Transcript
Page 1: Analysis of and numerical schemes for a mouse population model in Hantavirus epidemics

This article was downloaded by: [University of Chicago Library]On: 20 November 2014, At: 11:24Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Journal of Difference Equations and ApplicationsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/gdea20

Analysis of and numerical schemes for a mousepopulation model in Hantavirus epidemicsMingxiang Chen a & Dominic P. Clemence aa Department of Mathematics , North Carolina A&T State University , Greenboro, NC, 27411,USAPublished online: 25 Jan 2007.

To cite this article: Mingxiang Chen & Dominic P. Clemence (2006) Analysis of and numerical schemes for a mousepopulation model in Hantavirus epidemics, Journal of Difference Equations and Applications, 12:9, 887-899, DOI:10.1080/10236190600779791

To link to this article: http://dx.doi.org/10.1080/10236190600779791

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Page 2: Analysis of and numerical schemes for a mouse population model in Hantavirus epidemics

Analysis of and numerical schemes for a mousepopulation model in Hantavirus epidemics

MINGXIANG CHEN* and DOMINIC P. CLEMENCE†

Department of Mathematics, North Carolina A&T State University, Greenboro, NC 27411, USA

(Received 20 February 2006; revised 14 April 2006; in final form 27 April 2006)

This paper considers a non-linear system of ordinary differential equations, which arises in the study ofhantavirus epidemics. The system has the property that the total population obeys the logistic equation.For this system, we use linearization and the dynamical properties of the logistic equation to analyze thedynamics of the subpopulation system. In view of the usual numerical instabilities associated withstandard finite difference methods used for simulating such systems, we construct non-standard finitedifference (NSFD) schemes, which preserve the dynamic properties of the system, and may therefore beused for its simulation.

Keywords: Nonstandard finite difference schemes; Hantavirus epidemics; Finite difference methods;Equilibrium points

1. Introduction

The following basic model of mouse populations has been used in the study of Hantavirus

epidemics [1]:

dMs

dt¼ bM 2 cMs 2

MsM

K2 aMsMI ð1Þ

dMI

dt¼ 2cMI 2

MIM

Kþ aMsMI ð2Þ

where Ms $ 0 and MI $ 0 are, respectively, the populations of susceptible and infected

mice, b is the birth rate, c is the death rate, a is the infection rate, and K is related to the

carrying capacity of the environment. Note that for system (1) and (2), the conservation law

describing the dynamics of the total population, M ¼ Ms þ MI, is the logistic equation

dM

dt¼ ðb2 cÞM 2

M 2

K: ð3Þ

Therefore, the carrying capacity for the total population is M* ¼ (b 2 c)K.

Since the closed-form general solutions of the non-linear system (1) and (2), are

not possible, any useful information from the system must be deduced from dynamical

analysis and numerical simulation. The purpose of this note is two-fold: (i) to describe

the dynamical properties of the system (1) and (2), and (ii) to propose two

Journal of Difference Equations and Applications

ISSN 1023-6198 print/ISSN 1563-5120 online q 2006 Taylor & Francis

http://www.tandf.co.uk/journals

DOI: 10.1080/10236190600779791

*Corresponding author. Email: [email protected]†Email: [email protected]

Journal of Difference Equations and Applications,

Vol. 12, No. 9, September 2006, 887–899

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 1

1:24

20

Nov

embe

r 20

14

Page 3: Analysis of and numerical schemes for a mouse population model in Hantavirus epidemics

non-standard finite difference (NSFD) [9] schemes which may be used for the simulation of

(1)–(3).

It is reasonable to expect that, in order to give reliable simulation results, any finite

difference scheme used for equations (1) and (2) should preserve their dynamic properties.

Standard finite difference numerical schemes, such as the Euler and Runge–Kutta methods,

are known to yield unstable, or simply incorrect, results even for equation (3) [3,7]. In fact,

numerical simulations show that, even with small step sizes and for some positive initial data,

it takes only a few iterations for Euler and Runge–Kutta schemes to achieve negative values

and then blow up (see table 1 in section 4). However, a novel class of finite difference

schemes, commonly known as NSFD schemes [6,7], have been shown to remove such

method-dependent instabilities (see, for example, [5] and [7]).

