a study of a fractional-order cholera model · a study of a fractional-order cholera model mohammad...

Appl. Math. Inf. Sci.8, No. 5, 2195-2206 (2014) 2195

Applied Mathematics & Information SciencesAn International Journal

http://dx.doi.org/10.12785/amis/080513

A Study of a Fractional-Order Cholera Model

Mohammad Javidi1,∗ and Bashir Ahmad2,∗

1 Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran2 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

Received: 16 Aug. 2013, Revised: 13 Nov. 2013, Accepted: 14 Nov. 2013Published online: 1 Sep. 2014

Abstract: In this work, we investigate the dynamical behavior of a fractional ordercholera model. All the feasible equilibria for thesystem are obtained and the conditions for the existence of interior equilibrium are determined. Local stability analysis of the choleramodel is studied by using the fractional Routh-Hurwitz stability conditions. Our results indicate the potential of fractional-order choleramodels to cope with modern epidemics.

Keywords: Cholera model, stability, numerical simulation, fractional order, dynamical system

1 Introduction

Cholera is an acute intestinal infection caused byingestion of food or water contaminated with thebacterium Vibrio cholerae. It has a short incubationperiod, from less than one day to five days, and producesan enterotoxin that causes a copious, painless, waterydiarrhoea that can quickly lead to severe dehydration anddeath if treatment is not promptly given. Vomiting alsooccurs in most patients. Cholera is an ancient disease thatcontinues to cause epidemic and pandemic infectiondespite ongoing efforts to limit its spread ([1]–[7]).Historically, six out of the seven cholera pandemics haveswept the globe since 1816 [8,9]. Most recently, theseventh pandemic started from Indonesia in 1961, spreadinto Europe, South Pacific and Japan in the late 1970s,reached South America in 1990s, and has continued(though much diminished) to the present. The last fewyears have witnessed many cholera outbreaks indeveloping countries, including Liberia (2002), Mali(2003), Senegal and Chad (2004), West Africa (2005),Angola and Sudan (2006), India (2007), Iraq and Congo(2008), Zimbabwe (2008–2009), Vietnam (2009),Nigeria, Central Africa, Pakistan and Haiti (2010), SierraLeone (2012). Every year there are an estimated 3 to 5million cholera cases and 100 000 to 120 000 deaths dueto cholera [9,10]. Particularly, cholera represents asignificant public health burden to developing countriesand cholera continues receiving worldwide attention.Cholera is an extremely virulent disease. It affects both

children and adults and can kill within hours. About 75%of people infected with Vibrio cholerae do not developany symptoms, although the bacteria are present in theirfaeces for 7-14 days after infection and are shed back intothe environment, potentially infecting other people.

Many mathematical models have been proposed toinvestigate the complex epidemic and endemic behaviorof cholera. The earliest mathematical model was proposedby Capasso and Paveri-Fontana [11] to study a choleraepidemic occurred in the Mediterranean in 1973. Codeco[12] in 2001 extended the work in [11] and explicitlyaccounted for the role of the aquatic reservoir in choleradynamics. Using similar non-linear incidence in Codeosmodel, Hartley et al. [13] incorporated a hyper-infectivestage of V. cholerae in 2006. This model emphasizes thestage of explosive infectivity of V. cholerae, based on thelaboratory measurements that freshly shed V. choleraefrom human intestines outcompeted other V. cholerae byas much as 700-fold for the first few hours in theenvironment [1,5]. Recently, Mukandavire et al. [14]proposed a model to estimate the reproduction number forthe 2008-2009 cholera outbreak in Zimbabwe. Theirmodel includes both environment-to-human andhuman-to-human transmission pathways. In 2010, Tienand Earn [15] published a water-borne disease modelwhich also included the dual transmission pathways, withbilinear incidence rates employed for both theenvironment-to-human and human-to-human infectionroutes. Jensen et al. [16] proposed a mathematical model

∗ Corresponding author e-mail:mo [email protected], [email protected]

c© 2014 NSPNatural Sciences Publishing Cor.

