Research ArticlePartial Differential Equations of an EpidemicModel with Spatial Diffusion

El Mehdi Lotfi1 Mehdi Maziane1 Khalid Hattaf12 and Noura Yousfi1

1 Department of Mathematics and Computer Science Faculty of Sciences Ben Mrsquosik Hassan II University PO Box 7955Sidi Othman Casablanca Morocco

2 Centre Regional des Metiers de lrsquoEducation et de la Formation 20340 Derb Ghalef Casablanca Morocco

Correspondence should be addressed to Khalid Hattaf khattafyahoofr

Received 30 August 2013 Revised 6 December 2013 Accepted 20 December 2013 Published 10 February 2014

Academic Editor William E Fitzgibbon

Copyright copy 2014 El Mehdi Lotfi et al This 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

The aimof this paper is to study the dynamics of a reaction-diffusion SIR epidemicmodel with specific nonlinear incidence rateTheglobal existence positivity and boundedness of solutions for a reaction-diffusion system with homogeneous Neumann boundaryconditions are proved The local stability of the disease-free equilibrium and endemic equilibrium is obtained via characteristicequations By means of Lyapunov functional the global stability of both equilibria is investigated More precisely our results showthat the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than or equal to unitywhich leads to the eradication of disease from populationWhen the basic reproduction number is greater than unity then disease-free equilibrium becomes unstable and the endemic equilibrium is globally asymptotically stable in this case the disease persistsin the population Numerical simulations are presented to illustrate our theoretical results

1 Introduction

In this paper we consider the following SIR epidemic modelwith a specific nonlinear incidence rate described by

119889119878

119889119905= Λ minus 120583119878 minus

120573119878119868

1 + 1205721119878 + 1205722119868 + 1205723119878119868

119889119868

119889119905=

120573119878119868

1 + 1205721119878 + 1205722119868 + 1205723119878119868

minus (120583 + 119889 + 119903) 119868

119889119877

119889119905= 119903119868 minus 120583119877

(1)

where 119878 119868 and 119877 are susceptible infectious and recoveredclasses respectively Λ is the recruitment rate of the popu-lation 120583 is the natural death rate of the population 119889 is thedeath rate due to disease 119903 is the recovery rate of the infectiveindividuals 120573 is the infection coefficient and 120573119878119868(1 + 120572

1119878 +

1205722119868 + 1205723119878119868) is the incidence rate where 120572

1 1205722 1205723

ge 0 areconstants It is very important to note that this incidence ratebecomes the bilinear incidence rate if 120572

1= 1205722= 1205723= 0 the

saturated incidence rate if 1205721

= 1205723

= 0 or 1205722

= 1205723

= 0

the Beddington-DeAngelis functional response introducedin [1 2] and used in [3] when 120572

3= 0 and Crowley-

Martin functional response presented in [4ndash6] if 1205723= 12057211205722

Moreover the function 120573119878(1 + 1205721119878 + 120572

2119868 + 1205723119878119868) satisfies

the hypotheses (1198671) (1198672) and (1198673) of general incidencerate presented by Hattaf et al in [7] From the biologicalpoint of view the transmission rate of infectious diseasesremains unknown in detail and may be different from onedisease to another In the classical epidemic models this ratewas assumed to be linear with respect to the numbers ofsusceptible and infected individualsThis assumption is basedon the law of mass action which is more appropriate forcommunicable diseases such as influenza but not for sexuallytransmitted diseases such as HIVAIDS For one reason thetransmission rate in system (1) is assumed to be nonlinearand has the form 120573119868(1 + 120572

1119878 + 1205722119868 + 1205723119878119868) that measures

the saturation effect which represents that the number ofindividual contacts reaches a certain maximum value dueto social or spatial distribution of the population For moredetails on the choice of the nonlinearity of the incidence ratewe refer the reader to the book of Capasso [8]

Hindawi Publishing CorporationInternational Journal of Partial Differential EquationsVolume 2014 Article ID 186437 6 pageshttpdxdoiorg1011552014186437

2 International Journal of Partial Differential Equations

On the other hand the spatial content of the environmenthas been ignored in the model (1) However due to thelarge mobility of people within a country or even worldwidespatially uniform models are not sufficient to give a realisticpicture of disease diffusion For this reason the spatial effectscannot be neglected in studying the spread of epidemics

Therefore we consider the following SIR epidemic modelwith specific nonlinear incidence rate and spatial diffusion

120597119878

120597119905= 119889119878Δ119878 + Λ minus 120583119878 (119909 119905)

minus120573119878 (119909 119905) 119868 (119909 119905)

1 + 1205721119878 (119909 119905) + 120572

2119868 (119909 119905) + 120572

3119878 (119909 119905) 119868 (119909 119905)

120597119868

120597119905= 119889119868Δ119868 +

120573119878 (119909 119905) 119868 (119909 119905)

1 + 1205721119878 (119909 119905) + 120572

2119868 (119909 119905) + 120572

3119878 (119909 119905) 119868 (119909 119905)

minus (120583 + 119889 + 119903) 119868 (119909 119905)

120597119877

120597119905= 119889119877Δ119877 + 119903119868 (119909 119905) minus 120583119877 (119909 119905)

(2)

where 119878(119909 119905) 119868(119909 119905) and 119877(119909 119905) represent the numbers ofsusceptible infected and removed individuals at location 119909

and time 119905 respectively The positive constants 119889119878 119889119868 and

119889119877denote the corresponding diffusion rates for these three

classes of individualsThe aim of this work is to investigate the global dynamics

of the reaction-diffusion system (2) Note that 119877 does notappear in the first two equations this allows us to study thesystem

120597119878

120597119905= 119889119878Δ119878 + Λ minus 120583119878 (119909 119905)

minus120573119878 (119909 119905) 119868 (119909 119905)

1 + 1205721119878 (119909 119905) + 120572

2119868 (119909 119905) + 120572

3119878 (119909 119905) 119868 (119909 119905)

120597119868

120597119905= 119889119868Δ119868 +

120573119878 (119909 119905) 119868 (119909 119905)

1 + 1205721119878 (119909 119905) + 120572

2119868 (119909 119905) + 120572

3119878 (119909 119905) 119868 (119909 119905)

minus (120583 + 119889 + 119903) 119868 (119909 119905)

(3)

with homogeneous Neumann boundary conditions

120597119878

120597]=

120597119868

120597]= 0 on 120597Ω times (0 +infin) (4)

and initial conditions

119878 (119909 0) = 1206011(119909) ge 0 119868 (119909 0) = 120601

2(119909) ge 0 119909 isin Ω (5)

Here Ω is a bounded domain in R119899 with smooth boundary120597Ω 120597119878120597] and 120597119868120597] are respectively the normal derivativesof 119878 and 119868 on 120597Ω

The rest of paper is organized as followsThe next sectiondeals with the global existence positivity and boundednessof solutions of problem (3)ndash(5) In Section 3 we discussthe stability analysis of equilibria In Section 4 we present

the numerical simulation to illustrate our result Finally theconclusion of our paper is in Section 5

2 Global Existence Positivity andBoundedness of Solutions

In this section we establish the global existence positivityand boundedness of solutions of problem (3)ndash(5) becausethis model describes the population Hence the populationshould remain nonnegative and bounded

Proposition 1 For any given initial data satisfying the con-dition (5) there exists a unique solution of problem (3)ndash(5)defined on [0 +infin) and this solution remains nonnegative andbounded for all 119905 ge 0

Proof System (3)ndash(5) can be written abstractly in the Banachspace 119883 = 119862(Ω) times 119862(Ω) of the form

1199061015840(119905) = 119860119906 (119905) + 119865 (119906 (119905)) 119905 gt 0

119906 (0) = 1199060isin 119883

(6)

where 119906= col(119878 119868) 1199060= col(120601

1 1206012) 119860119906(119905) = col(119889

119878Δ119878 119889119868Δ119868)

and

119865 (119906 (119905)) = (

Λ minus 120583119878 minus120573119878119868

1 + 1205721119878 + 1205722119868 + 1205723119878119868

120573119878119868

1 + 1205721119878 + 1205722119868 + 1205723119878119868

minus (120583 + 119889 + 119903) 119868

)

(7)

It is clear that 119865 is locally Lipschitz in119883 From [9] we deducethat system (6) admits a unique local solution on [0 119879max)where 119879max is the maximal existence time for solution ofsystem (6)

In addition system (3) can be written in the form120597119878

120597119905minus 119889119878Δ119878 = 119865

1(119878 119868)

120597119868

120597119905minus 119889119868Δ119868 = 119865

2(119878 119868)

(8)

It is easy to see that the functions 1198651(119878 119868) and 119865

2(119878 119868) are

continuously differentiable satisfying 1198651(0 119868) = Λ ge 0 and

1198652(119878 0) = 0 ge 0 for all 119878 119868 ge 0 Since initial data of system (3)

are nonnegative we deduce the positivity of the local solution(see the book of Smoller [10])

Now we show the boundedness of solution From (3)ndash(5)we have

120597119878

120597119905minus 119889119878Δ119878 le Λ minus 120583119878

120597119878

120597]= 0

119878 (119909 0) = 1206011(119909) le

100381710038171003817100381712060111003817100381710038171003817infin = max

119909isinΩ

1206011(119909)

(9)

By the comparison principle [11] we have 119878(119909 119905) le 1198781(119905)

International Journal of Partial Differential Equations 3

where 1198781(119905) = 120601

1(119909)119890minus120583119905

+ (Λ120583)(1 minus 119890minus120583119905

) is the solution ofthe problem

1198891198781

119889119905= Λ minus 120583119878

1

1198781(0) =

100381710038171003817100381712060111003817100381710038171003817infin

(10)

Since 1198781(119905) le maxΛ120583 120601

1infin for 119905 isin [0infin) we have that

119878 (119909 119905) le maxΛ

12058310038171003817100381710038171206011

1003817100381710038171003817infin forall (119909 119905) isin Ω times [0 119879max)

(11)FromTheorem2 given byAlikakos in [12] to establish the

119871infin uniform boundedness of 119868(119909 119905) it is sufficient to show the

1198711 uniform boundedness of 119868(119909 119905)Since 120597119878120597] = 120597119868120597] = 0 and (120597120597119905)(119878+119868)minusΔ(119889

