# Partial Differential Equations of an Epidemic Model with Spatial Diffusion

Authors: El Mehdi Lotfi; Mehdi Maziane; Khalid Hattaf; Noura Yousfi
Journal: International Journal of Partial Differential Equations
 (2014)

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## Abstract

The aim of this paper is to study the dynamics of a reaction-diffusion SIR epidemic model with specific nonlinear incidence rate. The global existence, positivity, and boundedness of solutions for a reaction-diffusion system with homogeneous Neumann boundary conditions are proved. The local stability of the disease-free equilibrium and endemic equilibrium is obtained via characteristic equations. By means of Lyapunov functional, the global stability of both equilibria is investigated. More precisely, our results show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than or equal to unity, which leads to the eradication of disease from population.
When 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 persists in the population. Numerical simulations are presented to illustrate our theoretical
results.


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## Body

## 1. Introduction

In this paper, we consider the following SIR epidemic model with a specific nonlinear incidence rate described by(1)dSdt=Λ-μS-βSI1+α1S+α2I+α3SI,dIdt=βSI1+α1S+α2I+α3SI-(μ+d+r)I,dRdt=rI-μR,
where S, I, and R are susceptible, infectious, and recovered classes, respectively. Λ is the recruitment rate of the population, μ is the natural death rate of the population, d is the death rate due to disease, r is the recovery rate of the infective individuals, β is the infection coefficient, and βSI/(1+α1S+α2I+α3SI) is the incidence rate, where α1,α2,α3≥0 are constants. It is very important to note that this incidence rate becomes the bilinear incidence rate if α1=α2=α3=0, the saturated incidence rate if α1=α3=0 or α2=α3=0, the Beddington-DeAngelis functional response introduced in [1, 2] and used in [3] when α3=0, and Crowley-Martin functional response presented in [4–6] if α3=α1α2. Moreover, the function βS/(1+α1S+α2I+α3SI) satisfies the hypotheses (H1), (H2), and (H3) of general incidence rate presented by Hattaf et al. in [7]. From the biological point of view, the transmission rate of infectious diseases remains unknown in detail and may be different from one disease to another. In the classical epidemic models, this rate was assumed to be linear with respect to the numbers of susceptible and infected individuals. This assumption is based on the law of mass action which is more appropriate for communicable diseases such as influenza but not for sexually transmitted diseases such as HIV/AIDS. For one reason, the transmission rate in system (1) is assumed to be nonlinear and has the form βI/(1+α1S+α2I+α3SI) that measures the saturation effect which represents that the number of individual contacts reaches a certain maximum value due to social or spatial distribution of the population. For more details on the choice of the nonlinearity of the incidence rate, we refer the reader to the book of Capasso [8].On the other hand, the spatial content of the environment has been ignored in the model (1). However, due to the large mobility of people within a country or even worldwide, spatially uniform models are not sufficient to give a realistic picture of disease diffusion. For this reason, the spatial effects cannot be neglected in studying the spread of epidemics.Therefore, we consider the following SIR epidemic model with specific nonlinear incidence rate and spatial diffusion:(2)∂S∂t=dSΔS+Λ-μS(x,t)-βS(x,t)I(x,t)1+α1S(x,t)+α2I(x,t)+α3S(x,t)I(x,t),∂I∂t=dIΔI+βS(x,t)I(x,t)1+α1S(x,t)+α2I(x,t)+α3S(x,t)I(x,t)-(μ+d+r)I(x,t),∂R∂t=dRΔR+rI(x,t)-μR(x,t),
where S(x,t), I(x,t), and R(x,t) represent the numbers of susceptible, infected, and removed individuals at location x and time t, respectively. The positive constants dS, dI, and dR denote the corresponding diffusion rates for these three classes of individuals.The aim of this work is to investigate the global dynamics of the reaction-diffusion system (2). Note that R does not appear in the first two equations; this allows us to study the system
(3)∂S∂t=dSΔS+Λ-μS(x,t)-βS(x,t)I(x,t)1+α1S(x,t)+α2I(x,t)+α3S(x,t)I(x,t),∂I∂t=dIΔI+βS(x,t)I(x,t)1+α1S(x,t)+α2I(x,t)+α3S(x,t)I(x,t)-(μ+d+r)I(x,t),
with homogeneous Neumann boundary conditions
(4)∂S∂ν=∂I∂ν=0,on⁡∂Ω×(0,+∞),
and initial conditions
(5)S(x,0)=ϕ1(x)≥0,I(x,0)=ϕ2(x)≥0,x∈Ω-.
Here, Ω is a bounded domain in ℝn with smooth boundary ∂Ω. ∂S/∂ν and ∂I/∂ν are, respectively, the normal derivatives of S and I on ∂Ω.The rest of paper is organized as follows. The next section deals with the global existence, positivity, and boundedness of solutions of problem (3)–(5). In Section 3, we discuss the stability analysis of equilibria. In Section 4, we present the numerical simulation to illustrate our result. Finally, the conclusion of our paper is in Section 5.


