSFDPC method with h 001

12 Journal of Mathematics

times107

NSFD-PC

Disease-free equilibrium

20 40 60 80 100 120 140 160 180 2000Time (years) h = 01

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

(a)

times107

NSFD-PC

Disease-free equilibrium

500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(b)

Disease-free equilibriumtimes107

times105

NSFD-PC

05 1 15 2 25 3 35 40Time (years) h = 1000

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(c)

X 5827Y 7022

Infe

cted

frac

tion

Endemic equilibriumtimes104

NSFD-PC

100 200 300 400 500 6000Time (years) h = 01

0

2

4

6

8

10

12

14

16

18

(d)

Endemic equilibriumtimes104

NSFD-PC

X 9600Y 7022

Infe

cted

frac

tion

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10

0

05

1

15

2

25

3

(e)

Endemic equilibriumtimes104

times105

NSFD-PC

X 861e + 005Y 7022

Infe

cted

frac

tion

1 2 3 4 5 6 7 8 9 100Time (years) h = 1000

0

05

1

15

2

25

3

(f )

Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000

Journal of Mathematics 13

Table 2 Parameters value of the measles model

Parameters N μ σ c

5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1

DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5

EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5

Disease-free equilibrium

RK4

times107

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Expo

sed

fract

ion

Disease-free equilibrium

RK4

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

times104

Infe

cted

frac

tion

Disease-free equilibrium

RK4

0

05

1

15

2

25

3

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

times107

X 5578Y 2435e + 007

Endemic equilibrium

RK4

1

15

2

25

3

35

Susc

eptib

le F

ract

ion

100 200 300 400 500 6000Time (years) h = 001

(d)

Figure 5 Continued

14 Journal of Mathematics

Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively

Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces

overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8

-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent

times105

X 5772Y 1125e + 004

Endemic equilibrium

RK4

Expo

sed

fract

ion

0

1

2

3

4

5

6

7

8

9

10

100 200 300 400 500 6000Time (years) h = 001

(e)

times105

X 7659Y 7022

RK4

Endemic equilibrium

Infe

cted

frac

tion

0

1

2

3

4

5

6

100 200 300 400 500 600 700 8000Time (years) h = 001

(f )

Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001

times1087 RK-4 scheme (disease-free equilibrium)

SusceptibleExposedInfected

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

Popu

latio

n fra

ctio

ns

(a)

RK-4 scheme (endemic equilibrium)

SusceptibleExposedInfected

times10145

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

Popu

latio

n fra

ctio

ns

(b)

Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01

Journal of Mathematics 15

Euler-PC

Disease-free equilibriumtimes107

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Euler-PC

Disease-free equilibrium

Expo

sed

fract

ion

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

Euler-PC

Disease-free equilibrium

Infe

cted

frac

tion

times104

0

05

1

15

2

25

3

35

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

Euler-PC

X 5733Y 2435e + 007

Endemic equilibriumtimes107

1

15

2

25

3

35Su

scep

tible

frac

tion

100 200 300 400 500 6000Time (years) h = 001

(d)

Euler-PC

X 5662Y 1118e + 004

Endemic equilibrium

Expo

sed

fract

ion

times105

0

2

4

6

8

10

12

100 200 300 400 500 6000Time (years) h = 001

(e)

Euler-PC

X 5427Y 7025

Endemic equilibrium

Infe

cted

frac

tion

times105

0

1

2

3

4

5

6

7

100 200 300 400 500 6000Time (years) h = 001

(f )

Figure 7 Time plots of population using the Euler-PC method with h 001

16 Journal of Mathematics

7 Conclusions

In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the

positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory

Euler-PC scheme (disease-free equilibrium)times1041

ndash3

ndash2

ndash1

0

1

2

3

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(a)

Euler-PC scheme (endemic equilibrium)times1053

ndash6

ndash4

ndash2

0

2

4

6

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(b)

Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01

NSFD-PCRK-4

times1092

Infe

cted

frac

tion

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(a)

NSFD-PCEuler-PC

times10213

Infe

cted

frac

tion

ndash4

ndash35

ndash3

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(b)

Figure 9 Comparative convergence graphs of the NSFDPC method

Journal of Mathematics 17

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6

References

[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018

[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012

[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019

[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018

[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007

[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020

[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994

[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000

[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007

[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973

[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003

[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002

[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005

[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007

[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005

[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003

[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001

[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009

[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005

[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008

[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008

[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003

[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002

[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010

[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001

18 Journal of Mathematics

zF1 Se Ee Ie( 1113857

zS

11 + μϕ(h)R0

zF1 Se Ee Ie( 1113857

zE 0

zF1 Se Ee Ie( 1113857

zI

minus Nβϕ(h)

1 + μϕ(h)R0

zF2 Se Ee Ie( 1113857

zS

μϕ(h) R0 minus 1( 1113857

1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + σ)]

zF2 Se Ee Ie( 1113857

zE

1[1 + ϕ(h)(μ + σ)]

zF2 Se Ee Ie( 1113857

zI

minus Nβϕ(h) μϕ(l)R20 minus 2μϕ(h)R0 minus 11113960 1113961

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

zF3 Se Ee Ie( 1113857

zS

μσϕ2(h) R0 minus 1( 1113857

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

zF3 Se Ee Ie( 1113857

zE

σϕ(h)

[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

zF3 Se Ee Ie( 1113857

zI

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h) minus 11113960 1113961

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

(25)

zG1 Se Ee Ie( 1113857

zS

11 + μϕ(h)R01113858 1113859

minusNμβσϕ3(h) R0 minus 1( 1113857

R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

zG1 Se Ee Ie( 1113857

zE

minus Nβσϕ2(h)

R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]

Eige

n va

lues

Jacobian at DFE

0492

0494

0496

0498

05

0502

0504

0506

0508

051

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size

(a)

098

0981

0982

0983

0984

0985

0986

0987

0988

0989Jacobian at EE

Eige

n va

lues

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size

(b)

Figure 2 Spectral radius of the Jacobian at equilibrium points

Journal of Mathematics 9

zG1 Se Ee Ie( 1113857

zI

minus Nβϕ(h)

1 + ϵminus 1 + μϕ(h)R01113960 11139611 + μϕ(h)R0

middot ϵminus 1+

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

⎡⎣ ⎤⎦

zG2 Se Ee Ie( 1113857

zS

μβϕ(h) R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890

minusNμβ2σϕ4(h)μ R0 minus 1( 1113857

2

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nμβσϕ3(h) R0 minus 1( 1113857

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

1113891

zG2 Se Ee Ie( 1113857

zE

1[1 + ϕ(h)(μ + σ)]

+Nβσϕ2(h)