119878119878+119889119868119868) le

Λ minus 120583(119878 + 119868) we get120597

120597119905(intΩ

(119878 + 119868) 119889119909) le mes (Ω)Λ minus 120583(intΩ

(119878 + 119868) 119889119909)

(12)Hence

intΩ

(119878 + 119868) 119889119909 le mes (Ω)maxΛ

12058310038171003817100381710038171206011 + 120601

2

1003817100381710038171003817infin (13)

which implies that sup119905ge0

intΩ119868(119909 119905)119889119909 le 119870 =

mes(Ω)maxΛ120583 1206011+ 1206012infin Using [12 Theorem 31]

we deduce that there exists a positive constant 119870lowast that

depends on 119870 and on 1206011+ 1206012infin

such that

sup119905ge0

119868 (sdot 119905)infin le 119870lowast (14)

From the above we have proved that 119878(119909 119905) and 119868(119909 119905) are119871infin bounded on Ω times [0 119879max) Therefore it follows from the

standard theory for semilinear parabolic systems (see [13])that 119879max = +infin This completes the proof of the proposition

3 Qualitative Analysis of the Spatial Model

Using the results presented by Hattaf et al in [7] it is easyto get that the basic reproduction number of disease in theabsence of spatial dependence is given by

1198770=

120573Λ

(120583 + 1205721Λ) (120583 + 119889 + 119903)

(15)

which describes the average number of secondary infectionsproduced by a single infectious individual during the entireinfectious period

It is not hard to show that the system (3) is always adisease-free equilibrium of the form 119864

119891(Λ120583 0) Further if

1198770

gt 1 the system (3) has an endemic stationary state119864lowast(119878lowast 119868lowast) where

119878lowast=

2 (119886 + 1205722Λ)

120573 minus 1205721119886 + 1205722120583 minus 1205723Λ + radic120575

119868lowast=

Λ minus 120583119878lowast

119886

(16)

with 119886 = 120583 + 119889+119903 and 120575 = (120573minus1205721119886+1205722120583minus1205723Λ)2+41205723120583(119886+

1205722Λ)

The objective of this section is to discuss the local andglobal stability of the equilibria

31 Local Stability of the Equilibria First we linearize thedynamical system (3) around arbitrary spatially homoge-neous fixed point119864(119878 119868) for small space- and time-dependentfluctuations and expand them in Fourier space For this let

119878 ( 119905) sim 119878119890120582119905119890119894sdot 119909

119868 ( 119905) sim 119868119890120582119905119890119894sdot 119909

(17)

where = (119909 119910) and sdot = ⟨ ⟩ = 1198962 and 120582 are the

wavenumber vector and frequency respectivelyThen we canobtain the corresponding characteristic equation as follows

det (119869 minus 1198962119863 minus 120582119868

2) = 0 (18)

where 1198682is the identity matrix 119863 = diag(119889

119878 119889119868) is the dif-

fusion matrix and 119869 is the Jacobian matrix of (3) withoutdiffusion (119889

119878= 119889119868= 0) at 119864 which is given by

119869 =(

(

minus120583 minus

120573119868 (1 + 1205722119868)

(1 + 1205721119878 + 1205722119868 + 1205723119878 119868)2minus

120573119878 (1 + 1205721119878)

(1 + 1205721119878 + 1205722119868 + 1205723119878 119868)2

120573119868 (1 + 1205722119868)

(1 + 1205721119878 + 1205722119868 + 1205723119878 119868)2

120573119878 (1 + 1205721119878)

(1 + 1205721119878 + 1205722119868 + 1205723119878 119868)2minus 119886

)

)

(19)The characterization of the local stability of disease-freeequilibrium 119864

119891is given by the following result

Theorem 2 The disease-free equilibrium 119864119891is locally asymp-

totically stable if 1198770lt 1 and it is unstable if 119877

0gt 1

Proof Evaluating (18) at 119864119891 we have

(120583 + 1198962119889119878+ 120582) (119886 (119877

0minus 1) minus 119896

2119889119868minus 120582) = 0 (20)

Clearly the roots of (20) are 1205821(119896) = minus(120583 + 119896

2119889119878) lt 0 and

1205822(119896) = 119886(119877

0minus1) minus 119896

2119889119868 Note that 120582

2(119896) is negative if 119877

0lt 1

for all 119896 Hence 119864119891is locally asymptotically stable if 119877

0lt 1 If

1198770gt 1 120582

2(0) is positive So 119864

119891is unstable

Next we focus on the local stability of the endemicequilibrium 119864

lowast

Theorem 3 The endemic equilibrium 119864lowast is locally asymptoti-

cally stable if 1198770gt 1

Proof Evaluating (18) at 119864lowast(119878lowast 119868lowast) we have

1205822+ 1198861(119896) 120582 + 119886

2(119896) = 0 (21)

where1198861(119896) = 120583 + 119896

2(119889119878+ 119889119868)

+ 119886(119868lowast(1 + 120572

2119868lowast)

(1 + 1205721119878lowast + 120572

2119868lowast + 120572

3119878lowast119868lowast) 119878lowast

+1205722119868lowast+ 1205723119878lowast119868lowast

1 + 1205721119878lowast + 120572

2119868lowast + 120572

3119878lowast119868lowast

)

4 International Journal of Partial Differential Equations

1198862(119896) = (120583 +

120573119868lowast(1 + 120572

2119868lowast)

(1 + 1205721119878lowast + 120572

2119868lowast + 120572

3119878lowast119868lowast)

2+ 1198962119889119878)

times (1198962119889119868+

120573119878lowast(1205722119868lowast+ 1205723119878lowast119868lowast)

(1 + 1205721119878lowast + 120572

2119868lowast + 120572

3119878lowast119868lowast)

2)

+1205732119878lowast119868lowast(1 + 120572

2119868lowast) (1 + 120572

1119878lowast)

(1 + 1205721119878lowast + 120572

2119868lowast + 120572

3119878lowast119868lowast)

4

(22)

We have 1198861gt 0 and 119886

2gt 0 then 119864

lowast is locally asymptoticallystable

32 Global Stability of the Equilibria The purpose of thissubsection is to determine the global stability for reaction-diffusion equations (3)ndash(5) by constructing Lyapunov func-tionals These Lyapunov functionals are obtained from thosefor ordinary differential equations (1) by applying themethodof Hattaf and Yousfi presented in [14]

The system (1) is particular case of themodel proposed byHattaf et al [7] with 119891(119878 119868) = 120573119878(1 + 120572

1119878 + 1205722119868 + 1205723119878119868) To

study the global stability of 119864119891for (1) the authors Hattaf et al

[7] proposed the following Lyapunov functional

1198811= 119878 minus 119878

0minus int

119878

1198780

119891 (1198780 0)

119891 (119883 0)119889119883 + 119868 (23)

where 1198780= Λ120583

From [14] we construct the Lyapunov functional forsystem (3)ndash(5) at 119864

119891as follows

1198821= intΩ

1198811(119878 (119909 119905) 119868 (119909 119905)) 119889119909 (24)

Calculating the time derivative of 1198821along the solution of

system (3)ndash(5) we have

1198891198821

119889119905= intΩ

minus120583(119878 minus 119878

0)2

119878 (1 + 12057211198780)

+119886119868 (1 + 1205721119878

1 + 1205721119878 + 1205722119868 + 1205723119878119868

1198770minus 1)119889119909

minus1198891198781198780

1 + 12057211198780

intΩ

|nabla119878|2

1198782119889119909

le intΩ

minus120583(119878 minus 119878

0)2

119878 (1 + 12057211198780)+ 119886119868 (119877

0minus 1)119889119909

minus1198891198781198780

1 + 12057211198780

intΩ

|nabla119878|2

1198782119889119909

(25)

Since 1198770

le 1 we have 1198891198821119889119905 le 0 Thus the disease-free

equilibrium 119864119891is stable and 119889119882

1119889119905 = 0 if and only if 119878 = 119878

0

and 119868(1198770minus 1) = 0 We discuss two cases as follows

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

Space

Susceptible

Infected

Figure 1The initial distributions of the numbers of susceptibles andinfectious individuals for Figures 2 and 3

(i) If 1198770lt 1 then 119868 = 0

(ii) If 1198770= 1 from 119878 = 119878

0and the first equation of (3) we

have

120597119878

120597119905=

1205971198780

120597119905= 119889119878Δ1198780+ Λ minus 120583119878

0minus 119891 (119878

0 119868) 119868 = 0 (26)

Then 119891(1198780 119868)119868 = (120573119878

0119868)(1 +120572

11198780+1205722119868 +12057231198780119868) = 0

Since 120573 gt 0 and 1198780gt 0 then 119868 = 0

By the above discussion we deduce that the largest compactinvariant set in Γ = (119878 119868) | (119889119882

1)119889119905 = 0 is just the single-

ton 119864119891 From LaSalle invariance principle [15] we conclude

that 119864119891is globally asymptotically stable

Using same technique we construct a Lyapunov func-tional 119882

2for system (3)ndash(5) at 119864

lowast from the Lyapunovfunctional 119881

2defined by Hattaf et al in [7] It is easy to

show that 1198812verifies the condition (15) given in [14] Hence

it follows from [14 Proposition 21] that 1198822is a Lyapunov

functional for the reaction-diffusion system (3)ndash(5) at 119864lowast

when1198770gt 1We summarize the above in the following result

Theorem 4 (i) If 1198770

le 1 the disease-free equilibrium 119864119891

of (3)ndash(5) is globally asymptotically stable for all diffusioncoefficients

(ii) If 1198770

gt 1 the endemic equilibrium 119864lowast of (3)ndash(5) is

globally asymptotically stable for all diffusion coefficients

4 Numerical Simulations

In this section we present the numerical simulations toillustrate our theoretical results To simplify we considersystem (3) under Neumann boundary conditions

120597119878

120597]=

120597119868

120597]= 0 119905 gt 0 119909 = 0 1 (27)

International Journal of Partial Differential Equations 5

002

0406

081

020

4060

80100

0

1

2

3

4

5

Num

ber o

f sus

cept

ible

indi

vidu

als

0 204

0608

2040

6080

Space xTime

t

0

005

01

015

02

025

Num

ber o

f inf

ecte

d in

divi

dual

s

002

0406

081

020

4060

80100

Space xTime

t

Figure 2The temporal solution found by numerical of problem (3) with the Neumann boundary conditions (27) and initial conditions (28)