## 2. Global Existence, Positivity, and Boundedness of Solutions

In this section, we establish the global existence, positivity, and boundedness of solutions of problem (3)–(5) because this model describes the population. Hence, the population should remain nonnegative and bounded.Proposition 1.
For any given initial data satisfying the condition (5), there exists a unique solution of problem (3)–(5) defined on [0,+∞) and this solution remains nonnegative and bounded for all t≥0.Proof.
System (3)–(5) can be written abstractly in the Banach space X=C(Ω-)×C(Ω-) of the form
(6)u′(t)=Au(t)+F(u(t)),t>0,u(0)=u0∈X,
where u=col(S,I), u0=col(ϕ1,ϕ2), Au(t)=col(dSΔS,dIΔI), and
(7)F(u(t))=(Λ-μS-βSI1+α1S+α2I+α3SIβSI1+α1S+α2I+α3SI-(μ+d+r)I).
It is clear that F is locally Lipschitz in X. From [9], we deduce that system (6) admits a unique local solution on [0,Tmax⁡), where Tmax⁡ is the maximal existence time for solution of system (6).
In addition, system (3) can be written in the form
(8)∂S∂t-dSΔS=F1(S,I),∂I∂t-dIΔI=F2(S,I).
It is easy to see that the functions F1(S,I) and F2(S,I) are continuously differentiable satisfying F1(0,I)=Λ≥0 and F2(S,0)=0≥0 for all S,I≥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)–(5) we have
(9)∂S∂t-dSΔS≤Λ-μS,∂S∂ν=0,S(x,0)=ϕ1(x)≤∥ϕ1∥∞=max⁡x∈Ω¯⁡ϕ1(x).
By the comparison principle [11], we have S(x,t)≤S1(t), where S1(t)=ϕ1(x)e-μt+(Λ/μ)(1-e-μt) is the solution of the problem
(10)dS1dt=Λ-μS1,S1(0)=∥ϕ1∥∞.
Since S1(t)≤max⁡{Λ/μ,∥ϕ1∥∞} for t∈[0,∞), we have that
(11)S(x,t)≤max⁡{Λμ,∥ϕ1∥∞},∀(x,t)∈Ω-×[0,Tmax⁡).
From Theorem2 given by Alikakos in [12], to establish the L∞ uniform boundedness of I(x,t), it is sufficient to show the L1 uniform boundedness of I(x,t).
Since∂S/∂ν=∂I/∂ν=0 and (∂/∂t)(S+I)-Δ(dSS+dII)≤Λ-μ(S+I), we get
(12)∂∂t(∫Ω(S+I)dx)≤mes(Ω)Λ-μ(∫Ω(S+I)dx).
Hence,
(13)∫Ω(S+I)dx≤mes(Ω)max⁡{Λμ,∥ϕ1+ϕ2∥∞},
which implies that sup⁡t≥0⁡∫ΩI(x,t)dx≤K:=mes(Ω)max⁡{Λ/μ,∥ϕ1+ϕ2∥∞}. Using [12, Theorem 3.1], we deduce that there exists a positive constant K* that depends on K and on ∥ϕ1+ϕ2∥∞ such that
(14)sup⁡t≥0⁡∥I(·,t)∥∞≤K*.
From the above, we have proved thatS(x,t) and I(x,t) are L∞ bounded on Ω-×[0,Tmax⁡). Therefore, it follows from the standard theory for semilinear parabolic systems (see [13]) that Tmax⁡=+∞. 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 easy to get that the basic reproduction number of disease in the absence of spatial dependence is given by
(15)R0=βΛ(μ+α1Λ)(μ+d+r),
which describes the average number of secondary infections produced by a single infectious individual during the entire infectious period.It is not hard to show that the system (3) is always a disease-free equilibrium of the form Ef(Λ/μ,0). Further, if R0>1, the system (3) has an endemic stationary state E*(S*,I*) where
(16)S*=2(a+α2Λ)β-α1a+α2μ-α3Λ+δ,I*=Λ-μS*a,
with a=μ+d+r and δ=(β-α1a+α2μ-α3Λ)2+4α3μ(a+α2Λ).The objective of this section is to discuss the local and global stability of the equilibria.