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]

minusNμβ2σϕ3

(h) R0 minus 1( 1113857

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961

zG2 Se Ee Ie( 1113857

zI

minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961

+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967

βR20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nβϕ(h)

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus

N2β2σϕ3(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

zG3 Se Ee Ie( 1113857

zS

σϕ(h)

[1 + ϕ(h)(μ + c)]

μβϕ(h) R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890

minusNμβ2σϕ4(h)μ R0 minus 1( 1113857

2

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nμβσϕ3(h) R0 minus 1( 1113857

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

1113891

zG3 Se Ee Ie( 1113857

zE

σϕ(h)

[1 + ϕ(h)(μ + c)]

1[1 + ϕ(h)(μ + σ)]

minusNμβ2σϕ3(h) R0 minus 1( 1113857

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961

⎡⎢⎣

+Nβσϕ2

(h)

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]

1113891

zG3 Se Ee Ie( 1113857

zI

σϕ(h)

[1 + ϕ(h)(μ + c)]

minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961⎡⎢⎣

+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967

βR20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

⎤⎥⎦

10 Journal of Mathematics

Table 1 Comparison of NSFD scheme with Euler and RK-4 schemes

l 001 01 100 1000Euler PC Convergence Divergence Divergence DivergenceRK-4 Convergence Divergence Divergence DivergenceNSFDPC Convergence Convergence Convergence Convergence

times107

NSFD-PC

Disease-free equilibrium

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Disease-free equilibriumtimes104

NSFD-PC

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

0

05

1

15

2

25

3

35

4

45

5

Expo

sed

fract

ion

(b)

times104

NSFD-PC

Disease-free equilibrium

0

05

1

15

2

25

3

Infe

cted

frac

tion

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

times107

NSFD-PC

X 5675Y 2435e + 007

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

1

15

2

25

3

35

Susc

eptib

le fr

actio

n

(d)

Figure 3 Continued

Journal of Mathematics 11

+Nβϕ(h)

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus

N2β2σϕ3(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

(26)

-e Jacobian evaluated at an endemic point(Slowaste Elowaste Ilowaste ) (Se Ee Ie) (NR0 μNμ + σ(1 minus 1R0) μβ(R0 minus 1)) is evaluated as

JG Slowaste Elowaste Ilowaste( 1113857

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(27)

It is tedious to calculate the eigenvalues of the Jacobian ofendemic equilibrium analytically so we have plotted thelargest eigenvalue against each step size and Figure 2(b)shows that for all step sizes the spectral radius of the Ja-cobian of EE remains less than one if R0 gt 1 which impliesthat the numerical scheme in equations (8) and (10) isunconditionally convergent if R0 gt 1 for all step sizes

6 Numerical Results and Discussion

All the methods showed the convergence for small step sizesin Table 1 However for large values of step sizes only thenormalized NSFDPC converge to the correct disease-freepoint for β 01 times 10minus 5 as well as the correct endemic pointfor β 03 times 10minus 5 -is means that for the value of the basic

reproductive number (threshold parameter) R0 lt 1 thenDFE is stable ie solution to the system (S E I R) areconverging to it -is can be seen from Figures 3(a)ndash3(c) forh 001 Also for the value of the basic reproductivenumber (threshold parameter) R0 gt 1 then EE is stable iesolution to the system (S E I R) is converging to it-is canbe seen from Figures 3(d)ndash3(f ) for h 001

Figures 4(a)ndash4(c) show how the NSFDPC method ofDFE converges to the equilibrium point for different stepsizes Figures 4(d)ndash4(f) show how the NSFDPC method ofEE converges to the equilibrium point for different step sizes

For comparison with the well-known RK-4 method tosystem (1) using the parameter values given in Table 2 it isfound that in Figures 5(a)ndash5(f) the numerical solutionconverges for h 001 for both equilibrium points and

times105

NSFD-PC

X 562Y 1125e + 004

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

0

1

2

3

4

5

6

7

8Ex

pose

d fra

ctio

n

(e)

times105

NSFD-PC

X 5523Y 7023

Infe

cted

frac

tion

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

0

05

1

15

2

25

3

35

4

45

5

(f )

Figure 3 Time plots of susceptible exposed and infected population using the NSFDPC method with h 001

12 Journal of Mathematics

times107

NSFD-PC

Disease-free equilibrium

20 40 60 80 100 120 140 160 180 2000Time (years) h = 01

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

(a)

times107

NSFD-PC

Disease-free equilibrium

500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(b)

Disease-free equilibriumtimes107

times105

NSFD-PC

05 1 15 2 25 3 35 40Time (years) h = 1000

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(c)

X 5827Y 7022

Infe

cted

frac

tion

Endemic equilibriumtimes104

NSFD-PC

100 200 300 400 500 6000Time (years) h = 01

0

2

4

6

8

10

12

14

16

18

(d)

Endemic equilibriumtimes104

NSFD-PC

X 9600Y 7022

Infe

cted

frac

tion

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10

0

05

1

15

2

25

3

(e)

Endemic equilibriumtimes104

times105

NSFD-PC

X 861e + 005Y 7022

Infe

cted

frac

tion

1 2 3 4 5 6 7 8 9 100Time (years) h = 1000

0

05

1

15

2

25

3

(f )

Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000

Journal of Mathematics 13

Table 2 Parameters value of the measles model

Parameters N μ σ c

5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1

DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5

EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5

Disease-free equilibrium

RK4

times107

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Expo

sed

fract

ion

Disease-free equilibrium

RK4

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

times104

Infe

cted

frac

tion

Disease-free equilibrium

RK4

0

05

1

15

2

25

3

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

times107

X 5578Y 2435e + 007

Endemic equilibrium

RK4

1

15

2

25

3

35

Susc

eptib

le F

ract

ion

100 200 300 400 500 6000Time (years) h = 001

(d)

Figure 5 Continued

14 Journal of Mathematics

Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively

Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces

overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8

-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent

times105

X 5772Y 1125e + 004

Endemic equilibrium

RK4

Expo

sed

fract

ion

0

1

2

3

4

5

6

7

8

9

10

100 200 300 400 500 6000Time (years) h = 001

(e)

times105

X 7659Y 7022

RK4

Endemic equilibrium

Infe

cted

frac

tion

0

1

2

3

4

5

6

100 200 300 400 500 600 700 8000Time (years) h = 001

(f )

Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001

times1087 RK-4 scheme (disease-free equilibrium)

SusceptibleExposedInfected

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

Popu

latio

n fra

ctio

ns

(a)

RK-4 scheme (endemic equilibrium)

SusceptibleExposedInfected

times10145

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

Popu

latio

n fra

ctio

ns

(b)

Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01

Journal of Mathematics 15

Euler-PC

Disease-free equilibriumtimes107

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Euler-PC

Disease-free equilibrium

Expo

sed

fract

ion

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

Euler-PC

Disease-free equilibrium

Infe

cted

frac

tion

times104

0

05

1

15

2

25

3

35

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

Euler-PC

X 5733Y 2435e + 007

Endemic equilibriumtimes107

1

15

2

25

3

35Su

scep

tible

frac

tion

100 200 300 400 500 6000Time (years) h = 001

(d)

Euler-PC

X 5662Y 1118e + 004

Endemic equilibrium

Expo

sed

fract

ion

times105

0

2

4

6

8

10

12

100 200 300 400 500 6000Time (years) h = 001

(e)

Euler-PC

X 5427Y 7025

Endemic equilibrium

Infe

cted

frac

tion

times105

0

1

2

3

4

5

6

7

100 200 300 400 500 6000Time (years) h = 001

(f )

Figure 7 Time plots of population using the Euler-PC method with h 001

16 Journal of Mathematics

7 Conclusions

In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the

positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory

Euler-PC scheme (disease-free equilibrium)times1041

ndash3

ndash2

ndash1

0

1

2

3

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(a)

Euler-PC scheme (endemic equilibrium)times1053

ndash6

ndash4

ndash2

0

2

4

6

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(b)

Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01

NSFD-PCRK-4

times1092

Infe

cted

frac

tion

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(a)

NSFD-PCEuler-PC

times10213

Infe

cted

frac

tion

ndash4

ndash35

ndash3

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(b)

Figure 9 Comparative convergence graphs of the NSFDPC method

Journal of Mathematics 17

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6

References

[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018

[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012

[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019

[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018

[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007

[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020

[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994

[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000

[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007

[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973

[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003

[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002

[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005

[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007

[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005

[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003

[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001

[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009

[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005

[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008

[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008

[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003

[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002

[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010

[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001

18 Journal of Mathematics

zG1 Se Ee Ie( 1113857

zI

minus Nβϕ(h)

1 + ϵminus 1 + μϕ(h)R01113960 11139611 + μϕ(h)R0

middot ϵminus 1+

R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]

⎡⎣ ⎤⎦

zG2 Se Ee Ie( 1113857

zS

μβϕ(h) R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890

minusNμβ2σϕ4(h)μ R0 minus 1( 1113857

2

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nμβσϕ3(h) R0 minus 1( 1113857

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

1113891

zG2 Se Ee Ie( 1113857

zE

1[1 + ϕ(h)(μ + σ)]

+Nβσϕ2(h)

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]

minusNμβ2σϕ3

(h) R0 minus 1( 1113857

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961

zG2 Se Ee Ie( 1113857

zI

minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961

+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967

βR20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nβϕ(h)

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus

N2β2σϕ3(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

zG3 Se Ee Ie( 1113857

zS

σϕ(h)

[1 + ϕ(h)(μ + c)]

μβϕ(h) R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890

minusNμβ2σϕ4(h)μ R0 minus 1( 1113857

2

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

+Nμβσϕ3(h) R0 minus 1( 1113857

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

1113891

zG3 Se Ee Ie( 1113857

zE

σϕ(h)

[1 + ϕ(h)(μ + c)]

1[1 + ϕ(h)(μ + σ)]

minusNμβ2σϕ3(h) R0 minus 1( 1113857

βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961

⎡⎢⎣

+Nβσϕ2

(h)

R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]

1113891

zG3 Se Ee Ie( 1113857

zI

σϕ(h)

[1 + ϕ(h)(μ + c)]

minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857

β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961⎡⎢⎣

+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967

βR20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859

⎤⎥⎦

10 Journal of Mathematics

Table 1 Comparison of NSFD scheme with Euler and RK-4 schemes

l 001 01 100 1000Euler PC Convergence Divergence Divergence DivergenceRK-4 Convergence Divergence Divergence DivergenceNSFDPC Convergence Convergence Convergence Convergence

times107

NSFD-PC

Disease-free equilibrium

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Disease-free equilibriumtimes104

NSFD-PC

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

0

05

1

15

2

25

3

35

4

45

5

Expo

sed

fract

ion

(b)

times104

NSFD-PC

Disease-free equilibrium

0

05

1

15

2

25

3

Infe

cted

frac

tion

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

times107

NSFD-PC

X 5675Y 2435e + 007

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

1

15

2

25

3

35

Susc

eptib

le fr

actio

n

(d)

Figure 3 Continued

Journal of Mathematics 11

+Nβϕ(h)

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus

N2β2σϕ3(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

(26)

-e Jacobian evaluated at an endemic point(Slowaste Elowaste Ilowaste ) (Se Ee Ie) (NR0 μNμ + σ(1 minus 1R0) μβ(R0 minus 1)) is evaluated as

JG Slowaste Elowaste Ilowaste( 1113857

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(27)

It is tedious to calculate the eigenvalues of the Jacobian ofendemic equilibrium analytically so we have plotted thelargest eigenvalue against each step size and Figure 2(b)shows that for all step sizes the spectral radius of the Ja-cobian of EE remains less than one if R0 gt 1 which impliesthat the numerical scheme in equations (8) and (10) isunconditionally convergent if R0 gt 1 for all step sizes

6 Numerical Results and Discussion

All the methods showed the convergence for small step sizesin Table 1 However for large values of step sizes only thenormalized NSFDPC converge to the correct disease-freepoint for β 01 times 10minus 5 as well as the correct endemic pointfor β 03 times 10minus 5 -is means that for the value of the basic

reproductive number (threshold parameter) R0 lt 1 thenDFE is stable ie solution to the system (S E I R) areconverging to it -is can be seen from Figures 3(a)ndash3(c) forh 001 Also for the value of the basic reproductivenumber (threshold parameter) R0 gt 1 then EE is stable iesolution to the system (S E I R) is converging to it-is canbe seen from Figures 3(d)ndash3(f ) for h 001

Figures 4(a)ndash4(c) show how the NSFDPC method ofDFE converges to the equilibrium point for different stepsizes Figures 4(d)ndash4(f) show how the NSFDPC method ofEE converges to the equilibrium point for different step sizes

For comparison with the well-known RK-4 method tosystem (1) using the parameter values given in Table 2 it isfound that in Figures 5(a)ndash5(f) the numerical solutionconverges for h 001 for both equilibrium points and

times105

NSFD-PC

X 562Y 1125e + 004

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

0

1

2

3

4

5

6

7

8Ex

pose

d fra

ctio

n

(e)

times105

NSFD-PC

X 5523Y 7023

Infe

cted

frac

tion

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

0

05

1

15

2

25

3

35

4

45

5

(f )