Figure 3The temporal solution found by numerical of problem (3) with the Neumann boundary conditions (27) and initial conditions (28)

and initial conditions

119878 (119909 0) = 11119909 0 le 119909 lt 05

11 (1 minus 119909) 05 le 119909 le 1

119868 (119909 0) = 05119909 0 le 119909 lt 05

05 (1 minus 119909) 05 le 119909 le 1

(28)

In Figure 1 we show that themaximum value of the num-bers of susceptibles and infectious individuals is concentratedat themiddle of the interval [0 1] and these numbers decreaselinearly to zero at the boundaries 119909 = 0 and 119909 = 1

Now we choose the following data set of system (3) 119889119878=

01 119889119868

= 05 Λ = 05 120573 = 02 120583 = 01 1205721

= 011205722= 002 120572

3= 003 119889 = 01 and 119903 = 05 By calculation we

have 1198770= 09524 In this case system (3) has a disease-free

equilibrium 119864119891(5 0) Hence by Theorem 4(i) 119864

119891is globally

asymptotically stable Numerical simulation illustrates ourresult (see Figure 2)

In Figure 3 we choose 120573 = 06 and do not change theother parameter values By calculation we have 119877

0= 28571

which satisfyTheorem4(ii) then the disease-free equilibriumis still present and the system (2) has a unique endemic

equilibrium 119864lowast(13625 05196) Therefore by Theorem 2

and Theorem 4(ii) 119864119891

is unstable while 119864lowast is globally

asymptotically stable Numerical simulation illustrates ourresult (see Figure 3)

5 Conclusion

In this paper we investigated the dynamics of a reaction-diffusion epidemic model with specific nonlinear incidencerateThis specific nonlinear incidence rate includes the tradi-tional bilinear incidence rate the saturated incidence rate theBeddington-DeAngelis functional response and Crowley-Martin functional response The global dynamics of themodel are completely determined by the basic reproductionnumber 119877

0 We proved that the disease-free equilibrium is

globally asymptotically stable if 1198770

le 1 which leads tothe eradication of disease from population When 119877

0gt 1

then disease-free equilibrium becomes unstable and a uniqueendemic equilibrium exists and is globally asymptotically sta-ble which means that the disease persists in the population

From our theoretical and numerical results we concludethat the spatial diffusion has no effect on the stability behavior

6 International Journal of Partial Differential Equations

of equilibria in the case of Neumann conditions and spatiallyconstant coefficients

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the editor and the anony-mous referees for their valuable remarks and commentswhich have led to the improvement of the quality of theirstudy

References

[1] J R Beddington ldquoMutual interference between parasites orpredators and its effect on searching efficiencyrdquo Journal ofAnimal Ecology pp 331ndash341 1975

[2] D L DeAngelis R A Goldsten and R Neill ldquoA model fortrophic interactionrdquo Ecology vol 56 pp 881ndash892 1975

[3] R S Cantrell and C Cosner ldquoOn the Dynamics of Predator-Prey Models with the Beddington-Deangelis FunctionalResponserdquo Journal of Mathematical Analysis and Applicationsvol 257 no 1 pp 206ndash222 2001

[4] P H Crowley and E K Martin ldquoFunctional responses andinterferencewithin and between year classes of a dragonfly pop-ulationrdquo Journal of the North American Benthological Societyvol 8 pp 211ndash221 1989

[5] X Zhou and J Cui ldquoGlobal stability of the viral dynamics withcrowley-martin functional responserdquo Bulletin of the KoreanMathematical Society vol 48 no 3 pp 555ndash574 2011

[6] X Q Liu S M Zhong B D Tian and F X Zheng ldquoAsymptoticproperties of a stochastic predator-prey model with Crowley-Martin functional responserdquo Journal of Applied Mathematicsand Computing vol 43 no 1-2 pp 479ndash490 2013

[7] K Hattaf A A Lashari Y Louartassi and N Yousfi ldquoA delayedSIR epidemic model with general incidence raterdquo ElectronicJournal of QualitativeTheory of Differential Equations vol 3 pp1ndash9 2013

[8] V Capasso Mathematical Structures of Epidemic Systems Lec-ture Notes in Biomathematics Springer Germany 1993

[9] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983

[10] J Smoller Shock Waves and Reaction-Diffusion EquationsSpringer Berlin Germany 1983

[11] M H Protter and H F Weinberger Maximum Principles inDifferential Equations Prentice Hall Englewood Cliffs NJUSA 1967

[12] N D Alikakos ldquoAn application of the invariance principle toreaction-diffusion equationsrdquo Journal of Differential Equationsvol 33 no 2 pp 201ndash225 1979

[13] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer BerlinGermany 1993

[14] K Hattaf and N Yousfi ldquoGlobal stability for reaction-diffusionequations in biologyrdquo Computers and Mathematics with Appli-cations vol 66 pp 1488ndash1497 2013

[15] J P LaSalleThe Stability of Dynamical Systems Regional Con-ference Series in AppliedMathematics SIAM Philadelphia PaUSA 1976

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 International Journal of Partial Differential Equations

On the other hand the spatial content of the environmenthas been ignored in the model (1) However due to thelarge mobility of people within a country or even worldwidespatially uniform models are not sufficient to give a realisticpicture of disease diffusion For this reason the spatial effectscannot be neglected in studying the spread of epidemics

Therefore we consider the following SIR epidemic modelwith specific nonlinear incidence rate and spatial diffusion

120597119878

120597119905= 119889119878Δ119878 + Λ minus 120583119878 (119909 119905)

minus120573119878 (119909 119905) 119868 (119909 119905)

1 + 1205721119878 (119909 119905) + 120572

2119868 (119909 119905) + 120572

3119878 (119909 119905) 119868 (119909 119905)

120597119868

120597119905= 119889119868Δ119868 +

120573119878 (119909 119905) 119868 (119909 119905)

1 + 1205721119878 (119909 119905) + 120572

2119868 (119909 119905) + 120572

3119878 (119909 119905) 119868 (119909 119905)

minus (120583 + 119889 + 119903) 119868 (119909 119905)

120597119877

120597119905= 119889119877Δ119877 + 119903119868 (119909 119905) minus 120583119877 (119909 119905)

(2)

where 119878(119909 119905) 119868(119909 119905) and 119877(119909 119905) represent the numbers ofsusceptible infected and removed individuals at location 119909

and time 119905 respectively The positive constants 119889119878 119889119868 and

119889119877denote the corresponding diffusion rates for these three

classes of individualsThe aim of this work is to investigate the global dynamics

of the reaction-diffusion system (2) Note that 119877 does notappear in the first two equations this allows us to study thesystem

120597119878

120597119905= 119889119878Δ119878 + Λ minus 120583119878 (119909 119905)

minus120573119878 (119909 119905) 119868 (119909 119905)

1 + 1205721119878 (119909 119905) + 120572

2119868 (119909 119905) + 120572

3119878 (119909 119905) 119868 (119909 119905)

120597119868

120597119905= 119889119868Δ119868 +

120573119878 (119909 119905) 119868 (119909 119905)

1 + 1205721119878 (119909 119905) + 120572

2119868 (119909 119905) + 120572

3119878 (119909 119905) 119868 (119909 119905)

minus (120583 + 119889 + 119903) 119868 (119909 119905)

(3)

with homogeneous Neumann boundary conditions

120597119878

120597]=

120597119868

120597]= 0 on 120597Ω times (0 +infin) (4)

and initial conditions

119878 (119909 0) = 1206011(119909) ge 0 119868 (119909 0) = 120601

2(119909) ge 0 119909 isin Ω (5)

Here Ω is a bounded domain in R119899 with smooth boundary120597Ω 120597119878120597] and 120597119868120597] are respectively the normal derivativesof 119878 and 119868 on 120597Ω

The rest of paper is organized as followsThe next sectiondeals with the global existence positivity and boundednessof solutions of problem (3)ndash(5) In Section 3 we discussthe stability analysis of equilibria In Section 4 we present

the numerical simulation to illustrate our result Finally theconclusion of our paper is in Section 5

2 Global Existence Positivity andBoundedness of Solutions

In this section we establish the global existence positivityand boundedness of solutions of problem (3)ndash(5) becausethis model describes the population Hence the populationshould remain nonnegative and bounded

Proposition 1 For any given initial data satisfying the con-dition (5) there exists a unique solution of problem (3)ndash(5)defined on [0 +infin) and this solution remains nonnegative andbounded for all 119905 ge 0

Proof System (3)ndash(5) can be written abstractly in the Banachspace 119883 = 119862(Ω) times 119862(Ω) of the form

1199061015840(119905) = 119860119906 (119905) + 119865 (119906 (119905)) 119905 gt 0

119906 (0) = 1199060isin 119883

(6)

where 119906= col(119878 119868) 1199060= col(120601

1 1206012) 119860119906(119905) = col(119889

119878Δ119878 119889119868Δ119868)

and

119865 (119906 (119905)) = (

Λ minus 120583119878 minus120573119878119868

1 + 1205721119878 + 1205722119868 + 1205723119878119868

120573119878119868

1 + 1205721119878 + 1205722119868 + 1205723119878119868

minus (120583 + 119889 + 119903) 119868

)

(7)

It is clear that 119865 is locally Lipschitz in119883 From [9] we deducethat system (6) admits a unique local solution on [0 119879max)where 119879max is the maximal existence time for solution ofsystem (6)

In addition system (3) can be written in the form120597119878

120597119905minus 119889119878Δ119878 = 119865

1(119878 119868)

120597119868

120597119905minus 119889119868Δ119868 = 119865

2(119878 119868)

(8)

It is easy to see that the functions 1198651(119878 119868) and 119865

2(119878 119868) are

continuously differentiable satisfying 1198651(0 119868) = Λ ge 0 and

1198652(119878 0) = 0 ge 0 for all 119878 119868 ge 0 Since initial data of system (3)

are nonnegative we deduce the positivity of the local solution(see the book of Smoller [10])