### 3.1. Local Stability of the Equilibria

First, we linearize the dynamical system (3) around arbitrary spatially homogeneous fixed point E-(S-,I-) for small space- and time-dependent fluctuations and expand them in Fourier space. For this, let
(17)S(x→,t)~S-eλteik→·x→,I(x→,t)~I-eλteik→·x→,
where x→=(x,y) and k→·k→:=〈k→,k→〉:=k2; k→ and λ are the wavenumber vector and frequency, respectively. Then we can obtain the corresponding characteristic equation as follows:
(18)det⁡(J-k2D-λI2)=0,
where I2 is the identity matrix, D=diag⁡(dS,dI) is the diffusion matrix, and J is the Jacobian matrix of (3) without diffusion (dS=dI=0) atE-which is given by
(19)J=(-μ-βI¯(1+α2I¯)(1+α1S¯+α2I¯+α3S-I-)2-βS¯(1+α1S¯)(1+α1S¯+α2I¯+α3S-I-)2βI¯(1+α2I¯)(1+α1S¯+α2I¯+α3S-I-)2βS¯(1+α1S¯)(1+α1S¯+α2I¯+α3S-I-)2-a).
The characterization of the local stability of disease-free equilibrium Ef is given by the following result.Theorem 2.
The disease-free equilibriumEf is locally asymptotically stable if R0<1 and it is unstable if R0>1.Proof.
Evaluating (18) at Ef, we have
(20)(μ+k2dS+λ)(a(R0-1)-k2dI-λ)=0.
Clearly, the roots of (20) are λ1(k)=-(μ+k2dS)<0 and λ2(k)=a(R0-1)-k2dI. Note that λ2(k) is negative if R0<1 for all k. Hence Ef is locally asymptotically stable if R0<1. If R0>1, λ2(0) is positive. So Ef is unstable.Next, we focus on the local stability of the endemic equilibriumE*.Theorem 3.
The endemic equilibriumE* is locally asymptotically stable if R0>1.Proof.
Evaluating (18) at E*(S*,I*), we have
(21)λ2+a1(k)λ+a2(k)=0,
where
(22)a1(k)=μ+k2(dS+dI)+a(I*(1+α2I*)(1+α1S*+α2I*+α3S*I*)S*+α2I*+α3S*I*1+α1S*+α2I*+α3S*I*),a2(k)=(μ+βI*(1+α2I*)(1+α1S*+α2I*+α3S*I*)2+k2dS)×(k2dI+βS*(α2I*+α3S*I*)(1+α1S*+α2I*+α3S*I*)2)+β2S*I*(1+α2I*)(1+α1S*)(1+α1S*+α2I*+α3S*I*)4.
We have a1>0 and a2>0; then E* is locally asymptotically stable.