Figure 3 Time plots of susceptible exposed and infected population using the NSFDPC method with h 001

12 Journal of Mathematics

times107

NSFD-PC

Disease-free equilibrium

20 40 60 80 100 120 140 160 180 2000Time (years) h = 01

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

(a)

times107

NSFD-PC

Disease-free equilibrium

500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(b)

Disease-free equilibriumtimes107

times105

NSFD-PC

05 1 15 2 25 3 35 40Time (years) h = 1000

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(c)

X 5827Y 7022

Infe

cted

frac

tion

Endemic equilibriumtimes104

NSFD-PC

100 200 300 400 500 6000Time (years) h = 01

0

2

4

6

8

10

12

14

16

18

(d)

Endemic equilibriumtimes104

NSFD-PC

X 9600Y 7022

Infe

cted

frac

tion

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10

0

05

1

15

2

25

3

(e)

Endemic equilibriumtimes104

times105

NSFD-PC

X 861e + 005Y 7022

Infe

cted

frac

tion

1 2 3 4 5 6 7 8 9 100Time (years) h = 1000

0

05

1

15

2

25

3

(f )

Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000

Journal of Mathematics 13

Table 2 Parameters value of the measles model

Parameters N μ σ c

5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1

DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5

EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5

Disease-free equilibrium

RK4

times107

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Expo

sed

fract

ion

Disease-free equilibrium

RK4

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

times104

Infe

cted

frac

tion

Disease-free equilibrium

RK4

0

05

1

15

2

25

3

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

times107

X 5578Y 2435e + 007

Endemic equilibrium

RK4

1

15

2

25

3

35

Susc

eptib

le F

ract

ion

100 200 300 400 500 6000Time (years) h = 001

(d)

Figure 5 Continued

14 Journal of Mathematics

Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively

Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces

overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8

-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent

times105

X 5772Y 1125e + 004

Endemic equilibrium

RK4

Expo

sed

fract

ion

0

1

2

3

4

5

6

7

8

9

10

100 200 300 400 500 6000Time (years) h = 001

(e)

times105

X 7659Y 7022

RK4

Endemic equilibrium

Infe

cted

frac

tion

0

1

2

3

4

5

6

100 200 300 400 500 600 700 8000Time (years) h = 001

(f )

Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001

times1087 RK-4 scheme (disease-free equilibrium)

SusceptibleExposedInfected

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

Popu

latio

n fra

ctio

ns

(a)

RK-4 scheme (endemic equilibrium)

SusceptibleExposedInfected

times10145

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

Popu

latio

n fra

ctio

ns

(b)

Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01

Journal of Mathematics 15

Euler-PC

Disease-free equilibriumtimes107

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Euler-PC

Disease-free equilibrium

Expo

sed

fract

ion

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

Euler-PC

Disease-free equilibrium

Infe

cted

frac

tion

times104

0

05

1

15

2

25

3

35

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

Euler-PC

X 5733Y 2435e + 007

Endemic equilibriumtimes107

1

15

2

25

3

35Su

scep

tible

frac

tion

100 200 300 400 500 6000Time (years) h = 001

(d)

Euler-PC

X 5662Y 1118e + 004

Endemic equilibrium

Expo

sed

fract

ion

times105

0

2

4

6

8

10

12

100 200 300 400 500 6000Time (years) h = 001

(e)

Euler-PC

X 5427Y 7025

Endemic equilibrium

Infe

cted

frac

tion

times105

0

1

2

3

4

5

6

7

100 200 300 400 500 6000Time (years) h = 001

(f )

Figure 7 Time plots of population using the Euler-PC method with h 001

16 Journal of Mathematics

7 Conclusions

In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the

positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory

Euler-PC scheme (disease-free equilibrium)times1041

ndash3

ndash2

ndash1

0

1

2

3

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(a)

Euler-PC scheme (endemic equilibrium)times1053

ndash6

ndash4

ndash2

0

2

4

6

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(b)

Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01

NSFD-PCRK-4

times1092

Infe

cted

frac

tion

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(a)

NSFD-PCEuler-PC

times10213

Infe

cted

frac

tion

ndash4

ndash35

ndash3

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(b)

Figure 9 Comparative convergence graphs of the NSFDPC method

Journal of Mathematics 17

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6

References

[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018

[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012

[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019

[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018

[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007

[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020

[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994

[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000

[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007

[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973

[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003

[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002

[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005

[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007

[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005

[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003

[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001

[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009

[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005

[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008

[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008

[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003

[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002

[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010

[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001

18 Journal of Mathematics

Table 1 Comparison of NSFD scheme with Euler and RK-4 schemes

l 001 01 100 1000Euler PC Convergence Divergence Divergence DivergenceRK-4 Convergence Divergence Divergence DivergenceNSFDPC Convergence Convergence Convergence Convergence

times107

NSFD-PC

Disease-free equilibrium

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Disease-free equilibriumtimes104

NSFD-PC

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

0

05

1

15

2

25

3

35

4

45

5

Expo

sed

fract

ion

(b)

times104

NSFD-PC

Disease-free equilibrium

0

05

1

15

2

25

3

Infe

cted

frac

tion

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

times107

NSFD-PC

X 5675Y 2435e + 007

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

1

15

2

25

3

35

Susc

eptib

le fr

actio

n

(d)

Figure 3 Continued

Journal of Mathematics 11

+Nβϕ(h)

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus

N2β2σϕ3(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

(26)

-e Jacobian evaluated at an endemic point(Slowaste Elowaste Ilowaste ) (Se Ee Ie) (NR0 μNμ + σ(1 minus 1R0) μβ(R0 minus 1)) is evaluated as

JG Slowaste Elowaste Ilowaste( 1113857

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(27)

It is tedious to calculate the eigenvalues of the Jacobian ofendemic equilibrium analytically so we have plotted thelargest eigenvalue against each step size and Figure 2(b)shows that for all step sizes the spectral radius of the Ja-cobian of EE remains less than one if R0 gt 1 which impliesthat the numerical scheme in equations (8) and (10) isunconditionally convergent if R0 gt 1 for all step sizes

6 Numerical Results and Discussion

All the methods showed the convergence for small step sizesin Table 1 However for large values of step sizes only thenormalized NSFDPC converge to the correct disease-freepoint for β 01 times 10minus 5 as well as the correct endemic pointfor β 03 times 10minus 5 -is means that for the value of the basic

reproductive number (threshold parameter) R0 lt 1 thenDFE is stable ie solution to the system (S E I R) areconverging to it -is can be seen from Figures 3(a)ndash3(c) forh 001 Also for the value of the basic reproductivenumber (threshold parameter) R0 gt 1 then EE is stable iesolution to the system (S E I R) is converging to it-is canbe seen from Figures 3(d)ndash3(f ) for h 001