Now we show the boundedness of solution From (3)ndash(5)we have

120597119878

120597119905minus 119889119878Δ119878 le Λ minus 120583119878

120597119878

120597]= 0

119878 (119909 0) = 1206011(119909) le

100381710038171003817100381712060111003817100381710038171003817infin = max

119909isinΩ

1206011(119909)

(9)

By the comparison principle [11] we have 119878(119909 119905) le 1198781(119905)

International Journal of Partial Differential Equations 3

where 1198781(119905) = 120601

1(119909)119890minus120583119905

+ (Λ120583)(1 minus 119890minus120583119905

) is the solution ofthe problem

1198891198781

119889119905= Λ minus 120583119878

1

1198781(0) =

100381710038171003817100381712060111003817100381710038171003817infin

(10)

Since 1198781(119905) le maxΛ120583 120601

1infin for 119905 isin [0infin) we have that

119878 (119909 119905) le maxΛ

12058310038171003817100381710038171206011

1003817100381710038171003817infin forall (119909 119905) isin Ω times [0 119879max)

(11)FromTheorem2 given byAlikakos in [12] to establish the

119871infin uniform boundedness of 119868(119909 119905) it is sufficient to show the

1198711 uniform boundedness of 119868(119909 119905)Since 120597119878120597] = 120597119868120597] = 0 and (120597120597119905)(119878+119868)minusΔ(119889

119878119878+119889119868119868) le

Λ minus 120583(119878 + 119868) we get120597

120597119905(intΩ

(119878 + 119868) 119889119909) le mes (Ω)Λ minus 120583(intΩ

(119878 + 119868) 119889119909)

(12)Hence

intΩ

(119878 + 119868) 119889119909 le mes (Ω)maxΛ

12058310038171003817100381710038171206011 + 120601

2

1003817100381710038171003817infin (13)

which implies that sup119905ge0

intΩ119868(119909 119905)119889119909 le 119870 =

mes(Ω)maxΛ120583 1206011+ 1206012infin Using [12 Theorem 31]

we deduce that there exists a positive constant 119870lowast that

depends on 119870 and on 1206011+ 1206012infin

such that

sup119905ge0

119868 (sdot 119905)infin le 119870lowast (14)

From the above we have proved that 119878(119909 119905) and 119868(119909 119905) are119871infin bounded on Ω times [0 119879max) Therefore it follows from the

standard theory for semilinear parabolic systems (see [13])that 119879max = +infin This completes the proof of the proposition

3 Qualitative Analysis of the Spatial Model

Using the results presented by Hattaf et al in [7] it is easyto get that the basic reproduction number of disease in theabsence of spatial dependence is given by

1198770=

120573Λ

(120583 + 1205721Λ) (120583 + 119889 + 119903)

(15)

which describes the average number of secondary infectionsproduced by a single infectious individual during the entireinfectious period

It is not hard to show that the system (3) is always adisease-free equilibrium of the form 119864

119891(Λ120583 0) Further if

1198770

gt 1 the system (3) has an endemic stationary state119864lowast(119878lowast 119868lowast) where

119878lowast=

2 (119886 + 1205722Λ)

120573 minus 1205721119886 + 1205722120583 minus 1205723Λ + radic120575

119868lowast=

Λ minus 120583119878lowast

119886

(16)

with 119886 = 120583 + 119889+119903 and 120575 = (120573minus1205721119886+1205722120583minus1205723Λ)2+41205723120583(119886+

1205722Λ)

The objective of this section is to discuss the local andglobal stability of the equilibria

31 Local Stability of the Equilibria First we linearize thedynamical system (3) around arbitrary spatially homoge-neous fixed point119864(119878 119868) for small space- and time-dependentfluctuations and expand them in Fourier space For this let

119878 ( 119905) sim 119878119890120582119905119890119894sdot 119909

119868 ( 119905) sim 119868119890120582119905119890119894sdot 119909

(17)

where = (119909 119910) and sdot = ⟨ ⟩ = 1198962 and 120582 are the

wavenumber vector and frequency respectivelyThen we canobtain the corresponding characteristic equation as follows

det (119869 minus 1198962119863 minus 120582119868

2) = 0 (18)

where 1198682is the identity matrix 119863 = diag(119889

119878 119889119868) is the dif-

fusion matrix and 119869 is the Jacobian matrix of (3) withoutdiffusion (119889

119878= 119889119868= 0) at 119864 which is given by

119869 =(

(

minus120583 minus

120573119868 (1 + 1205722119868)

(1 + 1205721119878 + 1205722119868 + 1205723119878 119868)2minus

120573119878 (1 + 1205721119878)

(1 + 1205721119878 + 1205722119868 + 1205723119878 119868)2

120573119868 (1 + 1205722119868)

(1 + 1205721119878 + 1205722119868 + 1205723119878 119868)2

120573119878 (1 + 1205721119878)

(1 + 1205721119878 + 1205722119868 + 1205723119878 119868)2minus 119886

)

)

(19)The characterization of the local stability of disease-freeequilibrium 119864

119891is given by the following result

Theorem 2 The disease-free equilibrium 119864119891is locally asymp-

totically stable if 1198770lt 1 and it is unstable if 119877

0gt 1

Proof Evaluating (18) at 119864119891 we have

(120583 + 1198962119889119878+ 120582) (119886 (119877

0minus 1) minus 119896

2119889119868minus 120582) = 0 (20)

Clearly the roots of (20) are 1205821(119896) = minus(120583 + 119896

2119889119878) lt 0 and

1205822(119896) = 119886(119877

0minus1) minus 119896

2119889119868 Note that 120582

2(119896) is negative if 119877

0lt 1

for all 119896 Hence 119864119891is locally asymptotically stable if 119877

0lt 1 If

1198770gt 1 120582

2(0) is positive So 119864

119891is unstable

Next we focus on the local stability of the endemicequilibrium 119864

lowast

Theorem 3 The endemic equilibrium 119864lowast is locally asymptoti-

cally stable if 1198770gt 1

Proof Evaluating (18) at 119864lowast(119878lowast 119868lowast) we have

1205822+ 1198861(119896) 120582 + 119886

2(119896) = 0 (21)

where1198861(119896) = 120583 + 119896

2(119889119878+ 119889119868)

+ 119886(119868lowast(1 + 120572

2119868lowast)

(1 + 1205721119878lowast + 120572

2119868lowast + 120572

3119878lowast119868lowast) 119878lowast

+1205722119868lowast+ 1205723119878lowast119868lowast

1 + 1205721119878lowast + 120572

2119868lowast + 120572

3119878lowast119868lowast

)

4 International Journal of Partial Differential Equations

1198862(119896) = (120583 +

120573119868lowast(1 + 120572

2119868lowast)

(1 + 1205721119878lowast + 120572

2119868lowast + 120572

3119878lowast119868lowast)

2+ 1198962119889119878)

times (1198962119889119868+

120573119878lowast(1205722119868lowast+ 1205723119878lowast119868lowast)

(1 + 1205721119878lowast + 120572

2119868lowast + 120572

3119878lowast119868lowast)

2)

+1205732119878lowast119868lowast(1 + 120572

2119868lowast) (1 + 120572

1119878lowast)

(1 + 1205721119878lowast + 120572

2119868lowast + 120572

3119878lowast119868lowast)

4

(22)

We have 1198861gt 0 and 119886

2gt 0 then 119864

lowast is locally asymptoticallystable

32 Global Stability of the Equilibria The purpose of thissubsection is to determine the global stability for reaction-diffusion equations (3)ndash(5) by constructing Lyapunov func-tionals These Lyapunov functionals are obtained from thosefor ordinary differential equations (1) by applying themethodof Hattaf and Yousfi presented in [14]

The system (1) is particular case of themodel proposed byHattaf et al [7] with 119891(119878 119868) = 120573119878(1 + 120572

1119878 + 1205722119868 + 1205723119878119868) To

study the global stability of 119864119891for (1) the authors Hattaf et al

[7] proposed the following Lyapunov functional

1198811= 119878 minus 119878

0minus int

119878

1198780

119891 (1198780 0)

119891 (119883 0)119889119883 + 119868 (23)

where 1198780= Λ120583

From [14] we construct the Lyapunov functional forsystem (3)ndash(5) at 119864

119891as follows

1198821= intΩ

1198811(119878 (119909 119905) 119868 (119909 119905)) 119889119909 (24)

Calculating the time derivative of 1198821along the solution of

system (3)ndash(5) we have

1198891198821

119889119905= intΩ

minus120583(119878 minus 119878

0)2

119878 (1 + 12057211198780)

+119886119868 (1 + 1205721119878

1 + 1205721119878 + 1205722119868 + 1205723119878119868

1198770minus 1)119889119909

minus1198891198781198780

1 + 12057211198780

intΩ

|nabla119878|2

1198782119889119909

le intΩ

minus120583(119878 minus 119878

0)2

119878 (1 + 12057211198780)+ 119886119868 (119877

0minus 1)119889119909

minus1198891198781198780

1 + 12057211198780

intΩ

|nabla119878|2

1198782119889119909

(25)

Since 1198770

le 1 we have 1198891198821119889119905 le 0 Thus the disease-free

equilibrium 119864119891is stable and 119889119882

1119889119905 = 0 if and only if 119878 = 119878

0

and 119868(1198770minus 1) = 0 We discuss two cases as follows

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

Space

Susceptible

Infected

Figure 1The initial distributions of the numbers of susceptibles andinfectious individuals for Figures 2 and 3

(i) If 1198770lt 1 then 119868 = 0

(ii) If 1198770= 1 from 119878 = 119878

0and the first equation of (3) we

have

120597119878

120597119905=

1205971198780

120597119905= 119889119878Δ1198780+ Λ minus 120583119878

0minus 119891 (119878

0 119868) 119868 = 0 (26)

Then 119891(1198780 119868)119868 = (120573119878

0119868)(1 +120572

11198780+1205722119868 +12057231198780119868) = 0

Since 120573 gt 0 and 1198780gt 0 then 119868 = 0

By the above discussion we deduce that the largest compactinvariant set in Γ = (119878 119868) | (119889119882