http://dx.doi.org/10.12785/amis/080513

2196 M. Javidi, B. Ahmad: A Study of a Fractional-Order Cholera Model

to study how lytic bacteriophage specific for V. choleraeaffects cholera outbreaks. In [17], the authors proposed anew and unified deterministic model that incorporates ageneral incidence rate and a general formulation of thepathogen concentration to analyse the dynamics ofcholera. This work unifies many existing cholera modelsproposed by different authors. In [18], the authorsproposed global stability analysis for severaldeterministic cholera epidemic models. These models,incorporating both human population and pathogen V.cholerae concentration, constitute four-dimensionalnon-linear autonomous systems where the classicalPoincar-Bendixson theory is not applicable.

Fractional-order differentiation is regarded as thegeneralization of classical integer-order differentiation toreal or complex orders. There has been much interest indeveloping the theoretical analysis and numericalmethods for fractional differential equations as fractionalcalculus is found to be a valuable tool in various fields ofscience and engineering. Indeed, we can find numerousapplications in polymer rheology, regular variation inthermodynamics, biophysics, blood flow phenomena,aerodynamics, electro-dynamics of complex medium,viscoelasticity, Bode analysis of feedback amplifiers,capacitor theory, electrical circuits, electro-analyticalchemistry, biology, control theory, fitting of experimentaldata, etc. ([19]–[21]). For some recent work on fractionaldifferential equations and inclusions, see ([22]–[34]) andthe references therein.

Recently, several investigators have studied thequalitative properties and numerical solutions offractional-order biological models, for instance, see [35].It has been mainly due to the reason that fractional-orderequations are naturally related to systems with memorywhich exists in most biological systems. Also they areclosely related to fractals which are abundant inbiological systems. Yan and Kou [36] investigatedstability properties of fractional-order differentialequations and applied their results to analyze the stabilityof the equilibria for the model of HIV-1 infection.

In [37], the existence and uniqueness of solutions,stability of equilibria and numerical solutions for thefractional-order predator-prey model and rabies modelwere investigated. In [38], stability properties for afractional-order model of nonlocal epidemics werestudied and the results were found to be relevant tofoot-and-mouth disease, SARS and avian flu. In addition,Ding and Ye have also introduced some kinds of modelsfor HIV infection and discussed the stability of equilibriafor the corresponding systems [39,40].

The objective of this paper is to investigate afractional-order cholera model by means of an efficientnumerical method, based on an idea of transforming thegiven model to a system of ordinary differential equationsof integer order. All the feasible equilibria for the systemare discussed. The conditions ensuring the existence ofinterior equilibrium are also given. Local stabilityanalysis of the cholera model is carried out by applying

the fractional Routh–Hurwitz criterion. Our results showthe worth of fractional-order cholera models to representmodern epidemics.

The paper is organized as follows. First of all, wedescribe our model. Section 3 contains some preliminaryconcepts. The stability for equilibria of the system isdiscussed in Section 4. In Section 5, the given model isstudied numerically and the graphical results arepresented in Section 6.

2 The Model

We consider the fractional-order Codeco model involvingCaputo derivative given by [12]:

dα Sdtα = n(H −S)−a SB

K+B ,dα Idtα = a SB

K+B − rI,dα Bdtα = B(nb−mb)+ eI, nb < mb,

(1)

S(δ ) = S0, I(δ ) = I0 > 0, B(δ ) = B0 > 0,

where the symbols appearing in this model are listed inTable 1. The first equation of (1) describes the dynamicsof susceptibles in a community of constant size H.Susceptible individuals are renewed at a raten. Renewalmay occur as result of birth, immigration and/or loss ofacquired immunity (cholera apparently does not conferlife-long immunity). Susceptible people becomes infectedat a ratea B

K+B , where a is the rate of contact withuntreated water andB

K+B is the probability of a person tocatch cholera. Probability of catching cholera depends onthe concentration ofV. cholerae in the consumed water.