Figures 4(a)ndash4(c) show how the NSFDPC method ofDFE converges to the equilibrium point for different stepsizes Figures 4(d)ndash4(f) show how the NSFDPC method ofEE converges to the equilibrium point for different step sizes

For comparison with the well-known RK-4 method tosystem (1) using the parameter values given in Table 2 it isfound that in Figures 5(a)ndash5(f) the numerical solutionconverges for h 001 for both equilibrium points and

times105

NSFD-PC

X 562Y 1125e + 004

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

0

1

2

3

4

5

6

7

8Ex

pose

d fra

ctio

n

(e)

times105

NSFD-PC

X 5523Y 7023

Infe

cted

frac

tion

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

0

05

1

15

2

25

3

35

4

45

5

(f )

Figure 3 Time plots of susceptible exposed and infected population using the NSFDPC method with h 001

12 Journal of Mathematics

times107

NSFD-PC

Disease-free equilibrium

20 40 60 80 100 120 140 160 180 2000Time (years) h = 01

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

(a)

times107

NSFD-PC

Disease-free equilibrium

500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(b)

Disease-free equilibriumtimes107

times105

NSFD-PC

05 1 15 2 25 3 35 40Time (years) h = 1000

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(c)

X 5827Y 7022

Infe

cted

frac

tion

Endemic equilibriumtimes104

NSFD-PC

100 200 300 400 500 6000Time (years) h = 01

0

2

4

6

8

10

12

14

16

18

(d)

Endemic equilibriumtimes104

NSFD-PC

X 9600Y 7022

Infe

cted

frac

tion

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10

0

05

1

15

2

25

3

(e)

Endemic equilibriumtimes104

times105

NSFD-PC

X 861e + 005Y 7022

Infe

cted

frac

tion

1 2 3 4 5 6 7 8 9 100Time (years) h = 1000

0

05

1

15

2

25

3

(f )

Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000

Journal of Mathematics 13

Table 2 Parameters value of the measles model

Parameters N μ σ c

5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1

DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5

EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5

Disease-free equilibrium

RK4

times107

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Expo

sed

fract

ion

Disease-free equilibrium

RK4

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

times104

Infe

cted

frac

tion

Disease-free equilibrium

RK4

0

05

1

15

2

25

3

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

times107

X 5578Y 2435e + 007

Endemic equilibrium

RK4

1

15

2

25

3

35

Susc

eptib

le F

ract

ion

100 200 300 400 500 6000Time (years) h = 001

(d)

Figure 5 Continued

14 Journal of Mathematics

Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively

Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces

overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8

-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent

times105

X 5772Y 1125e + 004

Endemic equilibrium

RK4

Expo

sed

fract

ion

0

1

2

3

4

5

6

7

8

9

10

100 200 300 400 500 6000Time (years) h = 001

(e)

times105

X 7659Y 7022

RK4

Endemic equilibrium

Infe

cted

frac

tion

0

1

2

3

4

5

6

100 200 300 400 500 600 700 8000Time (years) h = 001

(f )

Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001

times1087 RK-4 scheme (disease-free equilibrium)

SusceptibleExposedInfected

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

Popu

latio

n fra

ctio

ns

(a)

RK-4 scheme (endemic equilibrium)

SusceptibleExposedInfected

times10145

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

Popu

latio

n fra

ctio

ns

(b)

Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01

Journal of Mathematics 15

Euler-PC

Disease-free equilibriumtimes107

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Euler-PC

Disease-free equilibrium

Expo

sed

fract

ion

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

Euler-PC

Disease-free equilibrium

Infe

cted

frac

tion

times104

0

05

1

15

2

25

3

35

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

Euler-PC

X 5733Y 2435e + 007

Endemic equilibriumtimes107

1

15

2

25

3

35Su

scep

tible

frac

tion

100 200 300 400 500 6000Time (years) h = 001

(d)

Euler-PC

X 5662Y 1118e + 004

Endemic equilibrium

Expo

sed

fract

ion

times105

0

2

4

6

8

10

12

100 200 300 400 500 6000Time (years) h = 001

(e)

Euler-PC

X 5427Y 7025

Endemic equilibrium

Infe

cted

frac

tion

times105

0

1

2

3

4

5

6

7

100 200 300 400 500 6000Time (years) h = 001

(f )

Figure 7 Time plots of population using the Euler-PC method with h 001

16 Journal of Mathematics

7 Conclusions

In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the

positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory

Euler-PC scheme (disease-free equilibrium)times1041

ndash3

ndash2

ndash1

0

1

2

3

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(a)

Euler-PC scheme (endemic equilibrium)times1053

ndash6

ndash4

ndash2

0

2

4

6

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(b)

Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01

NSFD-PCRK-4

times1092

Infe

cted

frac

tion

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(a)

NSFD-PCEuler-PC

times10213

Infe

cted

frac

tion

ndash4

ndash35

ndash3

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(b)

Figure 9 Comparative convergence graphs of the NSFDPC method

Journal of Mathematics 17

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6

References

[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018

[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012

[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019

[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018

[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007

[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020

[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994

[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000

[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007

[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973

[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003

[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002

[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005

[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007

[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005

[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003

[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001

[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009

[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005

[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008

[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008

[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003

[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002

[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010

[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001

18 Journal of Mathematics

+Nβϕ(h)

R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus

N2β2σϕ3(h) μϕ(h)R

20 minus 2μϕ(h)R0 minus 11113960 1113961

R20[1 + ϕ(h)(μ + σ)]

2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859

(26)

-e Jacobian evaluated at an endemic point(Slowaste Elowaste Ilowaste ) (Se Ee Ie) (NR0 μNμ + σ(1 minus 1R0) μβ(R0 minus 1)) is evaluated as

JG Slowaste Elowaste Ilowaste( 1113857

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

zG1 Slowaste Elowaste Ilowaste( 1113857

zS

zG1 Slowaste Elowaste Ilowaste( 1113857

zE

zG1 Slowaste Elowaste Ilowaste( 1113857

zI

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(27)

It is tedious to calculate the eigenvalues of the Jacobian ofendemic equilibrium analytically so we have plotted thelargest eigenvalue against each step size and Figure 2(b)shows that for all step sizes the spectral radius of the Ja-cobian of EE remains less than one if R0 gt 1 which impliesthat the numerical scheme in equations (8) and (10) isunconditionally convergent if R0 gt 1 for all step sizes