In this note, the proposed NSFD schemes are compared to the standard Euler, RK-2, and

RK-4 methods on their performance in preserving the following properties of the system

(1)–(3) for various time-step sizes h and initial conditions:

(P1), Positivity: The scheme does not admit negative solutions with non-negative initial data,

i.e. Ms(0) $ 0, MI(0) $ 0 implies that Ms(t) $ 0, MI(t) $ 0 for all t . 0.; (P2),

Convergence: The scheme produce data that converge to correct equilibria.; (P3), Non-

periodicity: The scheme does not possess solutions which are periodic.

The NSFD schemes are shown to remain faithful to the dynamics of equations (1) and (2) for

much larger h than the standard methods, with one in particular retaining system dynamics

for all h . 0.

In the next section, we describe the details of the global dynamics of the system (1), (2) and

(3), and give the local stability analysis of its equilibria. The biological significance of the

results presented is also discussed. In section 3, we construct two NSFD schemes, which may

be used to simulate equations (1) and (2). Finally, in section 4, the results of the numerical

simulation are given and discussed.

2. Analysis of equilibria

The dynamics of equation (3) with various parameter values are well-known and have the

following features:

. If b ¼ c, then equation (3) becomes dM=dt ¼ 2M 2=K, which has one stable equilibrium

M ¼ 0.

. If b – c, then equation (3) can be written as dM/dt ¼ (b 2 c)M[1 2 (M/K(b 2 c))]. The

equation then has one stable equilibrium M ¼ 0 if b , c; and the equation has two

equilibria if b . c: M ¼ 0, which is unstable, and M ¼ K(b 2 c), which is stable.

Table 1. Comparison of RK-4, RK-2, Euler and NSFD.

Method RK-4 RK-2 Euler NSFD

Result When n ¼ 6,(Ms, MI) ! (1, 21)

When n ¼ 4,(Ms, MI) ! (1, 21)

When n ¼ 7,(Ms, MI) ! (1, 21)

(Ms, MI) ! (10, 10)

K ¼ 40, a ¼ 0.1, b ¼ 1, c ¼ 0.5, h ¼ 0.1 M0s ¼ 200, M0

I ¼ 230.

M. Chen and D. P. Clemence888

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 1

1:24

20

Nov

embe

r 20

14

Page 4: Analysis of and numerical schemes for a mouse population model in Hantavirus epidemics

Using the equilibria M ¼ 0 and M ¼ K(b 2 c) of equation (3), we observe that the system

(1) and (2) has the following non-negative equilibria:

(1) If b # c, then equations (1) and (2) has only a single equilibrium (0, 0);

(2) If b . c, and

† if K(b 2 c) 2 (b/a) # 0, then equations (1) and (2) has equilibria (0, 0) and

(K(b 2 c), 0);

† if K(b 2 c) 2 (b/a) . 0, then equations (1) and (2) has equilibria (0, 0), (K(b 2 c),

0) and ((b/a), K(b 2 c) 2 (b/a)).

2.1 Global analysis

To examine the global behavior of the system (1) and (2), we first examine its dynamical flow

on the boundary of the first quadrant.

On the positive MI-axis, Ms ¼ 0, MI . 0, so by equation (1), dMs/dt ¼ bMI . 0 and the

flow is into the first quadrant. On the positive Ms-axis, MI ¼ 0, Ms . 0, so by equations (1)

and (2), ðdMs=dtÞ ¼ ðb2 cÞMs 2 ðM2s=KÞ, and (dMI/dt) ¼ 0, so the flow stays on the Ms-axis

and the dynamics is the same as that of the logistic equation (3). By continuity and

uniqueness, the flow is thus invariant in the first quadrant.

We then use the Liapunov function V(Ms, MI) ¼ Ms þ MI $ 0 in the first quadrant. Note

that the derivative of V through equations (1) and (2) is

dV

dt¼ ðb2 cÞV 2

V 2

K:

If b # c, then (dV/dt) , 0 for all V . 0, from which we conclude that (0, 0) is globally

asymptotically stable.

For the case b . c, in the region where V . K(b 2 c), (dV/dt) , 0, and where

V . K(b 2 c), (dV/dt) , 0, so the limit set of any trajectory is on the line segment

Ms þ MI ¼ K(b 2 c) with 0 , Ms # K(b 2 c). By uniqueness of solutions, the limit set

must be an equilibrium point on this line segment.