### 3.2. Global Stability of the Equilibria

The purpose of this subsection is to determine the global stability for reaction-diffusion equations (3)–(5) by constructing Lyapunov functionals. These Lyapunov functionals are obtained from those for ordinary differential equations (1) by applying the method of Hattaf and Yousfi presented in [14].The system (1) is particular case of the model proposed by Hattaf et al. [7] with f(S,I)=βS/(1+α1S+α2I+α3SI). To study the global stability of Ef for (1), the authors Hattaf et al. [7] proposed the following Lyapunov functional:
(23)V1=S-S0-∫S0Sf(S0,0)f(X,0)dX+I,
where S0=Λ/μ.From [14], we construct the Lyapunov functional for system (3)–(5) at Ef as follows:
(24)W1=∫ΩV1(S(x,t),I(x,t))dx.
Calculating the time derivative of W1 along the solution of system (3)–(5), we have
(25)dW1dt=∫Ω{-μ(S-S0)2S(1+α1S0)+aI(1+α1S1+α1S+α2I+α3SIR0-1)μ(S-S0)2S(1+α1S0)}dx-dSS01+α1S0∫Ω|∇S|2S2dx≤∫Ω{-μ(S-S0)2S(1+α1S0)+aI(R0-1)}dx-dSS01+α1S0∫Ω|∇S|2S2dx.
Since R0≤1, we have dW1/dt≤0. Thus, the disease-free equilibrium Ef is stable, and dW1/dt=0 if and only if S=S0 and I(R0-1)=0. We discuss two cases as follows.(i)
IfR0<1, then I=0.(ii)
IfR0=1, from S=S0 and the first equation of (3), we have
(26)∂S∂t=∂S0∂t=dSΔS0+Λ-μS0-f(S0,I)I=0.
Then, f(S0,I)I=(βS0I)/(1+α1S0+α2I+α3S0I)=0. Since β>0 and S0>0, then I=0. By the above discussion, we deduce that the largest compact invariant set in Γ={(S,I)∣(dW1)/dt=0} is just the singleton Ef. From LaSalle invariance principle [15], we conclude that Ef is globally asymptotically stable.Using same technique, we construct a Lyapunov functionalW2 for system (3)–(5) at E* from the Lyapunov functional V2 defined by Hattaf et al. in [7]. It is easy to show that V2 verifies the condition (15) given in [14]. Hence, it follows from [14, Proposition 2.1] that W2 is a Lyapunov functional for the reaction-diffusion system (3)–(5) at E* when R0>1. We summarize the above in the following result.Theorem 4.
(i) IfR0≤1, the disease-free equilibrium Ef of (3)–(5) is globally asymptotically stable for all diffusion coefficients.
(ii) IfR0>1, the endemic equilibrium E* of (3)–(5) is globally asymptotically stable for all diffusion coefficients.


## 3.1. Local Stability of the Equilibria

First, we linearize the dynamical system (3) around arbitrary spatially homogeneous fixed point E-(S-,I-) for small space- and time-dependent fluctuations and expand them in Fourier space. For this, let
(17)S(x→,t)~S-eλteik→·x→,I(x→,t)~I-eλteik→·x→,
where x→=(x,y) and k→·k→:=〈k→,k→〉:=k2; k→ and λ are the wavenumber vector and frequency, respectively. Then we can obtain the corresponding characteristic equation as follows:
(18)det⁡(J-k2D-λI2)=0,
where I2 is the identity matrix, D=diag⁡(dS,dI) is the diffusion matrix, and J is the Jacobian matrix of (3) without diffusion (dS=dI=0) atE-which is given by
(19)J=(-μ-βI¯(1+α2I¯)(1+α1S¯+α2I¯+α3S-I-)2-βS¯(1+α1S¯)(1+α1S¯+α2I¯+α3S-I-)2βI¯(1+α2I¯)(1+α1S¯+α2I¯+α3S-I-)2βS¯(1+α1S¯)(1+α1S¯+α2I¯+α3S-I-)2-a).
The characterization of the local stability of disease-free equilibrium Ef is given by the following result.Theorem 2.
The disease-free equilibriumEf is locally asymptotically stable if R0<1 and it is unstable if R0>1.Proof.
Evaluating (18) at Ef, we have
(20)(μ+k2dS+λ)(a(R0-1)-k2dI-λ)=0.
Clearly, the roots of (20) are λ1(k)=-(μ+k2dS)<0 and λ2(k)=a(R0-1)-k2dI. Note that λ2(k) is negative if R0<1 for all k. Hence Ef is locally asymptotically stable if R0<1. If R0>1, λ2(0) is positive. So Ef is unstable.Next, we focus on the local stability of the endemic equilibriumE*.Theorem 3.
The endemic equilibriumE* is locally asymptotically stable if R0>1.Proof.
Evaluating (18) at E*(S*,I*), we have
(21)λ2+a1(k)λ+a2(k)=0,
where
(22)a1(k)=μ+k2(dS+dI)+a(I*(1+α2I*)(1+α1S*+α2I*+α3S*I*)S*+α2I*+α3S*I*1+α1S*+α2I*+α3S*I*),a2(k)=(μ+βI*(1+α2I*)(1+α1S*+α2I*+α3S*I*)2+k2dS)×(k2dI+βS*(α2I*+α3S*I*)(1+α1S*+α2I*+α3S*I*)2)+β2S*I*(1+α2I*)(1+α1S*)(1+α1S*+α2I*+α3S*I*)4.
We have a1>0 and a2>0; then E* is locally asymptotically stable.