Table 1. Symbols used in the modelSymbol Description

State VariablesS number of susceptiblesI number of infectedB concentration of toxigenic V. cholerae in water (cells/ml)

ParametersH total human populationn Human birth and death rates (day-1)a rate of exposure to contaminated water (day-1)K concentration of V. cholerae in water that yields 50%

chance of catching cholera (cells/ml)r rate at which people recover from cholera (day-1)

nb growth rate of V. cholerae in the aquatic environment (day-1)mb loss rate of V. cholerae in the aquatic environment (day-1)e contribution of each infected person to the population of

V. cholerae in the aquatic environment (cell/ml day-1 person-1)

3 Preliminaries

The Riemann-Liouville fractional integral operator oforderα > 0, of function f ∈ L1(R+) is defined as

Iα0 f (t) =

1Γ (α)

∫ t

0(t − s)α−1 f (s)ds,

whereΓ (·) is the Euler gamma function.


Appl. Math. Inf. Sci.8, No. 5, 2195-2206 (2014) /www.naturalspublishing.com/Journals.asp 2197

The Riemann-Liouville and Caputo fractionalderivative of orderα > 0, n − 1 < α < n, n ∈ N for agiven continuous functionf are defined by

Dαt f (t) =

1Γ (n−α)

( ddt

)n ∫ t

a(t − s)n−α−1 f (s)ds,

Dαt0 f (t) =

1Γ (n−α)

∫ t

a

f (n)(s)(t − s)α+1−n ds.

In case of Caputo derivative, the functionf ∈ ACn−1. Theinitial value problem related to the above defnition is

Dα x(t) = f (t,x(t)),x(t)|t=0+ = x0,

(2)

where 0< α < 1 andDα = Dα0 .

Now, we recall some stability theorems on fractional-order systems.

Theorem 1 ([41]). The following autonomous system:

dα xdtα = Ax, x(0) = x0, (3)

with 0 < α ≤ 1, x ∈ Rn andA ∈ R

n×n, is asymptoticallystable if and only if|arg(λ )| > απ

2 is satisfied for alleigenvalues of matrixA. Also, this system is stable if andonly if |arg(λ )| ≥ απ

2 is satisfied for all eigenvalues ofmatrix A with those critical eigenvalues satisfying|arg(λ )| = απ

2 and having geometric multiplicity of one.The geometric multiplicity of an eigenvalueλ of thematrix A is the dimension of the subspace of vectorsv forwhich Av = λv.

Theorem 2 ([42]). Consider the followingcommensurate fractional-order system:

dα xdtα = f (x), x(0) = x0, (4)

with 0 < α ≤ 1 andx ∈ Rn. The equilibrium points of

system (4) are calculated by solving the equation:f (x) = 0. These points are locally asymptotically stable ifall eigenvaluesλi of the Jacobian matrixJ = ∂ f

∂x evaluatedat the equilibrium points satisfy:|arg(λi)|>

απ2 .

4 Stability of equilibrium

In this section, we analyze model (1) by finding itsequilibria and studying their stability. Steady states of themodel satisfy the following equations:

dα Sdtα = 0, dα I

dtα = 0, dα Bdtα = 0. (5)

(5) has a trivial equilibrium E0 = (H,0,0). LetR0 = aeH

rK(mb−nb) be the basic reproduction number. If

R0 > 1 then (5) has a nontrivial equilibriumE1 = (S, I,B),where

S =rK(mb−nb)+neH

e(n+a)=

H(a+R0)

R0(a+n),

B =n(aeH − rK(mb−nb)

er(n+a)=

nK(mb−nb)(R0−1)e(n+a)

,

I =(mb−nb)B

e=

nK(mb−nb)2(R0−1)e2(n+a)

.