Since for K # b/(a(b 2 c)) there is only one equilibrium (K(b 2 c), 0) on this line

segment, the limit set must be (K(b 2 c), 0) (figures 2(b), 3(b)). For the case K . b/a(b 2 c),

there are two equilibria on the line segment: (K(b 2 c), 0) and ((b/a), K(b 2 c) 2 (b/a)).

Next, consider the flow of the system on the line segment. Using Ms þ MI ¼ K(b 2 c) in

equation (2), we have

dMI

dt¼ 2bMI þ aMsMI ¼ MIðaMs 2 bÞ:

So (dMI/dt) , 0 if Ms , (b/a) and (dMI/dt) . 0 if Ms . (b/a). Thus ((b/a),K(b 2 c) 2 (b/a))

is attracting and it is a global attractor of the the system (1) and (2) (figure 1(b)).

2.2 Local analysis

To reveal more detail about the local behavior near the equilibria, we find the Jacobian of

equation (1) and (2) at each equilibrium point:

Jð0; 0Þ ¼b2 c b

0 2c

!; JðKðb2 cÞ; 0Þ ¼

2ðb2 cÞ c2 aKðb2 cÞ

0 2bþ aKðb2 cÞ

!;

Stability and NSFD schemes for a Hantavirus epidemics model 889

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 1

1:24

20

Nov

embe

r 20

14

Page 5: Analysis of and numerical schemes for a mouse population model in Hantavirus epidemics

and

Jb

a;Kðb2 cÞ2

b

a

� �¼

2aKðb2 cÞ þ b 1 2 1aK

� �2 b

aK

aKðb2 cÞ þ baK

2 2bþ c 2b 1 2 1aK

� �þ c

0@

1A:

The eigenvalues of J(0, 0) are (b 2 c) and 2c. If b , c, then (0, 0) is a stable node. If

b ¼ c, then (0, 0) is a degenerate node, and the global analysis above show that it is still

attracting in the first quadrant. If b . c, then (0, 0) is a saddle, and it is repelling in the first

quadrant.

Now consider the case b . c and the equilibrium (K(b 2 c), 0). It is easy to see that

J(K(b 2 c), 0) has eigenvalues 2 (b 2 c), which is negative, and 2b þ aK(b 2 c). If

K , b/(a(b 2 c)), then both eigenvalues are negative and (K(b 2 c), 0) is a stable node.

If K ¼ b/(a(b 2 c)), then (K(b 2 c), 0), is a degenerate node, and the global analysis shows

that it is still attracting in the first quadrant (figure 2(b)). If K . b/(a(b 2 c)) . 0, then

(K(b 2 c), 0) is a saddle with the positive Ms-axis being the stable manifold, and the

equilibrium is repelling for all other trajectories.

Finally consider the case b . c and K . b/(a(b 2 c)), and the equilibrium ((b/a),

K(b 2 c) 2 (b/a)). As J((b/a), K(b 2 c) 2 (b/a)) has trace 2aK(b 2 c) þ c , c 2 b , 0,

and determinant (b 2 c[aK(b 2 c) 2 b ] . 0, it has two eigenvalues with negative real

parts: in fact, the eigenvalues are 2 (b 2 c) , 0 and 2aK(b 2 c) þ b , 0. Therefore,

((b/a), (K(b 2 c) 2 (b/a)) is a stable node, and it is a global attractor by global analysis

(figure 1(b)).

We summarize the above analysis and discussion in the following theorem.

Theorem 1 The model given by equations (1) and (2) possesses the following stability

properties:

(i) If b # c, then the system has the unique equilibrium (0, 0), and it is globally

asymptotically stable.

(ii) If b . c and K # b/(a(b 2 c)), then the system has two equilibria: (0, 0), which is

unstable, and (K(b 2 c), 0), which is globally asymptotically stable.

Figure 1. Density–time profile for the case K . Kc: K ¼ 40, with a ¼ 0.1, b ¼ 1, c ¼ 0.5, h ¼ 0.1.

M. Chen and D. P. Clemence890

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 1

1:24

20

Nov

embe

r 20

14

Page 6: Analysis of and numerical schemes for a mouse population model in Hantavirus epidemics

(iii) If b . c and K . (b/a(b 2 c)), then the system has three equilibria: (0, 0) and

(K(b 2 c), 0), which are unstable; and ((b/a), K(b 2 c) 2 (b/a)), which is globally

asymptotically stable.