1)119889119905 = 0 is just the single-

ton 119864119891 From LaSalle invariance principle [15] we conclude

that 119864119891is globally asymptotically stable

Using same technique we construct a Lyapunov func-tional 119882

2for system (3)ndash(5) at 119864

lowast from the Lyapunovfunctional 119881

2defined by Hattaf et al in [7] It is easy to

show that 1198812verifies the condition (15) given in [14] Hence

it follows from [14 Proposition 21] that 1198822is a Lyapunov

functional for the reaction-diffusion system (3)ndash(5) at 119864lowast

when1198770gt 1We summarize the above in the following result

Theorem 4 (i) If 1198770

le 1 the disease-free equilibrium 119864119891

of (3)ndash(5) is globally asymptotically stable for all diffusioncoefficients

(ii) If 1198770

gt 1 the endemic equilibrium 119864lowast of (3)ndash(5) is

globally asymptotically stable for all diffusion coefficients

4 Numerical Simulations

In this section we present the numerical simulations toillustrate our theoretical results To simplify we considersystem (3) under Neumann boundary conditions

120597119878

120597]=

120597119868

120597]= 0 119905 gt 0 119909 = 0 1 (27)

International Journal of Partial Differential Equations 5

002

0406

081

020

4060

80100

0

1

2

3

4

5

Num

ber o

f sus

cept

ible

indi

vidu

als

0 204

0608

2040

6080

Space xTime

t

0

005

01

015

02

025

Num

ber o

f inf

ecte

d in

divi

dual

s

002

0406

081

020

4060

80100

Space xTime

t

Figure 2The temporal solution found by numerical of problem (3) with the Neumann boundary conditions (27) and initial conditions (28)

Figure 3The temporal solution found by numerical of problem (3) with the Neumann boundary conditions (27) and initial conditions (28)

and initial conditions

119878 (119909 0) = 11119909 0 le 119909 lt 05

11 (1 minus 119909) 05 le 119909 le 1

119868 (119909 0) = 05119909 0 le 119909 lt 05

05 (1 minus 119909) 05 le 119909 le 1

(28)

In Figure 1 we show that themaximum value of the num-bers of susceptibles and infectious individuals is concentratedat themiddle of the interval [0 1] and these numbers decreaselinearly to zero at the boundaries 119909 = 0 and 119909 = 1

Now we choose the following data set of system (3) 119889119878=

01 119889119868

= 05 Λ = 05 120573 = 02 120583 = 01 1205721

= 011205722= 002 120572

3= 003 119889 = 01 and 119903 = 05 By calculation we

have 1198770= 09524 In this case system (3) has a disease-free

equilibrium 119864119891(5 0) Hence by Theorem 4(i) 119864

119891is globally

asymptotically stable Numerical simulation illustrates ourresult (see Figure 2)

In Figure 3 we choose 120573 = 06 and do not change theother parameter values By calculation we have 119877

0= 28571

which satisfyTheorem4(ii) then the disease-free equilibriumis still present and the system (2) has a unique endemic

equilibrium 119864lowast(13625 05196) Therefore by Theorem 2

and Theorem 4(ii) 119864119891

is unstable while 119864lowast is globally

asymptotically stable Numerical simulation illustrates ourresult (see Figure 3)

5 Conclusion

In this paper we investigated the dynamics of a reaction-diffusion epidemic model with specific nonlinear incidencerateThis specific nonlinear incidence rate includes the tradi-tional bilinear incidence rate the saturated incidence rate theBeddington-DeAngelis functional response and Crowley-Martin functional response The global dynamics of themodel are completely determined by the basic reproductionnumber 119877

0 We proved that the disease-free equilibrium is

globally asymptotically stable if 1198770

le 1 which leads tothe eradication of disease from population When 119877

0gt 1

then disease-free equilibrium becomes unstable and a uniqueendemic equilibrium exists and is globally asymptotically sta-ble which means that the disease persists in the population

From our theoretical and numerical results we concludethat the spatial diffusion has no effect on the stability behavior

6 International Journal of Partial Differential Equations

of equilibria in the case of Neumann conditions and spatiallyconstant coefficients

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the editor and the anony-mous referees for their valuable remarks and commentswhich have led to the improvement of the quality of theirstudy

References

[1] J R Beddington ldquoMutual interference between parasites orpredators and its effect on searching efficiencyrdquo Journal ofAnimal Ecology pp 331ndash341 1975

[2] D L DeAngelis R A Goldsten and R Neill ldquoA model fortrophic interactionrdquo Ecology vol 56 pp 881ndash892 1975

[3] R S Cantrell and C Cosner ldquoOn the Dynamics of Predator-Prey Models with the Beddington-Deangelis FunctionalResponserdquo Journal of Mathematical Analysis and Applicationsvol 257 no 1 pp 206ndash222 2001

[4] P H Crowley and E K Martin ldquoFunctional responses andinterferencewithin and between year classes of a dragonfly pop-ulationrdquo Journal of the North American Benthological Societyvol 8 pp 211ndash221 1989

[5] X Zhou and J Cui ldquoGlobal stability of the viral dynamics withcrowley-martin functional responserdquo Bulletin of the KoreanMathematical Society vol 48 no 3 pp 555ndash574 2011

[6] X Q Liu S M Zhong B D Tian and F X Zheng ldquoAsymptoticproperties of a stochastic predator-prey model with Crowley-Martin functional responserdquo Journal of Applied Mathematicsand Computing vol 43 no 1-2 pp 479ndash490 2013

[7] K Hattaf A A Lashari Y Louartassi and N Yousfi ldquoA delayedSIR epidemic model with general incidence raterdquo ElectronicJournal of QualitativeTheory of Differential Equations vol 3 pp1ndash9 2013

[8] V Capasso Mathematical Structures of Epidemic Systems Lec-ture Notes in Biomathematics Springer Germany 1993

[9] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983

[10] J Smoller Shock Waves and Reaction-Diffusion EquationsSpringer Berlin Germany 1983

[11] M H Protter and H F Weinberger Maximum Principles inDifferential Equations Prentice Hall Englewood Cliffs NJUSA 1967

[12] N D Alikakos ldquoAn application of the invariance principle toreaction-diffusion equationsrdquo Journal of Differential Equationsvol 33 no 2 pp 201ndash225 1979

[13] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer BerlinGermany 1993

[14] K Hattaf and N Yousfi ldquoGlobal stability for reaction-diffusionequations in biologyrdquo Computers and Mathematics with Appli-cations vol 66 pp 1488ndash1497 2013

[15] J P LaSalleThe Stability of Dynamical Systems Regional Con-ference Series in AppliedMathematics SIAM Philadelphia PaUSA 1976

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Partial Differential Equations 3

where 1198781(119905) = 120601

1(119909)119890minus120583119905

+ (Λ120583)(1 minus 119890minus120583119905

) is the solution ofthe problem

1198891198781

119889119905= Λ minus 120583119878

1

1198781(0) =

100381710038171003817100381712060111003817100381710038171003817infin

(10)

Since 1198781(119905) le maxΛ120583 120601

1infin for 119905 isin [0infin) we have that

119878 (119909 119905) le maxΛ

12058310038171003817100381710038171206011

1003817100381710038171003817infin forall (119909 119905) isin Ω times [0 119879max)

(11)FromTheorem2 given byAlikakos in [12] to establish the

119871infin uniform boundedness of 119868(119909 119905) it is sufficient to show the

1198711 uniform boundedness of 119868(119909 119905)Since 120597119878120597] = 120597119868120597] = 0 and (120597120597119905)(119878+119868)minusΔ(119889

119878119878+119889119868119868) le

Λ minus 120583(119878 + 119868) we get120597

120597119905(intΩ

(119878 + 119868) 119889119909) le mes (Ω)Λ minus 120583(intΩ

(119878 + 119868) 119889119909)

(12)Hence

intΩ

(119878 + 119868) 119889119909 le mes (Ω)maxΛ

12058310038171003817100381710038171206011 + 120601

2

1003817100381710038171003817infin (13)

which implies that sup119905ge0

intΩ119868(119909 119905)119889119909 le 119870 =

mes(Ω)maxΛ120583 1206011+ 1206012infin Using [12 Theorem 31]

we deduce that there exists a positive constant 119870lowast that

depends on 119870 and on 1206011+ 1206012infin

such that

sup119905ge0

119868 (sdot 119905)infin le 119870lowast (14)

From the above we have proved that 119878(119909 119905) and 119868(119909 119905) are119871infin bounded on Ω times [0 119879max) Therefore it follows from the

standard theory for semilinear parabolic systems (see [13])that 119879max = +infin This completes the proof of the proposition

3 Qualitative Analysis of the Spatial Model

Using the results presented by Hattaf et al in [7] it is easyto get that the basic reproduction number of disease in theabsence of spatial dependence is given by

1198770=

120573Λ

(120583 + 1205721Λ) (120583 + 119889 + 119903)

(15)

which describes the average number of secondary infectionsproduced by a single infectious individual during the entireinfectious period

It is not hard to show that the system (3) is always adisease-free equilibrium of the form 119864

119891(Λ120583 0) Further if

1198770

gt 1 the system (3) has an endemic stationary state119864lowast(119878lowast 119868lowast) where

119878lowast=

2 (119886 + 1205722Λ)

120573 minus 1205721119886 + 1205722120583 minus 1205723Λ + radic120575

119868lowast=

Λ minus 120583119878lowast

119886

(16)

with 119886 = 120583 + 119889+119903 and 120575 = (120573minus1205721119886+1205722120583minus1205723Λ)2+41205723120583(119886+

1205722Λ)

The objective of this section is to discuss the local andglobal stability of the equilibria

31 Local Stability of the Equilibria First we linearize thedynamical system (3) around arbitrary spatially homoge-neous fixed point119864(119878 119868) for small space- and time-dependentfluctuations and expand them in Fourier space For this let

119878 ( 119905) sim 119878119890120582119905119890119894sdot 119909

119868 ( 119905) sim 119868119890120582119905119890119894sdot 119909

(17)

where = (119909 119910) and sdot = ⟨ ⟩ = 1198962 and 120582 are the

wavenumber vector and frequency respectivelyThen we canobtain the corresponding characteristic equation as follows

det (119869 minus 1198962119863 minus 120582119868

2) = 0 (18)

where 1198682is the identity matrix 119863 = diag(119889

119878 119889119868) is the dif-

fusion matrix and 119869 is the Jacobian matrix of (3) withoutdiffusion (119889

119878= 119889119868= 0) at 119864 which is given by

119869 =(

(

minus120583 minus

120573119868 (1 + 1205722119868)