To investigate the local behavior of system (1) about eachof the equilibrium points, the Jacobian matrix J of theequilibrium pointE = (S, I,B) is computed as

J(E) =

−n− aBK+B 0 − aSK

(K+B)2aB

K+B −r aSK(K+B)2

0 e −(nb-mb)

Now we consider the asymptotically stability of system (1)at the equilibrium pointE0. The equilibrium pointE0 isasymptotically stable ifR0 < 1.The Jacobian matrix of (1) at equilibrium pointE0 is

J(E0) =

−n 0 − aHK

0 −r aHK

0 e (nb-mb)

with the characteristic equation

P(λ ) = λ 3+b1λ 2+b2λ +b3 = 0, (6)

where

b1 = mb−nb+n+ r,

b2 = (mb−nb)(n+ r)+nr−eaH

K

= n(mb−nb+ r)+ r(mb−nb)(1−R0),

b3 =−eaH

K+nr(mb−nb)

= rn(mb−nb)(1−R0),

b1b2−b3 = n(mb−nb+n+ r)(mb−nb+ r)

+(mb−nb+ r)r(mb−nb)(1−R0).

Therefore, the eigenvalues corresponding to theequilibriumE0 are

λ1 =−n,

λ2,3 =−(mb−nb+ r)±

√

(mb−nb+ r)2+4eaHK

2.

(7)


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Clearly, if R0 < 1, thenλ1,λ2,λ3 < 0.Let D(P) denote the discriminant of a polynomialP(λ ) =λ 3+b1λ 2+b2λ +b3. Then

D(P) = 18b1b2b3+(b1b2)2−4b3b3

1−4b32−27b3

3.

The equilibrium pointE1 is asymptotically stable ifone of the following conditions holds for polynomialPandD(P):(i)D(P)> 0,b1 > 0,b3 > 0 andb1b2 > b3.

(ii)D(P)< 0,b1 ≥ 0,b2 ≥ 0,b3 > 0 andα <23.

The Jacobian matrix of (1) at equilibrium pointE1 is

J(E1) =

−n− aBK+B

0 − r(mb-nb)e

K(K+B)

aBK+B

−r r(mb-nb)e

K(K+B)

0 e −(nb-mb)

with the characteristic equation

P(λ ) = λ 3+b1λ 2+b2λ +b3 = 0, (8)

where

b1 = mb−nb+n+ r+aB

K +B,

b2 = r(n+aB

K +B)+(mb−nb)(r+n+

aB

K +B)−

rK(mb−nb)

K +B,

b3 = (r(mb−nb)−r(mb−nb)K

K +B)(n+

aB

K +B)+

r(mb−nb)KaB

K +B.

Observe thatb1 > 0. Manipulatingb2 andb3, we have

b2 = n(mb−nb)+rK(mb−nb)+ [r(n+mb−nb)+a(mb−nb+ r)]B

K +B,

b3 =C0+C1B+C2B

2

(K +B)2.

where

C0 = rn2K2+ r2nK2+K(mb−nb)r2n,

C1 = rnKa+nKr2+ rn2K +Kr(n+ r+mb−nb)(n+mb−nb)

+Ka(mb−nb+ r)(n+ r+mb−nb)+ r2n(mb−nb),

C2 = (r+mb−nb)[r(n+mb−nb)+a(mb−nb+ r)]+(a+n)[nr+a(mb−nb+ r)].

It is clear thatC0,C1,C2 > 0. Thereforeb2,b3 > 0.

5 Numerical method

Here, we shall use a numerical method introduced byAtanackovic and Stankovic [43,44] to solve thefractional-order nonlinear system (1). In [43], it wasshown that the fractional derivative of orderα(0< α ≤ 1) for a functionf (t) may be expressed as

Dα f (t) = 1Γ (2−α)

f (1)(t)tα−1 [1+∑∞

p=1Γ (p−1+α)Γ (α−1)p! ]

−[α−1tα f (t)+∑∞

p=2Γ (p−1+α)

Γ (α−1)(p−1)! (f (t)tα +

Vp( f )(t)t p−1+α )],

(9)

where

Vp( f )(t) =−(p−1)∫ t

0 τ p−2 f (τ)dτ , p = 2,3, · · · , (10)

ddt Vp( f ) =−(p−1)t p−2 f (t), p = 2,3, · · · . (11)

We approximateDα f (t) by using M terms in sumsappearing in (9) as follows

Dα f (t)≃ 1Γ (2−α)

f (1)(t)tα−1 [1+∑M

p=1Γ (p−1+α)Γ (α−1)p! ]

−[α−1tα f (t)+∑M

p=2Γ (p−1+α)

Γ (α−1)(p−1)! (f (t)tα +

Vp( f )(t)t p−1+α )].