2.3 Biological significance

It is worthy to note that the mathematically natural conditions in the theorem

have an equally natural biological interpretation in terms of the Ross threshold

theorem. In particular, the epidemic is known to persist if the basic reproduction number

R0 . 1, and to die out if R0 , 1. For the model studied in this note, direct

calculation using the next generation approach [2] shows that R0 ¼ (Ka(b 2 c)/b).

This dimensionless parameter R0 has interpretation as “the number of infected

mice resulting from each infected mouse during its infected lifetime.” It is also well-

known [4] that R0 , 1 is equivalent to l* , 0, where l* is the largest eigenvalue of

the Jacobian at the disease-free equilibrium. When b . c, system (1) and (2) has a

non-trivial disease-free equilibrium (K(b 2 c), 0) with l* ¼ max{2 (b 2 c),

2b þ aK(b 2 c)}. If K . b/(a(b 2 c)), then l* . 0 and the epidemic will persist.

If K , (b/a(b 2 c)), then l* , 0 and the epidemic will die out. Thus, Kc ¼ b/(a(b 2 c))

is the threshold capacity.

In the study of populations, Theorem 1 therefore has the following interpretation.

(i) When the birth rate is not higher than the death rate, all the mice will always

eventually die.

(ii) At low carrying capacity (under or at the threshold value K ¼ b/(a(b 2 c)), any mouse

population, infected or uninfected, will eventually converge to K(b 2 c) uninfected

mice.

(iii) When the carrying capacity is above the threshold value, and if there are any initially

infected mice, the mouse subpopulations will coexist, and eventually tend to (b/a)

uninfected and K(b 2 c) 2 (b/a) infected mice; if there are initially no infected

mice, then the whole mouse population will be uninfected and eventually tend to

K(b 2 c) mice.

Figure 2. Phase portrait for the case K . Kc: K ¼ 40, with a ¼ 0.1, b ¼ 1, c ¼ 0.5, h ¼ 0.1.

Stability and NSFD schemes for a Hantavirus epidemics model 891

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 1

1:24

20

Nov

embe

r 20

14

Page 7: Analysis of and numerical schemes for a mouse population model in Hantavirus epidemics

3. Construction of non-standard numerical schemes

The purpose of this section is to construct numerical schemes for the system (1) and (2). The

proposed schemes should faithfully preserve the dynamical properties (P1)–(P3) which are

not preserved by Euler and RK methods.

Our purpose is achieved by implementing the following non-standard rules of [7], which

are also formalized in [6]:

Definition 1 A finite difference scheme is said to be non-standard if one of the following

occurs:

(i) In the discretized expression of r(y) for the equation (dy/dt) ¼ r(y), all non-linear

terms are approximated in a non-local way by a suitable function of several mesh

points.

(ii) The usual denominator Dt ¼ h of the discrete derivative is replaced by a non-negative

function f(h), with f(h) ¼ h þ O(h 2) as h ! 0þ.

The results of implementing these rules are discrete models of equations (1) and (2), which

are stable in the following sense:

Definition 2 A finite difference scheme is said to be qualitatively stable with respect to a

property P of the exact solutions of the original differential equation(s), or P-stable, if the

discrete solutions replicate P for all step sizes.

3.1 NSFD scheme I

To construct the first scheme with the desired properties, we proceed as follows. We use a

forward-difference approximation for the derivatives. We rewrite the term bM 2 cMs in

equation (1) as (b 2 c)Ms þ bMI, and approximate the linear terms Ms, MI by Mns ;M

nI .

For the non-linear terms in equation (1), we approximate the terms MsM by Mnþ1s Mn

and MsMI by Mnþ1s Mn

I ; while for equation (2), we approximate the terms MIM by Mnþ1I Mn

and MsMI by MnsM

nþ1I . Note that the resulting scheme will be non-standard in the sense of

Definition 1 (i). Moreover, since the scheme is implicit, it has superior stability over any

explicit finite difference scheme.

The resulting discrete model for equations (1) and (2) is the following:

Mnþ1s 2Mn

s

h¼ ðb2 cÞMn

s þ bMnI 2

Mnþ1s Mn

K2 aMnþ1

s MnI ð4Þ

Mnþ1I 2Mn

I

h¼ 2cMn

I 2Mnþ1

I Mn

Kþ aMnþ1

s MnI ð5Þ

here and elsewhere Mns ¼ MsðtnÞ, M

nI ¼ MIðtnÞ, and h . 0 is the time step.