6 Numerical Results and Discussion

All the methods showed the convergence for small step sizesin Table 1 However for large values of step sizes only thenormalized NSFDPC converge to the correct disease-freepoint for β 01 times 10minus 5 as well as the correct endemic pointfor β 03 times 10minus 5 -is means that for the value of the basic

reproductive number (threshold parameter) R0 lt 1 thenDFE is stable ie solution to the system (S E I R) areconverging to it -is can be seen from Figures 3(a)ndash3(c) forh 001 Also for the value of the basic reproductivenumber (threshold parameter) R0 gt 1 then EE is stable iesolution to the system (S E I R) is converging to it-is canbe seen from Figures 3(d)ndash3(f ) for h 001

Figures 4(a)ndash4(c) show how the NSFDPC method ofDFE converges to the equilibrium point for different stepsizes Figures 4(d)ndash4(f) show how the NSFDPC method ofEE converges to the equilibrium point for different step sizes

For comparison with the well-known RK-4 method tosystem (1) using the parameter values given in Table 2 it isfound that in Figures 5(a)ndash5(f) the numerical solutionconverges for h 001 for both equilibrium points and

times105

NSFD-PC

X 562Y 1125e + 004

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

0

1

2

3

4

5

6

7

8Ex

pose

d fra

ctio

n

(e)

times105

NSFD-PC

X 5523Y 7023

Infe

cted

frac

tion

Endemic equilibrium

100 200 300 400 500 6000Time (years) h = 001

0

05

1

15

2

25

3

35

4

45

5

(f )

Figure 3 Time plots of susceptible exposed and infected population using the NSFDPC method with h 001

12 Journal of Mathematics

times107

NSFD-PC

Disease-free equilibrium

20 40 60 80 100 120 140 160 180 2000Time (years) h = 01

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

(a)

times107

NSFD-PC

Disease-free equilibrium

500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(b)

Disease-free equilibriumtimes107

times105

NSFD-PC

05 1 15 2 25 3 35 40Time (years) h = 1000

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(c)

X 5827Y 7022

Infe

cted

frac

tion

Endemic equilibriumtimes104

NSFD-PC

100 200 300 400 500 6000Time (years) h = 01

0

2

4

6

8

10

12

14

16

18

(d)

Endemic equilibriumtimes104

NSFD-PC

X 9600Y 7022

Infe

cted

frac

tion

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10

0

05

1

15

2

25

3

(e)

Endemic equilibriumtimes104

times105

NSFD-PC

X 861e + 005Y 7022

Infe

cted

frac

tion

1 2 3 4 5 6 7 8 9 100Time (years) h = 1000

0

05

1

15

2

25

3

(f )

Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000

Journal of Mathematics 13

Table 2 Parameters value of the measles model

Parameters N μ σ c

5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1

DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5

EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5

Disease-free equilibrium

RK4

times107

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Expo

sed

fract

ion

Disease-free equilibrium

RK4

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

times104

Infe

cted

frac

tion

Disease-free equilibrium

RK4

0

05

1

15

2

25

3

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

times107

X 5578Y 2435e + 007

Endemic equilibrium

RK4

1

15

2

25

3

35

Susc

eptib

le F

ract

ion

100 200 300 400 500 6000Time (years) h = 001

(d)

Figure 5 Continued

14 Journal of Mathematics

Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively

Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces

overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8

-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent

times105

X 5772Y 1125e + 004

Endemic equilibrium

RK4

Expo

sed

fract

ion

0

1

2

3

4

5

6

7

8

9

10

100 200 300 400 500 6000Time (years) h = 001

(e)

times105

X 7659Y 7022

RK4

Endemic equilibrium

Infe

cted

frac

tion

0

1

2

3

4

5

6

100 200 300 400 500 600 700 8000Time (years) h = 001

(f )

Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001

times1087 RK-4 scheme (disease-free equilibrium)

SusceptibleExposedInfected

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

Popu

latio

n fra

ctio

ns

(a)

RK-4 scheme (endemic equilibrium)

SusceptibleExposedInfected

times10145

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

Popu

latio

n fra

ctio

ns

(b)

Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01

Journal of Mathematics 15

Euler-PC

Disease-free equilibriumtimes107

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Euler-PC

Disease-free equilibrium

Expo

sed

fract

ion

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

Euler-PC

Disease-free equilibrium

Infe

cted

frac

tion

times104

0

05

1

15

2

25

3

35

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

Euler-PC

X 5733Y 2435e + 007

Endemic equilibriumtimes107

1

15

2

25

3

35Su

scep

tible

frac

tion

100 200 300 400 500 6000Time (years) h = 001

(d)

Euler-PC

X 5662Y 1118e + 004

Endemic equilibrium

Expo

sed

fract

ion

times105

0

2

4

6

8

10

12

100 200 300 400 500 6000Time (years) h = 001

(e)

Euler-PC

X 5427Y 7025

Endemic equilibrium

Infe

cted

frac

tion

times105

0

1

2

3

4

5

6

7

100 200 300 400 500 6000Time (years) h = 001

(f )

Figure 7 Time plots of population using the Euler-PC method with h 001

16 Journal of Mathematics

7 Conclusions

In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the

positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory

Euler-PC scheme (disease-free equilibrium)times1041

ndash3

ndash2

ndash1

0

1

2

3

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(a)

Euler-PC scheme (endemic equilibrium)times1053

ndash6

ndash4

ndash2

0

2

4

6

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(b)

Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01

NSFD-PCRK-4

times1092

Infe

cted

frac

tion

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(a)

NSFD-PCEuler-PC

times10213

Infe

cted

frac

tion

ndash4

ndash35

ndash3

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(b)

Figure 9 Comparative convergence graphs of the NSFDPC method

Journal of Mathematics 17

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6

References

[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018

[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012

[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019

[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018

[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007

[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020

[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994

[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000

[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007

[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973

[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003

[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002

[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005

[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007

[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005

[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003

[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001

[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009

[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005

[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008

[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008

[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003

[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002

[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010

[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001

18 Journal of Mathematics

times107

NSFD-PC

Disease-free equilibrium

20 40 60 80 100 120 140 160 180 2000Time (years) h = 01

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

(a)

times107

NSFD-PC

Disease-free equilibrium

500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(b)

Disease-free equilibriumtimes107

times105

NSFD-PC

05 1 15 2 25 3 35 40Time (years) h = 1000

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

(c)

X 5827Y 7022

Infe

cted

frac

tion

Endemic equilibriumtimes104

NSFD-PC

100 200 300 400 500 6000Time (years) h = 01

0

2

4

6

8

10

12

14

16

18

(d)

Endemic equilibriumtimes104

NSFD-PC

X 9600Y 7022

Infe

cted

frac

tion

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10

0

05

1

15

2

25

3

(e)

Endemic equilibriumtimes104

times105

NSFD-PC

X 861e + 005Y 7022

Infe

cted

frac

tion

1 2 3 4 5 6 7 8 9 100Time (years) h = 1000

0

05

1

15

2

25

3

(f )

Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000

Journal of Mathematics 13

Table 2 Parameters value of the measles model

Parameters N μ σ c

5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1

DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5

EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5

Disease-free equilibrium

RK4

times107

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Expo

sed

fract

ion

Disease-free equilibrium

RK4

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

times104

Infe

cted

frac

tion

Disease-free equilibrium

RK4

0

05

1

15

2

25

3

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

times107

X 5578Y 2435e + 007

Endemic equilibrium

RK4

1

15

2

25

3

35

Susc

eptib

le F

ract

ion

100 200 300 400 500 6000Time (years) h = 001

(d)

Figure 5 Continued

14 Journal of Mathematics

Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively

Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces

overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8

-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent

times105

X 5772Y 1125e + 004

Endemic equilibrium

RK4

Expo

sed

fract

ion

0

1

2

3

4

5

6

7

8

9

10

100 200 300 400 500 6000Time (years) h = 001

(e)

times105

X 7659Y 7022

RK4

Endemic equilibrium

Infe

cted

frac

tion

0

1

2

3

4

5

6

100 200 300 400 500 600 700 8000Time (years) h = 001

(f )

Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001

times1087 RK-4 scheme (disease-free equilibrium)

SusceptibleExposedInfected

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

Popu

latio

n fra

ctio

ns

(a)

RK-4 scheme (endemic equilibrium)

SusceptibleExposedInfected

times10145

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

Popu

latio

n fra

ctio

ns

(b)

Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01

Journal of Mathematics 15

Euler-PC

Disease-free equilibriumtimes107

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Euler-PC

Disease-free equilibrium

Expo

sed

fract

ion

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

Euler-PC

Disease-free equilibrium

Infe

cted

frac

tion

times104

0

05

1

15

2

25

3

35

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

Euler-PC

X 5733Y 2435e + 007

Endemic equilibriumtimes107

1

15

2

25

3

35Su

scep

tible

frac

tion

100 200 300 400 500 6000Time (years) h = 001

(d)

Euler-PC

X 5662Y 1118e + 004

Endemic equilibrium

Expo

sed

fract

ion

times105

0

2

4

6

8

10

12

100 200 300 400 500 6000Time (years) h = 001

(e)

Euler-PC

X 5427Y 7025

Endemic equilibrium

Infe

cted

frac

tion

times105

0

1

2

3

4

5

6

7

100 200 300 400 500 6000Time (years) h = 001

(f )

Figure 7 Time plots of population using the Euler-PC method with h 001

16 Journal of Mathematics

7 Conclusions

In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the

positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory

Euler-PC scheme (disease-free equilibrium)times1041

ndash3

ndash2

ndash1

0

1

2

3

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(a)

Euler-PC scheme (endemic equilibrium)times1053

ndash6

ndash4

ndash2

0

2

4

6

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(b)

Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01

NSFD-PCRK-4

times1092

Infe

cted

frac

tion

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(a)

NSFD-PCEuler-PC

times10213

Infe

cted

frac

tion

ndash4

ndash35

ndash3

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(b)

Figure 9 Comparative convergence graphs of the NSFDPC method

Journal of Mathematics 17

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6

References

[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018

[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012

[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019

[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018

[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007

[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020

[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994

[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000

[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007

[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973

[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003

[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002

[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005

[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007

[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005

[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003

[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001

[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009

[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005

[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008

[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008

[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003

[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002

[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010

[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001

18 Journal of Mathematics

Table 2 Parameters value of the measles model

Parameters N μ σ c

5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1

DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5

EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5

Disease-free equilibrium

RK4

times107

1

15

2

25

3

35

4

45

5

Susc

eptib

le fr

actio

n

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Expo

sed

fract

ion

Disease-free equilibrium

RK4

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

times104

Infe

cted

frac

tion

Disease-free equilibrium

RK4

0

05

1

15

2

25

3

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

times107

X 5578Y 2435e + 007

Endemic equilibrium

RK4

1

15

2

25

3

35

Susc

eptib

le F

ract

ion

100 200 300 400 500 6000Time (years) h = 001

(d)

Figure 5 Continued

14 Journal of Mathematics

Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively

Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces

overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8

-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent

times105

X 5772Y 1125e + 004

Endemic equilibrium

RK4

Expo

sed

fract

ion

0

1

2

3

4

5

6

7

8

9

10

100 200 300 400 500 6000Time (years) h = 001

(e)

times105

X 7659Y 7022

RK4

Endemic equilibrium

Infe

cted

frac

tion

0

1

2

3

4

5

6

100 200 300 400 500 600 700 8000Time (years) h = 001

(f )

Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001

times1087 RK-4 scheme (disease-free equilibrium)

SusceptibleExposedInfected

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

Popu

latio

n fra

ctio

ns

(a)

RK-4 scheme (endemic equilibrium)

SusceptibleExposedInfected

times10145

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

Popu

latio

n fra

ctio

ns

(b)

Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01

Journal of Mathematics 15

Euler-PC

Disease-free equilibriumtimes107

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Euler-PC

Disease-free equilibrium

Expo

sed

fract

ion

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

Euler-PC

Disease-free equilibrium

Infe

cted

frac

tion

times104

0

05

1

15

2

25

3

35

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

Euler-PC

X 5733Y 2435e + 007

Endemic equilibriumtimes107

1

15

2

25

3

35Su

scep

tible

frac

tion

100 200 300 400 500 6000Time (years) h = 001

(d)

Euler-PC

X 5662Y 1118e + 004

Endemic equilibrium

Expo

sed

fract

ion

times105

0

2

4

6

8

10

12

100 200 300 400 500 6000Time (years) h = 001

(e)

Euler-PC

X 5427Y 7025

Endemic equilibrium

Infe

cted

frac

tion

times105

0

1

2

3

4

5

6

7

100 200 300 400 500 6000Time (years) h = 001

(f )

Figure 7 Time plots of population using the Euler-PC method with h 001

16 Journal of Mathematics

7 Conclusions

In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the

positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory

Euler-PC scheme (disease-free equilibrium)times1041

ndash3

ndash2

ndash1

0

1

2

3

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(a)

Euler-PC scheme (endemic equilibrium)times1053

ndash6

ndash4

ndash2

0

2

4

6

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(b)

Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01

NSFD-PCRK-4

times1092

Infe

cted

frac

tion

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(a)

NSFD-PCEuler-PC

times10213

Infe

cted

frac

tion

ndash4

ndash35

ndash3

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(b)