(12)

We can rewrite (12) as

Dα f (t)≃ Ω(α, t,M) f (1)(t)+Φ(α, t,M) f (t)+∑Mp=2 A(α, t, p)Vp( f )(t)

t p−1+α , (13)

where

Ω(α, t,M) =1+∑M

p=1Γ (p−1+α)Γ (α−1)p!

Γ (2−α)tα−1 ,

R(α, t)=1−α

tαΓ (2−α),A(α, t, p)=−

Γ (p−1+α)

Γ (2−α)Γ (α −1)p!,

Φ(α, t,M) = R(α, t)+M

∑p=2

A(α, t, p)tα .

We set

Θ1(t) = S(t),ΘM+1(t) = I(t),

Θ2M+1(t) = B(t),Θp(t) =Vp(S)(t),

Θp+M(t) =Vp(I)(t),Θ2M+p(t) =Vp(B)(t),

for p = 2,3, · · · .We can rewrite system (1) in the following form

Ω(α, t,M)Θ ′1(t)+Φ(α, t,M)Θ1(t)+∑M

p=2 A(α, t, p) Θp(t)t p−1+α

= n(H −Θ1(t))−aΘ1(t)Θ2M+1(t)K+Θ2M+1(t)

,

Ω(α, t,M)Θ ′M+1(t)+Φ(α, t,M)ΘM+1(t)+∑M

p=2 A(α, t, p)ΘM+p(t)t p−1+α

= aΘ1(t)Θ2M+1(t)K+Θ2M+1(t)

− rΘM+1(t),

Ω(α, t,M)Θ ′2M+1(t)+Φ(α, t,M)Θ2M+1(t)+∑M

p=2 A(α, t, p)Θ2M+p(t)

t p−1+α

= (nb−mb)Θ2M+1+ eΘM+1(t),

(14)



where

Θp(t) =−(p−1)∫ t

0τ p−2Θ1(τ)dτ ,

ΘM+p(t) =−(p−1)∫ t

0τ p−2ΘM+1(τ)dτ ,

Θ2M+p(t) =−(p−1)∫ t

0τ p−2Θ2M+1(τ)dτ ,

p = 2,3, · · · ,M.

(15)

Finally (14) and (15) can be rewritten as

Θ ′1(t) =

1Ω(α, t,M)

(n(H −Θ1(t))−aΘ1(t)Θ2M+1(t)K +Θ2M+1(t)

)

−Φ(α, t,M)Θ1(t)−M

∑p=2

A(α, t, p)Θp(t)

t p−1+α ),

Θ ′p(t) =−(p−1)t p−2Θ1(t), p = 2,3, · · · ,M,

Θ ′M+1(t) =

1Ω(α, t,M)

(aΘ1(t)Θ2M+1(t)K +Θ2M+1(t)

− rΘM+1(t))

−Φ(α, t,M)ΘM+1(t)−M

∑p=2

A(α, t, p)ΘM+p(t)

t p−1+α ),

Θ ′M+p(t) =−(p−1)t p−2ΘM+1(t), p = 2,3, · · · ,M,

Θ ′2M+1(t) =

1Ω(α, t,M)

((nb−mb)Θ2M+1+ eΘM+1(t))

−Φ(α, t,M)Θ2M+1(t)−M

∑p=2

A(α, t, p)Θ2M+p(t)

t p−1+α ),

Θ ′2M+p(t) =−(p−1)t p−2Θ2M+1(t), p = 2,3, · · · ,M,

(16)

with the following initial conditions

Θ1(δ ) = S0,

Θp (δ ) =−p−1

2∆ t p−1S0, p = 2,3, · · · ,M,

ΘM+1(δ ) = I0,

ΘM+p(δ ) =−p−1

2∆ t p−1I0, p = 2,3, · · · ,M,

Θ2M+1(δ ) = B0,

Θ2M+p(δ ) =p−1

2∆ t p−1B0, p = 2,3, · · · ,M.