M. Chen and D. P. Clemence892

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 1

1:24

20

Nov

embe

r 20

14

Page 8: Analysis of and numerical schemes for a mouse population model in Hantavirus epidemics

It is not difficult to see that addition of equations (4) and (5) yields the following consistent

discretization of equation (3), with obvious notation:

Mnþ1 2Mn

h¼ ðb2 cÞMn 2

MnMnþ1

K; ð6Þ

which is known to possess the same stability as in equation (3) [9].

The implicit equations (4)–(6) can be solved to give the following explicit formulation of

the NSFD scheme:

Mnþ1s ¼

K½Mns þ ðb2 cÞhMn

s þ bhMnI �

K þ aKhMnI þ hðMn

s þMnI Þ

; ð7Þ

Mnþ1I ¼

K{ð1 2 chÞK þ ½ð1 2 chÞð1 þ aKÞ þ abhK�hðMns þMn

I Þ�}MnI

½K þ hðMns þMn

I Þ�½K þ aKhMnI þ hðMn

s þMnI Þ�

; ð8Þ

Mnþ1 ¼K½1 þ ðb2 cÞh�Mn

K þ hMn: ð9Þ

While it is clear that the right-hand sides of equations (7) and (9) do not admit negative

iterations for Mns and M n for b . c (the only biologically relevant case) and positive initial

conditions, this is not obvious of MnI for equation (8). However, we can see easily that MI will

be non-negative if we restrict the step size to h # (1/c). In fact, the scheme (7)–(9) displays

the correct dynamics for all h # (1/c). Observe that the condition h # (1/c) implies that the

lower the death rate, the larger the value of allowable step size for this scheme. Numerous

numerical simulations of equations (7)–(9) have shown that the scheme possesses the

required positivity and convergence properties even for very large step sizes (figures 5(b),

6(a),(b)).

Figure 3. Density–time profile for the case K ¼ Kc: K ¼ 20, with a ¼ 0.1, b ¼ 1, c ¼ 0.5, h ¼ 0.1.

Stability and NSFD schemes for a Hantavirus epidemics model 893

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 1

1:24

20

Nov

embe

r 20

14

Page 9: Analysis of and numerical schemes for a mouse population model in Hantavirus epidemics

3.2 NSFD scheme II

The positivity restriction h # (1/c) imposed by equation (8) may be eliminated by using

Definition 1 (ii) as follows. Since e2ch ¼ 1 2 chþ Oðh2Þ for 0 , h p 1, or

h ¼ ðð1 2 e2chÞ=cÞ þ Oðh2Þ, we see that the function

fðhÞ ¼1 2 e2ch

cð10Þ

clearly satisfies the condition in Definition 1 (ii). Replacing h by f(h) in equations (7)–(9),

we have the following NSFD scheme for (1)–(3):

Mnþ1s ¼

K½Mns þ ðb2 cÞfðhÞMn

s þ bfðhÞMnI �

K þ aKfðhÞMnI þ fðhÞðMn

s þMnI Þ

; ð11Þ

Mnþ1I ¼

K{ð1 2 cfðhÞÞK þ ½ð1 2 cfðhÞÞð1 þ aKÞ þ abfðhÞK�fðhÞðMns þMn

I Þ�}MnI

½K þ fðhÞðMns þMn

I Þ�½K þ aKfðhÞMnI þ fðhÞðMn

s þMnI Þ�

; ð12Þ

Mnþ1 ¼K½1 þ ðb2 cÞfðhÞ�Mn

K þ fðhÞMn: ð13Þ

Note that the scheme (11)–(13) is non-standard in terms of both (i) and (ii) of Definition 1.

Moreover, since 1 2 cf(h) ¼ e2ch . 0 for all h . 0, we see that the choice of f(h) given

by equation (10) in equations (11)–(13) yields a scheme which is stable with respect to the

properties (P1)–(P3).

4. Numerical experiments

In this section, we describe the results of numerical experiments conducted to assess the

competitiveness of the schemes presented. We used the same parameter values as in [1]:

a ¼ 0.1, b ¼ 1, c ¼ 0.5. The threshold capacity is Kc ¼ b/(a(b 2 c)) ¼ 20.