(1 + 1205721119878 + 1205722119868 + 1205723119878 119868)2minus

120573119878 (1 + 1205721119878)

(1 + 1205721119878 + 1205722119868 + 1205723119878 119868)2

120573119868 (1 + 1205722119868)

(1 + 1205721119878 + 1205722119868 + 1205723119878 119868)2

120573119878 (1 + 1205721119878)

(1 + 1205721119878 + 1205722119868 + 1205723119878 119868)2minus 119886

)

)

(19)The characterization of the local stability of disease-freeequilibrium 119864

119891is given by the following result

Theorem 2 The disease-free equilibrium 119864119891is locally asymp-

totically stable if 1198770lt 1 and it is unstable if 119877

0gt 1

Proof Evaluating (18) at 119864119891 we have

(120583 + 1198962119889119878+ 120582) (119886 (119877

0minus 1) minus 119896

2119889119868minus 120582) = 0 (20)

Clearly the roots of (20) are 1205821(119896) = minus(120583 + 119896

2119889119878) lt 0 and

1205822(119896) = 119886(119877

0minus1) minus 119896

2119889119868 Note that 120582

2(119896) is negative if 119877

0lt 1

for all 119896 Hence 119864119891is locally asymptotically stable if 119877

0lt 1 If

1198770gt 1 120582

2(0) is positive So 119864

119891is unstable

Next we focus on the local stability of the endemicequilibrium 119864

lowast

Theorem 3 The endemic equilibrium 119864lowast is locally asymptoti-

cally stable if 1198770gt 1

Proof Evaluating (18) at 119864lowast(119878lowast 119868lowast) we have

1205822+ 1198861(119896) 120582 + 119886

2(119896) = 0 (21)

where1198861(119896) = 120583 + 119896

2(119889119878+ 119889119868)

+ 119886(119868lowast(1 + 120572

2119868lowast)

(1 + 1205721119878lowast + 120572

2119868lowast + 120572

3119878lowast119868lowast) 119878lowast

+1205722119868lowast+ 1205723119878lowast119868lowast

1 + 1205721119878lowast + 120572

2119868lowast + 120572

3119878lowast119868lowast

)

4 International Journal of Partial Differential Equations

1198862(119896) = (120583 +

120573119868lowast(1 + 120572

2119868lowast)

(1 + 1205721119878lowast + 120572

2119868lowast + 120572

3119878lowast119868lowast)

2+ 1198962119889119878)

times (1198962119889119868+

120573119878lowast(1205722119868lowast+ 1205723119878lowast119868lowast)

(1 + 1205721119878lowast + 120572

2119868lowast + 120572

3119878lowast119868lowast)

2)

+1205732119878lowast119868lowast(1 + 120572

2119868lowast) (1 + 120572

1119878lowast)

(1 + 1205721119878lowast + 120572

2119868lowast + 120572

3119878lowast119868lowast)

4

(22)

We have 1198861gt 0 and 119886

2gt 0 then 119864

lowast is locally asymptoticallystable

32 Global Stability of the Equilibria The purpose of thissubsection is to determine the global stability for reaction-diffusion equations (3)ndash(5) by constructing Lyapunov func-tionals These Lyapunov functionals are obtained from thosefor ordinary differential equations (1) by applying themethodof Hattaf and Yousfi presented in [14]

The system (1) is particular case of themodel proposed byHattaf et al [7] with 119891(119878 119868) = 120573119878(1 + 120572

1119878 + 1205722119868 + 1205723119878119868) To

study the global stability of 119864119891for (1) the authors Hattaf et al

[7] proposed the following Lyapunov functional

1198811= 119878 minus 119878

0minus int

119878

1198780

119891 (1198780 0)

119891 (119883 0)119889119883 + 119868 (23)

where 1198780= Λ120583

From [14] we construct the Lyapunov functional forsystem (3)ndash(5) at 119864

119891as follows

1198821= intΩ

1198811(119878 (119909 119905) 119868 (119909 119905)) 119889119909 (24)

Calculating the time derivative of 1198821along the solution of

system (3)ndash(5) we have

1198891198821

119889119905= intΩ

minus120583(119878 minus 119878

0)2

119878 (1 + 12057211198780)

+119886119868 (1 + 1205721119878

1 + 1205721119878 + 1205722119868 + 1205723119878119868

1198770minus 1)119889119909

minus1198891198781198780

1 + 12057211198780

intΩ

|nabla119878|2

1198782119889119909

le intΩ

minus120583(119878 minus 119878

0)2

119878 (1 + 12057211198780)+ 119886119868 (119877

0minus 1)119889119909

minus1198891198781198780

1 + 12057211198780

intΩ

|nabla119878|2

1198782119889119909

(25)

Since 1198770

le 1 we have 1198891198821119889119905 le 0 Thus the disease-free

equilibrium 119864119891is stable and 119889119882

1119889119905 = 0 if and only if 119878 = 119878

0

and 119868(1198770minus 1) = 0 We discuss two cases as follows

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

Space

Susceptible

Infected

Figure 1The initial distributions of the numbers of susceptibles andinfectious individuals for Figures 2 and 3

(i) If 1198770lt 1 then 119868 = 0

(ii) If 1198770= 1 from 119878 = 119878

0and the first equation of (3) we

have

120597119878

120597119905=

1205971198780

120597119905= 119889119878Δ1198780+ Λ minus 120583119878

0minus 119891 (119878

0 119868) 119868 = 0 (26)

Then 119891(1198780 119868)119868 = (120573119878

0119868)(1 +120572

11198780+1205722119868 +12057231198780119868) = 0

Since 120573 gt 0 and 1198780gt 0 then 119868 = 0

By the above discussion we deduce that the largest compactinvariant set in Γ = (119878 119868) | (119889119882

1)119889119905 = 0 is just the single-

ton 119864119891 From LaSalle invariance principle [15] we conclude

that 119864119891is globally asymptotically stable

Using same technique we construct a Lyapunov func-tional 119882

2for system (3)ndash(5) at 119864

lowast from the Lyapunovfunctional 119881

2defined by Hattaf et al in [7] It is easy to

show that 1198812verifies the condition (15) given in [14] Hence

it follows from [14 Proposition 21] that 1198822is a Lyapunov

functional for the reaction-diffusion system (3)ndash(5) at 119864lowast

when1198770gt 1We summarize the above in the following result

Theorem 4 (i) If 1198770

le 1 the disease-free equilibrium 119864119891

of (3)ndash(5) is globally asymptotically stable for all diffusioncoefficients

(ii) If 1198770

gt 1 the endemic equilibrium 119864lowast of (3)ndash(5) is

globally asymptotically stable for all diffusion coefficients

4 Numerical Simulations

In this section we present the numerical simulations toillustrate our theoretical results To simplify we considersystem (3) under Neumann boundary conditions

120597119878

120597]=

120597119868

120597]= 0 119905 gt 0 119909 = 0 1 (27)

International Journal of Partial Differential Equations 5

002

0406

081

020

4060

80100

0

1

2

3

4

5

Num

ber o

f sus

cept

ible

indi

vidu

als

0 204

0608

2040

6080

Space xTime

t

0

005

01

015

02

025

Num

ber o

f inf

ecte

d in

divi

dual

s

002

0406

081

020

4060

80100

Space xTime

t

Figure 2The temporal solution found by numerical of problem (3) with the Neumann boundary conditions (27) and initial conditions (28)

Figure 3The temporal solution found by numerical of problem (3) with the Neumann boundary conditions (27) and initial conditions (28)

and initial conditions

119878 (119909 0) = 11119909 0 le 119909 lt 05

11 (1 minus 119909) 05 le 119909 le 1

119868 (119909 0) = 05119909 0 le 119909 lt 05

05 (1 minus 119909) 05 le 119909 le 1

(28)

In Figure 1 we show that themaximum value of the num-bers of susceptibles and infectious individuals is concentratedat themiddle of the interval [0 1] and these numbers decreaselinearly to zero at the boundaries 119909 = 0 and 119909 = 1

Now we choose the following data set of system (3) 119889119878=

01 119889119868

= 05 Λ = 05 120573 = 02 120583 = 01 1205721

= 011205722= 002 120572

3= 003 119889 = 01 and 119903 = 05 By calculation we

have 1198770= 09524 In this case system (3) has a disease-free

equilibrium 119864119891(5 0) Hence by Theorem 4(i) 119864

119891is globally

asymptotically stable Numerical simulation illustrates ourresult (see Figure 2)

In Figure 3 we choose 120573 = 06 and do not change theother parameter values By calculation we have 119877

0= 28571

which satisfyTheorem4(ii) then the disease-free equilibriumis still present and the system (2) has a unique endemic

equilibrium 119864lowast(13625 05196) Therefore by Theorem 2

and Theorem 4(ii) 119864119891

is unstable while 119864lowast is globally

asymptotically stable Numerical simulation illustrates ourresult (see Figure 3)

5 Conclusion

In this paper we investigated the dynamics of a reaction-diffusion epidemic model with specific nonlinear incidencerateThis specific nonlinear incidence rate includes the tradi-tional bilinear incidence rate the saturated incidence rate theBeddington-DeAngelis functional response and Crowley-Martin functional response The global dynamics of themodel are completely determined by the basic reproductionnumber 119877

0 We proved that the disease-free equilibrium is

globally asymptotically stable if 1198770

le 1 which leads tothe eradication of disease from population When 119877

0gt 1

then disease-free equilibrium becomes unstable and a uniqueendemic equilibrium exists and is globally asymptotically sta-ble which means that the disease persists in the population

From our theoretical and numerical results we concludethat the spatial diffusion has no effect on the stability behavior

6 International Journal of Partial Differential Equations

of equilibria in the case of Neumann conditions and spatiallyconstant coefficients

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the editor and the anony-mous referees for their valuable remarks and commentswhich have led to the improvement of the quality of theirstudy

References

[1] J R Beddington ldquoMutual interference between parasites orpredators and its effect on searching efficiencyrdquo Journal ofAnimal Ecology pp 331ndash341 1975

[2] D L DeAngelis R A Goldsten and R Neill ldquoA model fortrophic interactionrdquo Ecology vol 56 pp 881ndash892 1975