(17)

In the next section, we solve the system (16) with theinitial conditions (17) by using the well knownRunge–Kutta method of order fourth.

6 Numerical Simulation and discussion

To facilitate the interpretation of our mathematical resultsdeveloped for the model (1) so far, we proceed toinvestigate it by numerical simulations. We solve thesystem (1) numerically by using the method proposed inthe previous section. In all numerical runs, the solutionhas been approximated atδ = ∆ t = 0.01, M = 5, η = mb− nb. We illustrate ournumerical results by considering a variety of examples.

Example 1.We consider the following set of parameters:H = 1000,N = 0.001,a = 0.1,K = 1000,r = 0.4,η =0.4,e = 1.The stability of equilibriaE0 can be seen in Figs. 1-2,with the initial conditions [S0, I0,B0] =[986,10,4], [974,20,6], [965,30,5], [932,60,8] andsimulation timeT = 1200 for α = 0.95 andα = 0.99respectively. It is easy to compute thatR0 = 0.6250< 1.

Example 2.Let us choose a set of parameters:H = 100,N = 0.001,a = 0.5,K = 100,r = 0.4,η =0.02,e = 1,and the initial conditions[S0, I0,B0] == [81,15,4], [69,25,6], [60,35,5], [32,60,8]with simulation timeT = 1800 for α = 0.6 <

23 and

α = 0.5 <23. The stability of equilibria

E1 = (1.7964,0.2455,12.2754) can be observed in Figs.3–4. One can easily find thatR0 = 62.5 > 1,b1 =0.4757 > 0,b2 = 0.0243 > 0,b3 = 4.3821e − 004 >

0,b1b2−b3 = 0.0111> 0,D(P) =−2.1616e−005.Example 3.

In this case, we consider the following set of parameters:




0 10 20 30 40 50 600

5

10

15

20

25

30

35

40

45

50

I

B

550 600 650 700 750 800 850 900 950 10000

5

10

15

20

25

30

35

40

45

50

S

B

550 600 650 700 750 800 850 900 950 10000

10

20

30

40

50

60

S

I

500600

700800

9001000

0

10

20

30

40

50

600

10

20

30

40

50

SI

B

Fig. 1: Stability of the equilibriaE0. Consider the followingchoice of parametric values:H = 1000,N = 0.001,a = 0.1,K =1000,r = 0.4,η = 0.4,e = 1. Forα = 0.95 andR0 = 0.6250< 1.

0 10 20 30 40 50 600

10

20

30

40

50

60

70

I

B

800 820 840 860 880 900 920 940 960 980 10000

10

20

30

40

50

60

70

S

B

800 820 840 860 880 900 920 940 960 980 10000

10

20

30

40

50

60

S

I

800

850

900

950

1000

0

10

20

30

40

50

600

10

20

30

40

50

60

70

SI

B

Fig. 2: Stability of the equilibriaE0. Consider the followingchoice of parametric values:H = 1000,N = 0.001,a = 0.1,K =1000,r = 0.4,η = 0.4,e = 1. Forα = 0.99 andR0 = 0.6250< 1.



0 10 20 30 40 50 602

4

6

8

10

12

14

16

18

I

B

0 10 20 30 40 50 60 70 80 902

4

6

8

10

12

14

16

18

S

B

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

S

I

020

4060

80100

0

10

20

30

40

50

602

4

6

8

10

12

14

16

18

SI

B

Fig. 3: Stability of the equilibriaE1. Consider the followingchoice of parametric values:H = 100,N = 0.001,a = 0.5,K =100,r = 0.4,η = 0.02,e = 1. For α = 0.6 andR0 = 62.5.