Figure 9 Comparative convergence graphs of the NSFDPC method

Journal of Mathematics 17

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6

References

[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018

[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012

[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019

[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018

[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007

[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020

[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994

[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000

[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007

[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973

[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003

[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002

[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005

[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007

[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005

[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003

[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001

[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009

[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005

[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008

[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008

[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003

[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002

[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010

[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001

18 Journal of Mathematics

Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively

Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces

overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8

-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent

times105

X 5772Y 1125e + 004

Endemic equilibrium

RK4

Expo

sed

fract

ion

0

1

2

3

4

5

6

7

8

9

10

100 200 300 400 500 6000Time (years) h = 001

(e)

times105

X 7659Y 7022

RK4

Endemic equilibrium

Infe

cted

frac

tion

0

1

2

3

4

5

6

100 200 300 400 500 600 700 8000Time (years) h = 001

(f )

Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001

times1087 RK-4 scheme (disease-free equilibrium)

SusceptibleExposedInfected

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

Popu

latio

n fra

ctio

ns

(a)

RK-4 scheme (endemic equilibrium)

SusceptibleExposedInfected

times10145

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

Popu

latio

n fra

ctio

ns

(b)

Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01

Journal of Mathematics 15

Euler-PC

Disease-free equilibriumtimes107

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Euler-PC

Disease-free equilibrium

Expo

sed

fract

ion

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

Euler-PC

Disease-free equilibrium

Infe

cted

frac

tion

times104

0

05

1

15

2

25

3

35

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

Euler-PC

X 5733Y 2435e + 007

Endemic equilibriumtimes107

1

15

2

25

3

35Su

scep

tible

frac

tion

100 200 300 400 500 6000Time (years) h = 001

(d)

Euler-PC

X 5662Y 1118e + 004

Endemic equilibrium

Expo

sed

fract

ion

times105

0

2

4

6

8

10

12

100 200 300 400 500 6000Time (years) h = 001

(e)

Euler-PC

X 5427Y 7025

Endemic equilibrium

Infe

cted

frac

tion

times105

0

1

2

3

4

5

6

7

100 200 300 400 500 6000Time (years) h = 001

(f )

Figure 7 Time plots of population using the Euler-PC method with h 001

16 Journal of Mathematics

7 Conclusions

In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the

positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory

Euler-PC scheme (disease-free equilibrium)times1041

ndash3

ndash2

ndash1

0

1

2

3

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(a)

Euler-PC scheme (endemic equilibrium)times1053

ndash6

ndash4

ndash2

0

2

4

6

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(b)

Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01

NSFD-PCRK-4

times1092

Infe

cted

frac

tion

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(a)

NSFD-PCEuler-PC

times10213

Infe

cted

frac

tion

ndash4

ndash35

ndash3

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(b)

Figure 9 Comparative convergence graphs of the NSFDPC method

Journal of Mathematics 17

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6

References

[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018

[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012

[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019

[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018

[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007

[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020

[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994

[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000

[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007

[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973

[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003

[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002

[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005

[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007

[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005

[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003

[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001

[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009

[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005

[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008

[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008

[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003

[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002

[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010

[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001

18 Journal of Mathematics

Euler-PC

Disease-free equilibriumtimes107

1

15

2

25

3

35

4

45

5Su

scep

tible

frac

tion

20 40 60 80 100 120 140 160 180 2000Time (years) h = 001

(a)

Euler-PC

Disease-free equilibrium

Expo

sed

fract

ion

times104

0

05

1

15

2

25

3

35

4

45

5

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(b)

Euler-PC

Disease-free equilibrium

Infe

cted

frac

tion

times104

0

05

1

15

2

25

3

35

02 04 06 08 1 12 14 16 18 20Time (years) h = 001

(c)

Euler-PC

X 5733Y 2435e + 007

Endemic equilibriumtimes107

1

15

2

25

3

35Su

scep

tible

frac

tion

100 200 300 400 500 6000Time (years) h = 001

(d)

Euler-PC

X 5662Y 1118e + 004

Endemic equilibrium

Expo

sed

fract

ion

times105

0

2

4

6

8

10

12

100 200 300 400 500 6000Time (years) h = 001

(e)

Euler-PC

X 5427Y 7025

Endemic equilibrium

Infe

cted

frac

tion

times105

0

1

2

3

4

5

6

7

100 200 300 400 500 6000Time (years) h = 001

(f )

Figure 7 Time plots of population using the Euler-PC method with h 001

16 Journal of Mathematics

7 Conclusions

In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the

positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory

Euler-PC scheme (disease-free equilibrium)times1041

ndash3

ndash2

ndash1

0

1

2

3

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(a)

Euler-PC scheme (endemic equilibrium)times1053

ndash6

ndash4

ndash2

0

2

4

6

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(b)

Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01

NSFD-PCRK-4

times1092

Infe

cted

frac

tion

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(a)

NSFD-PCEuler-PC

times10213

Infe

cted

frac

tion

ndash4

ndash35

ndash3

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(b)

Figure 9 Comparative convergence graphs of the NSFDPC method

Journal of Mathematics 17

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6

References

[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018

[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012

[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019

[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018

[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007

[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020

[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994

[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000

[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007

[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973

[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003

[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002

[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005

[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007

[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005

[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003

[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001

[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009

[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005

[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008

[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008

[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003

[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002

[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010

[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001

18 Journal of Mathematics

7 Conclusions

In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the

positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory

Euler-PC scheme (disease-free equilibrium)times1041

ndash3

ndash2

ndash1

0

1

2

3

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(a)

Euler-PC scheme (endemic equilibrium)times1053

ndash6

ndash4

ndash2

0

2

4

6

Popu

latio

n fra

ctio

ns

01 02 03 04 05 06 07 08 09 10Time (years) h = 01

SusceptibleExposedInfected

(b)

Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01

NSFD-PCRK-4

times1092

Infe

cted

frac

tion

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(a)

NSFD-PCEuler-PC

times10213

Infe

cted

frac

tion

ndash4

ndash35

ndash3

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

02 04 06 08 1 12 14 16 18 20Time (years) h = 01

(b)

Figure 9 Comparative convergence graphs of the NSFDPC method

Journal of Mathematics 17

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6

References

[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018

[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012

[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019

[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018

[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007

[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020

[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994

[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000

[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007

[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973

[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003

[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002

[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005

[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007

[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005

[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003

[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001

[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009

[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005

[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008

[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008

[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003

[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002

[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010

[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001

18 Journal of Mathematics

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare no conflicts of interest

Acknowledgments

-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6

References

[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018

[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012

[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019

[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018

[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007

[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020

[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994

[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000

[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007

[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973

[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003

[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002

[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005

[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007

[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005

[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003

[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001

[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009

[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005

[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008

[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008

[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003

[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002

[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010

[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001

18 Journal of Mathematics