[3] R S Cantrell and C Cosner ldquoOn the Dynamics of Predator-Prey Models with the Beddington-Deangelis FunctionalResponserdquo Journal of Mathematical Analysis and Applicationsvol 257 no 1 pp 206ndash222 2001

[4] P H Crowley and E K Martin ldquoFunctional responses andinterferencewithin and between year classes of a dragonfly pop-ulationrdquo Journal of the North American Benthological Societyvol 8 pp 211ndash221 1989

[5] X Zhou and J Cui ldquoGlobal stability of the viral dynamics withcrowley-martin functional responserdquo Bulletin of the KoreanMathematical Society vol 48 no 3 pp 555ndash574 2011

[6] X Q Liu S M Zhong B D Tian and F X Zheng ldquoAsymptoticproperties of a stochastic predator-prey model with Crowley-Martin functional responserdquo Journal of Applied Mathematicsand Computing vol 43 no 1-2 pp 479ndash490 2013

[7] K Hattaf A A Lashari Y Louartassi and N Yousfi ldquoA delayedSIR epidemic model with general incidence raterdquo ElectronicJournal of QualitativeTheory of Differential Equations vol 3 pp1ndash9 2013

[8] V Capasso Mathematical Structures of Epidemic Systems Lec-ture Notes in Biomathematics Springer Germany 1993

[9] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983

[10] J Smoller Shock Waves and Reaction-Diffusion EquationsSpringer Berlin Germany 1983

[11] M H Protter and H F Weinberger Maximum Principles inDifferential Equations Prentice Hall Englewood Cliffs NJUSA 1967

[12] N D Alikakos ldquoAn application of the invariance principle toreaction-diffusion equationsrdquo Journal of Differential Equationsvol 33 no 2 pp 201ndash225 1979

[13] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer BerlinGermany 1993

[14] K Hattaf and N Yousfi ldquoGlobal stability for reaction-diffusionequations in biologyrdquo Computers and Mathematics with Appli-cations vol 66 pp 1488ndash1497 2013

[15] J P LaSalleThe Stability of Dynamical Systems Regional Con-ference Series in AppliedMathematics SIAM Philadelphia PaUSA 1976

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Journal of Partial Differential Equations

1198862(119896) = (120583 +

120573119868lowast(1 + 120572

2119868lowast)

(1 + 1205721119878lowast + 120572

2119868lowast + 120572

3119878lowast119868lowast)

2+ 1198962119889119878)

times (1198962119889119868+

120573119878lowast(1205722119868lowast+ 1205723119878lowast119868lowast)

(1 + 1205721119878lowast + 120572

2119868lowast + 120572

3119878lowast119868lowast)

2)

+1205732119878lowast119868lowast(1 + 120572

2119868lowast) (1 + 120572

1119878lowast)

(1 + 1205721119878lowast + 120572

2119868lowast + 120572

3119878lowast119868lowast)

4

(22)

We have 1198861gt 0 and 119886

2gt 0 then 119864

lowast is locally asymptoticallystable

32 Global Stability of the Equilibria The purpose of thissubsection is to determine the global stability for reaction-diffusion equations (3)ndash(5) by constructing Lyapunov func-tionals These Lyapunov functionals are obtained from thosefor ordinary differential equations (1) by applying themethodof Hattaf and Yousfi presented in [14]

The system (1) is particular case of themodel proposed byHattaf et al [7] with 119891(119878 119868) = 120573119878(1 + 120572

1119878 + 1205722119868 + 1205723119878119868) To

study the global stability of 119864119891for (1) the authors Hattaf et al

[7] proposed the following Lyapunov functional

1198811= 119878 minus 119878

0minus int

119878

1198780

119891 (1198780 0)

119891 (119883 0)119889119883 + 119868 (23)

where 1198780= Λ120583

From [14] we construct the Lyapunov functional forsystem (3)ndash(5) at 119864

119891as follows

1198821= intΩ

1198811(119878 (119909 119905) 119868 (119909 119905)) 119889119909 (24)

Calculating the time derivative of 1198821along the solution of

system (3)ndash(5) we have

1198891198821

119889119905= intΩ

minus120583(119878 minus 119878

0)2

119878 (1 + 12057211198780)

+119886119868 (1 + 1205721119878

1 + 1205721119878 + 1205722119868 + 1205723119878119868

1198770minus 1)119889119909

minus1198891198781198780

1 + 12057211198780

intΩ

|nabla119878|2

1198782119889119909

le intΩ

minus120583(119878 minus 119878

0)2

119878 (1 + 12057211198780)+ 119886119868 (119877

0minus 1)119889119909

minus1198891198781198780

1 + 12057211198780

intΩ

|nabla119878|2

1198782119889119909

(25)

Since 1198770

le 1 we have 1198891198821119889119905 le 0 Thus the disease-free

equilibrium 119864119891is stable and 119889119882

1119889119905 = 0 if and only if 119878 = 119878

0

and 119868(1198770minus 1) = 0 We discuss two cases as follows

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

Space

Susceptible

Infected

Figure 1The initial distributions of the numbers of susceptibles andinfectious individuals for Figures 2 and 3

(i) If 1198770lt 1 then 119868 = 0

(ii) If 1198770= 1 from 119878 = 119878

0and the first equation of (3) we

have

120597119878

120597119905=

1205971198780

120597119905= 119889119878Δ1198780+ Λ minus 120583119878

0minus 119891 (119878

0 119868) 119868 = 0 (26)

Then 119891(1198780 119868)119868 = (120573119878

0119868)(1 +120572

11198780+1205722119868 +12057231198780119868) = 0

Since 120573 gt 0 and 1198780gt 0 then 119868 = 0

By the above discussion we deduce that the largest compactinvariant set in Γ = (119878 119868) | (119889119882

1)119889119905 = 0 is just the single-

ton 119864119891 From LaSalle invariance principle [15] we conclude

that 119864119891is globally asymptotically stable

Using same technique we construct a Lyapunov func-tional 119882

2for system (3)ndash(5) at 119864

lowast from the Lyapunovfunctional 119881

2defined by Hattaf et al in [7] It is easy to

show that 1198812verifies the condition (15) given in [14] Hence

it follows from [14 Proposition 21] that 1198822is a Lyapunov

functional for the reaction-diffusion system (3)ndash(5) at 119864lowast

when1198770gt 1We summarize the above in the following result

Theorem 4 (i) If 1198770

le 1 the disease-free equilibrium 119864119891

of (3)ndash(5) is globally asymptotically stable for all diffusioncoefficients

(ii) If 1198770

gt 1 the endemic equilibrium 119864lowast of (3)ndash(5) is

globally asymptotically stable for all diffusion coefficients

4 Numerical Simulations

In this section we present the numerical simulations toillustrate our theoretical results To simplify we considersystem (3) under Neumann boundary conditions

120597119878

120597]=

120597119868

120597]= 0 119905 gt 0 119909 = 0 1 (27)

International Journal of Partial Differential Equations 5

002

0406

081

020

4060

80100

0

1

2

3

4

5

Num

ber o

f sus

cept

ible

indi

vidu

als

0 204

0608

2040

6080

Space xTime

t

0

005

01

015

02

025

Num

ber o

f inf

ecte

d in

divi

dual

s

002

0406

081

020

4060

80100

Space xTime

t

Figure 2The temporal solution found by numerical of problem (3) with the Neumann boundary conditions (27) and initial conditions (28)

Figure 3The temporal solution found by numerical of problem (3) with the Neumann boundary conditions (27) and initial conditions (28)

and initial conditions

119878 (119909 0) = 11119909 0 le 119909 lt 05

11 (1 minus 119909) 05 le 119909 le 1

119868 (119909 0) = 05119909 0 le 119909 lt 05

05 (1 minus 119909) 05 le 119909 le 1

(28)

In Figure 1 we show that themaximum value of the num-bers of susceptibles and infectious individuals is concentratedat themiddle of the interval [0 1] and these numbers decreaselinearly to zero at the boundaries 119909 = 0 and 119909 = 1

Now we choose the following data set of system (3) 119889119878=

01 119889119868

= 05 Λ = 05 120573 = 02 120583 = 01 1205721

= 011205722= 002 120572

3= 003 119889 = 01 and 119903 = 05 By calculation we

have 1198770= 09524 In this case system (3) has a disease-free

equilibrium 119864119891(5 0) Hence by Theorem 4(i) 119864

119891is globally

asymptotically stable Numerical simulation illustrates ourresult (see Figure 2)

In Figure 3 we choose 120573 = 06 and do not change theother parameter values By calculation we have 119877

0= 28571

which satisfyTheorem4(ii) then the disease-free equilibriumis still present and the system (2) has a unique endemic

equilibrium 119864lowast(13625 05196) Therefore by Theorem 2

and Theorem 4(ii) 119864119891

is unstable while 119864lowast is globally

asymptotically stable Numerical simulation illustrates ourresult (see Figure 3)

5 Conclusion

In this paper we investigated the dynamics of a reaction-diffusion epidemic model with specific nonlinear incidencerateThis specific nonlinear incidence rate includes the tradi-tional bilinear incidence rate the saturated incidence rate theBeddington-DeAngelis functional response and Crowley-Martin functional response The global dynamics of themodel are completely determined by the basic reproductionnumber 119877

0 We proved that the disease-free equilibrium is

globally asymptotically stable if 1198770

le 1 which leads tothe eradication of disease from population When 119877

0gt 1

then disease-free equilibrium becomes unstable and a uniqueendemic equilibrium exists and is globally asymptotically sta-ble which means that the disease persists in the population

From our theoretical and numerical results we concludethat the spatial diffusion has no effect on the stability behavior

6 International Journal of Partial Differential Equations

of equilibria in the case of Neumann conditions and spatiallyconstant coefficients

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the editor and the anony-mous referees for their valuable remarks and commentswhich have led to the improvement of the quality of theirstudy

References

[1] J R Beddington ldquoMutual interference between parasites orpredators and its effect on searching efficiencyrdquo Journal ofAnimal Ecology pp 331ndash341 1975

[2] D L DeAngelis R A Goldsten and R Neill ldquoA model fortrophic interactionrdquo Ecology vol 56 pp 881ndash892 1975