All schemes produce nearly identical profiles and phase portraits for small step size h and

initial data near the global attractor.

For the case K . Kc, figures 1 and 2 give the typical density–time profile and phase

portrait.

For the case K ¼ Kc, figures 3 and 4 give the typical density–time profile and phase

portrait.

For the case K , Kc, figures 5 and 6 give the typical density–time profile and phase

portrait.

From figures 1–6, it seems that the standard methods work well for small step sizes.

However, we find that they do not produce true approximations for even small step sizes and

some relatively large initial data. For example, let K ¼ 40 (K . Kc) and h ¼ 0.1, for the

initial data M0s ¼ 200;M0

I ¼ 230, it takes only a few iterations (n) for the Euler and RK

methods to diverge, while the NSFD scheme produces data approaching the correct

equilibrium (10, 10) (table 1).

Further, while for small step sizes, the Euler and RK methods can approximate the

dynamics of equations (1) and (2) for initial data near the attractor, the simulation yields false

dynamics if the step size is increased (figures 7, 8). Figure 7 shows that for the same initial

data M0s ¼ 15 and M0

s ¼ 7:5, RK-4 method with h ¼ 0.1 gives correct time–density profile

M. Chen and D. P. Clemence894

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 1

1:24

20

Nov

embe

r 20

14

Page 10: Analysis of and numerical schemes for a mouse population model in Hantavirus epidemics

Figure 5. Density–time profile for the case K , Kc: K ¼ 10, with a ¼ 0.1, b ¼ 1, c ¼ 0.5, h ¼ 0.1.

Figure 4. Phase portrait for the case K ¼ Kc: K ¼ 20, with a ¼ 0.1, b ¼ 1, c ¼ 0.5, h ¼ 0.1.

Figure 6. Phase portrait for the case K , Kc: K ¼ 10, with a ¼ 0.1, b ¼ 1, c ¼ 0.5, h ¼ 0.1. Phase portrait.

Stability and NSFD schemes for a Hantavirus epidemics model 895

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 1

1:24

20

Nov

embe

r 20

14

Page 11: Analysis of and numerical schemes for a mouse population model in Hantavirus epidemics

that shows the convergence to attractor (10, 10), but RK-4 method with h ¼ 3.5

produces a false asymptotically 2-periodic solution. The RK-2 method displays the similar

phenomena.

Figure 8 shows that the Euler method yields irregular dynamics with step size h ¼ 1.

As a comparison, with the same step sizes h ¼ 1 and h ¼ 3.5, the NSFDI still provides

truthful approximation (figures 9 and 10).

It is interesting that, even if we let h ¼ 10, h ¼ 1000, the NSFD-I still gives the similar

global attractor, but the eigenvectors slightly changed (figures 11 and 12).

Notice that the NSFD-II scheme is obtained by replacing h in NSFD-I by

f(h) ¼ (1 2 e2ch)/c, and 0 , f(h) , (1/c) for all h . 0, thus the NSFD-II scheme

corresponds to a NSFD-I with the effective step-size less than (1/c) and therefore preserves

positivity and give the correct dynamics for all h . 0. To illustrate this, we include a time–

density profile and a phase portrait using NSFD-II with K ¼ 40 and step size h ¼ 1000

(figures 13 and 14).

Figure 7. Density–time plot for K ¼ 40 with a ¼ 0.1, b ¼ 1, c ¼ 0.5, initial data M0s ¼ 15 and M0

s ¼ 7:5,using RK4 method. The plot shows correct density–time profile with h ¼ 0.1 but displays the appearance of a falseperiod-2 solution with h ¼ 3.5.

Figure 8. Phase portrait for K ¼ 40 with a ¼ 0.1, b ¼ 1, c ¼ 0.5, h ¼ 1 using Euler methods.

M. Chen and D. P. Clemence896

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 1

1:24

20

Nov

embe

r 20

14

Page 12: Analysis of and numerical schemes for a mouse population model in Hantavirus epidemics

Figure 9. Phase portrait for K ¼ 40 with a ¼ 0.1, b ¼ 1, c ¼ 0.5 using NSFD-I scheme and the step size h ¼ 1.

Figure 10. Phase portrait for K ¼ 40 with a ¼ 0.1, b ¼ 0.1, c ¼ 0.5 using NSFD-I scheme and the step sizeh ¼ 3.5.