0 10 20 30 40 50 60 70 80−40

−35

−30

−25

−20

−15

−10

−5

0

5

10

I

B

0 10 20 30 40 50 60 70 80 90−40

−35

−30

−25

−20

−15

−10

−5

0

5

10

S

B

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

S

I

020

4060

80100

0

20

40

60

80−40

−30

−20

−10

0

10

SI

B





H = 1000,N = 0.001,a = 0.7,K = 1000,r = 0.4,η =0.02,e = 1.with the initial conditions [S0, I0,B0] =[986,155,25], [705,235,60], [475,435,50], [860,60,80]and simulation timeT = 1000 forα = 0.95,0.9,0.8 andα = 0.7. The stability of equilibriaE1 = (12.8388,2.4679,123.3951) is shown in Figs. 5-8.With the given data, we find thatR0 = 87.5 > 1,b1 =0.4979 > 0,b2 = 0.0336 > 0,b3 = 6.1599e − 004 >

0,b1b2−b3 = 0.0161> 0,D(P) = 9.4334e−006.

Example 4.Consider the following values of parameters:N = 0.002,a = 0.2,K = H,r = 0.05,η = 0.13,e = 20,and initial conditions: [S0, I0,B0] = [H − 180,155,25]with simulation time: T = 250 for α = 0.9 andH = 1000,3000,5000,7000. It is easy to compute thatR0 = 615.3846> 1. The numerical solution of (1) isshown by Fig. 9.

7 Conclusions

In this paper, we have studied several features of afractional-order cholera model. These features can besummarized as follows.(i) We present criteria for theexistence of infected-free equilibria and concentration oftoxigenic V. cholerae in water equilibria.(ii) Stability ofthe equilibria for the system (1) has been discussed interms of the reproduction numberR0 = aeH

rK(mb−nb) .

Precisely, we have established the following facts: ifR0 < 1, then the equilibriumE0 of system (1) is locallyasymptotically stable for all 0< α < 1; the equilibriumE1 of system (1) is locally asymptotically stable ifR0 > 0.Also the stability analysis for the system (1) is carried outby applying the fractional Routh-Hurwitz method.(iii)The fractional-order model (1) is converted to a system ofordinary differential equations of integer-order and is thensolved numerically by using the fourth order Runge-Kuttamethod. The graphical solutions are presented for severalchoices of the parameters and conditions involved in themodel.

Acknowledgement

The authors are grateful to the anonymous referee for acareful checking of the details and for helpful commentsthat improved this paper.

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http://dx.doi.org/10.1016/j.matcom.2012.01.004.

http://dx.doi.org/10.1155/2009/378614.


M. Javidi receivedthe PhD degree inApplied Mathematicsat Iran University of Science& Technology in Tehran. Hisresearch interests are in theareas of applied mathematicsand numerical methodsfor fractional differentialequations. He has published

research articles in reputed international journals ofapplied mathematics.

Bashir Ahmad is FullProfessor of Mathematicsat King Abdulaziz University,Jeddah, Saudi Arabia.He received his Ph.D.degree from Quaid-i-AzamUniversity, Islamabad,Pakistan in 1995. Hisresearch interest includesapproximate/numerical

methods for nonlinear problems involving a variety ofdifferential equations, existence theory of fractionaldifferential equations, stability and instability propertiesof dynamical systems, impulsive systems, and scatteringtheory. He has published extensively on nonlinearboundary value problems. He was honored with ”BestResearcher of King Abdulaziz University” award in 2009.He is a reviewer of several international journals. He isAssistant Managing Editor of Bulletin of MathematicalSciences (Springer) and member of editorial boards ofseveral journals.


a study of a fractional-order cholera model · a study of a fractional-order cholera model mohammad...

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