[3] R S Cantrell and C Cosner ldquoOn the Dynamics of Predator-Prey Models with the Beddington-Deangelis FunctionalResponserdquo Journal of Mathematical Analysis and Applicationsvol 257 no 1 pp 206ndash222 2001

[4] P H Crowley and E K Martin ldquoFunctional responses andinterferencewithin and between year classes of a dragonfly pop-ulationrdquo Journal of the North American Benthological Societyvol 8 pp 211ndash221 1989

[5] X Zhou and J Cui ldquoGlobal stability of the viral dynamics withcrowley-martin functional responserdquo Bulletin of the KoreanMathematical Society vol 48 no 3 pp 555ndash574 2011

[6] X Q Liu S M Zhong B D Tian and F X Zheng ldquoAsymptoticproperties of a stochastic predator-prey model with Crowley-Martin functional responserdquo Journal of Applied Mathematicsand Computing vol 43 no 1-2 pp 479ndash490 2013

[7] K Hattaf A A Lashari Y Louartassi and N Yousfi ldquoA delayedSIR epidemic model with general incidence raterdquo ElectronicJournal of QualitativeTheory of Differential Equations vol 3 pp1ndash9 2013

[8] V Capasso Mathematical Structures of Epidemic Systems Lec-ture Notes in Biomathematics Springer Germany 1993

[9] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983

[10] J Smoller Shock Waves and Reaction-Diffusion EquationsSpringer Berlin Germany 1983

[11] M H Protter and H F Weinberger Maximum Principles inDifferential Equations Prentice Hall Englewood Cliffs NJUSA 1967

[12] N D Alikakos ldquoAn application of the invariance principle toreaction-diffusion equationsrdquo Journal of Differential Equationsvol 33 no 2 pp 201ndash225 1979

[13] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer BerlinGermany 1993

[14] K Hattaf and N Yousfi ldquoGlobal stability for reaction-diffusionequations in biologyrdquo Computers and Mathematics with Appli-cations vol 66 pp 1488ndash1497 2013

[15] J P LaSalleThe Stability of Dynamical Systems Regional Con-ference Series in AppliedMathematics SIAM Philadelphia PaUSA 1976

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Partial Differential Equations 5

002

0406

081

020

4060

80100

0

1

2

3

4

5

Num

ber o

f sus

cept

ible

indi

vidu

als

0 204

0608

2040

6080

Space xTime

t

0

005

01

015

02

025

Num

ber o

f inf

ecte

d in

divi

dual

s

002

0406

081

020

4060

80100

Space xTime

t

Figure 2The temporal solution found by numerical of problem (3) with the Neumann boundary conditions (27) and initial conditions (28)

Figure 3The temporal solution found by numerical of problem (3) with the Neumann boundary conditions (27) and initial conditions (28)

and initial conditions

119878 (119909 0) = 11119909 0 le 119909 lt 05

11 (1 minus 119909) 05 le 119909 le 1

119868 (119909 0) = 05119909 0 le 119909 lt 05

05 (1 minus 119909) 05 le 119909 le 1

(28)

In Figure 1 we show that themaximum value of the num-bers of susceptibles and infectious individuals is concentratedat themiddle of the interval [0 1] and these numbers decreaselinearly to zero at the boundaries 119909 = 0 and 119909 = 1

Now we choose the following data set of system (3) 119889119878=

01 119889119868

= 05 Λ = 05 120573 = 02 120583 = 01 1205721

= 011205722= 002 120572

3= 003 119889 = 01 and 119903 = 05 By calculation we

have 1198770= 09524 In this case system (3) has a disease-free

equilibrium 119864119891(5 0) Hence by Theorem 4(i) 119864

119891is globally

asymptotically stable Numerical simulation illustrates ourresult (see Figure 2)

In Figure 3 we choose 120573 = 06 and do not change theother parameter values By calculation we have 119877

0= 28571

which satisfyTheorem4(ii) then the disease-free equilibriumis still present and the system (2) has a unique endemic

equilibrium 119864lowast(13625 05196) Therefore by Theorem 2

and Theorem 4(ii) 119864119891

is unstable while 119864lowast is globally

asymptotically stable Numerical simulation illustrates ourresult (see Figure 3)

5 Conclusion

In this paper we investigated the dynamics of a reaction-diffusion epidemic model with specific nonlinear incidencerateThis specific nonlinear incidence rate includes the tradi-tional bilinear incidence rate the saturated incidence rate theBeddington-DeAngelis functional response and Crowley-Martin functional response The global dynamics of themodel are completely determined by the basic reproductionnumber 119877

0 We proved that the disease-free equilibrium is

globally asymptotically stable if 1198770

le 1 which leads tothe eradication of disease from population When 119877

0gt 1

then disease-free equilibrium becomes unstable and a uniqueendemic equilibrium exists and is globally asymptotically sta-ble which means that the disease persists in the population

From our theoretical and numerical results we concludethat the spatial diffusion has no effect on the stability behavior

6 International Journal of Partial Differential Equations

of equilibria in the case of Neumann conditions and spatiallyconstant coefficients

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the editor and the anony-mous referees for their valuable remarks and commentswhich have led to the improvement of the quality of theirstudy

References

[1] J R Beddington ldquoMutual interference between parasites orpredators and its effect on searching efficiencyrdquo Journal ofAnimal Ecology pp 331ndash341 1975

[2] D L DeAngelis R A Goldsten and R Neill ldquoA model fortrophic interactionrdquo Ecology vol 56 pp 881ndash892 1975

[3] R S Cantrell and C Cosner ldquoOn the Dynamics of Predator-Prey Models with the Beddington-Deangelis FunctionalResponserdquo Journal of Mathematical Analysis and Applicationsvol 257 no 1 pp 206ndash222 2001

[4] P H Crowley and E K Martin ldquoFunctional responses andinterferencewithin and between year classes of a dragonfly pop-ulationrdquo Journal of the North American Benthological Societyvol 8 pp 211ndash221 1989

[5] X Zhou and J Cui ldquoGlobal stability of the viral dynamics withcrowley-martin functional responserdquo Bulletin of the KoreanMathematical Society vol 48 no 3 pp 555ndash574 2011

[6] X Q Liu S M Zhong B D Tian and F X Zheng ldquoAsymptoticproperties of a stochastic predator-prey model with Crowley-Martin functional responserdquo Journal of Applied Mathematicsand Computing vol 43 no 1-2 pp 479ndash490 2013

[7] K Hattaf A A Lashari Y Louartassi and N Yousfi ldquoA delayedSIR epidemic model with general incidence raterdquo ElectronicJournal of QualitativeTheory of Differential Equations vol 3 pp1ndash9 2013

[8] V Capasso Mathematical Structures of Epidemic Systems Lec-ture Notes in Biomathematics Springer Germany 1993

[9] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983

[10] J Smoller Shock Waves and Reaction-Diffusion EquationsSpringer Berlin Germany 1983

[11] M H Protter and H F Weinberger Maximum Principles inDifferential Equations Prentice Hall Englewood Cliffs NJUSA 1967

[12] N D Alikakos ldquoAn application of the invariance principle toreaction-diffusion equationsrdquo Journal of Differential Equationsvol 33 no 2 pp 201ndash225 1979

[13] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer BerlinGermany 1993

[14] K Hattaf and N Yousfi ldquoGlobal stability for reaction-diffusionequations in biologyrdquo Computers and Mathematics with Appli-cations vol 66 pp 1488ndash1497 2013

[15] J P LaSalleThe Stability of Dynamical Systems Regional Con-ference Series in AppliedMathematics SIAM Philadelphia PaUSA 1976

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 International Journal of Partial Differential Equations

of equilibria in the case of Neumann conditions and spatiallyconstant coefficients

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the editor and the anony-mous referees for their valuable remarks and commentswhich have led to the improvement of the quality of theirstudy

References

[1] J R Beddington ldquoMutual interference between parasites orpredators and its effect on searching efficiencyrdquo Journal ofAnimal Ecology pp 331ndash341 1975

[2] D L DeAngelis R A Goldsten and R Neill ldquoA model fortrophic interactionrdquo Ecology vol 56 pp 881ndash892 1975

[3] R S Cantrell and C Cosner ldquoOn the Dynamics of Predator-Prey Models with the Beddington-Deangelis FunctionalResponserdquo Journal of Mathematical Analysis and Applicationsvol 257 no 1 pp 206ndash222 2001

[4] P H Crowley and E K Martin ldquoFunctional responses andinterferencewithin and between year classes of a dragonfly pop-ulationrdquo Journal of the North American Benthological Societyvol 8 pp 211ndash221 1989

[5] X Zhou and J Cui ldquoGlobal stability of the viral dynamics withcrowley-martin functional responserdquo Bulletin of the KoreanMathematical Society vol 48 no 3 pp 555ndash574 2011

[6] X Q Liu S M Zhong B D Tian and F X Zheng ldquoAsymptoticproperties of a stochastic predator-prey model with Crowley-Martin functional responserdquo Journal of Applied Mathematicsand Computing vol 43 no 1-2 pp 479ndash490 2013

[7] K Hattaf A A Lashari Y Louartassi and N Yousfi ldquoA delayedSIR epidemic model with general incidence raterdquo ElectronicJournal of QualitativeTheory of Differential Equations vol 3 pp1ndash9 2013

[8] V Capasso Mathematical Structures of Epidemic Systems Lec-ture Notes in Biomathematics Springer Germany 1993

[9] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983

[10] J Smoller Shock Waves and Reaction-Diffusion EquationsSpringer Berlin Germany 1983

[11] M H Protter and H F Weinberger Maximum Principles inDifferential Equations Prentice Hall Englewood Cliffs NJUSA 1967

[12] N D Alikakos ldquoAn application of the invariance principle toreaction-diffusion equationsrdquo Journal of Differential Equationsvol 33 no 2 pp 201ndash225 1979

[13] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer BerlinGermany 1993

[14] K Hattaf and N Yousfi ldquoGlobal stability for reaction-diffusionequations in biologyrdquo Computers and Mathematics with Appli-cations vol 66 pp 1488ndash1497 2013

[15] J P LaSalleThe Stability of Dynamical Systems Regional Con-ference Series in AppliedMathematics SIAM Philadelphia PaUSA 1976

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of