Figure 11. Phase portrait for K ¼ 40 with a ¼ 0.1, b ¼ 1, c ¼ 0.5 using NSFD-I scheme and the step size h ¼ 10.

Stability and NSFD schemes for a Hantavirus epidemics model 897

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 1

1:24

20

Nov

embe

r 20

14

Page 13: Analysis of and numerical schemes for a mouse population model in Hantavirus epidemics

Figure 12. Phase portrait for K ¼ 40 with a ¼ 0.1, b ¼ 1, c ¼ 0.5 using NSFD-I scheme and the step sizeh ¼ 1000.

Figure 13. Density–time profile for K ¼ 40 with a ¼ 0.1, b ¼ 1, c ¼ 0.5 using NSFD-II scheme and the step sizeh ¼ 1.

Figure 14. Phase portrait for K ¼ 40 with a ¼ 0.1, b ¼ 1, c ¼ 0.5 using NSFD-II scheme and the step sizeh ¼ 1000.

M. Chen and D. P. Clemence898

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 1

1:24

20

Nov

embe

r 20

14

Page 14: Analysis of and numerical schemes for a mouse population model in Hantavirus epidemics

5. Conclusion

The stability of a hantavirus epidemics mouse population model has been studied using

various mathematical techniques and different ways of forming discretization of the

differential equations. The biological significance of the various mathematical results was

discussed, particularly with respect to the basic reproduction number and the extinction or

persistence of epidemics, Ross threshold Theorem. In view of the difficulties that are often

encountered in employing standard finite difference methods forcalculating numerical

solutions to differential equations, we constructed and analyzed two NSFD schemes to

overcome the issues raised by these difficulties. Both schemes are O(h) convergent and were

found to be P1-stable and P2-stable as determined by the mathematical properties of these

discrete schemes and verified by numerous simulations. It should be noted that while NSFD-I

appears to have a natural ‘time scale ’and this leads to a restriction on the step-size, it, along

with NSFD-II, both possess the required stability with respect to P1 and P2. Our general

conclusion is that, with regard to the mouse hantavirus epidemic model, NSFD schemes

provide superior numerical integration methods as compared to standard procedures such as

the Euler and Runge–Kutta schemes.

Acknowledgements

We thank Professor Gregory Gibson for his help with the figures; we are also grateful to

Professors Abba Gumel and Ronald Mickens for useful discussions on the topics

investigated. We also wish to thank Professor for his advice and analysis of his NSFD

methods.

The work of D. P. Clemence was partially supported by NASA grant NAG 9-1402; he is

also grateful for the hospitality of the Africa Institute for Mathematical Sciences where some

of the work was completed under a Victor Rothschild Fellowship.

References

[1] Abramson, G. and Kenkre, V.M., 2002, Spatiotemporal patterns in the Hantavirus infection. Physical Review,66, 011912.

[2] Castillo-Chavez, C., Feng, Z. and Huang, W., 2002, In: C. Castillo-Chavez, P. Van den Driessche, D. Kirschnerand A.-A. Yakubu (Eds.) On the Computation R0 and its Role on Global Stability, Mathematical Approachesfor Emerging and Reemerging Infectious Diseases: An Introduction, IMA Volume 125 (Springer-Verlag),pp. 229–250.

[3] Chen, B. and Solis, F., 1998, Discretization of nonlinear differential equations using explicit finite ordermethods. Journal of Computation and Applied Mathematics, 90, 171–178.

[4] Diekmann, O., Hesterbeek, J.A.P. and Mets, A.J., 1990, On the definition of basic reproduction ration R0 inmodels for infections disease in heterogeneous populations, Journal of Mathematical Biology, 28, 365–382.

[5] Gumel, A.B., 2002, Removal of contrived chaos in finite difference methods. International Journal ofComputers and Mathematics, 79(9), 1033–1041.

[6] Anguelov, R. and Lubuma, J.M.-S., 2001, Contributions to the mathematics of the nonstandard finite differencemethod and applications. Numerical Methods of Partial Differential Equations, 17, 518–545.

[7] Mickens, R.E., 1994, Nonstandard Finite Difference Models of Differential Equations (Beijing: WorldScientific Publishing Co. Pte. Ltd).

Stability and NSFD schemes for a Hantavirus epidemics model 899

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 1

1:24

20

Nov

embe

r 20

14


Top Related