## 3.2. Global Stability of the Equilibria

The purpose of this subsection is to determine the global stability for reaction-diffusion equations (3)–(5) by constructing Lyapunov functionals. These Lyapunov functionals are obtained from those for ordinary differential equations (1) by applying the method of Hattaf and Yousfi presented in [14].The system (1) is particular case of the model proposed by Hattaf et al. [7] with f(S,I)=βS/(1+α1S+α2I+α3SI). To study the global stability of Ef for (1), the authors Hattaf et al. [7] proposed the following Lyapunov functional:
(23)V1=S-S0-∫S0Sf(S0,0)f(X,0)dX+I,
where S0=Λ/μ.From [14], we construct the Lyapunov functional for system (3)–(5) at Ef as follows:
(24)W1=∫ΩV1(S(x,t),I(x,t))dx.
Calculating the time derivative of W1 along the solution of system (3)–(5), we have
(25)dW1dt=∫Ω{-μ(S-S0)2S(1+α1S0)+aI(1+α1S1+α1S+α2I+α3SIR0-1)μ(S-S0)2S(1+α1S0)}dx-dSS01+α1S0∫Ω|∇S|2S2dx≤∫Ω{-μ(S-S0)2S(1+α1S0)+aI(R0-1)}dx-dSS01+α1S0∫Ω|∇S|2S2dx.
Since R0≤1, we have dW1/dt≤0. Thus, the disease-free equilibrium Ef is stable, and dW1/dt=0 if and only if S=S0 and I(R0-1)=0. We discuss two cases as follows.(i)
IfR0<1, then I=0.(ii)
IfR0=1, from S=S0 and the first equation of (3), we have
(26)∂S∂t=∂S0∂t=dSΔS0+Λ-μS0-f(S0,I)I=0.
Then, f(S0,I)I=(βS0I)/(1+α1S0+α2I+α3S0I)=0. Since β>0 and S0>0, then I=0. By the above discussion, we deduce that the largest compact invariant set in Γ={(S,I)∣(dW1)/dt=0} is just the singleton Ef. From LaSalle invariance principle [15], we conclude that Ef is globally asymptotically stable.Using same technique, we construct a Lyapunov functionalW2 for system (3)–(5) at E* from the Lyapunov functional V2 defined by Hattaf et al. in [7]. It is easy to show that V2 verifies the condition (15) given in [14]. Hence, it follows from [14, Proposition 2.1] that W2 is a Lyapunov functional for the reaction-diffusion system (3)–(5) at E* when R0>1. We summarize the above in the following result.Theorem 4.
(i) IfR0≤1, the disease-free equilibrium Ef of (3)–(5) is globally asymptotically stable for all diffusion coefficients.
(ii) IfR0>1, the endemic equilibrium E* of (3)–(5) is globally asymptotically stable for all diffusion coefficients.


## 4. Numerical Simulations

In this section, we present the numerical simulations to illustrate our theoretical results. To simplify, we consider system (3) under Neumann boundary conditions
(27)∂S∂ν=∂I∂ν=0,t>0,x=0,1,
and initial conditions
(28)S(x,0)={1.1x,0≤x<0.5,1.1(1-x),0.5≤x≤1,I(x,0)={0.5x,0≤x<0.5,0.5(1-x),0.5≤x≤1.In Figure1, we show that the maximum value of the numbers of susceptibles and infectious individuals is concentrated at the middle of the interval [0,1] and these numbers decrease linearly to zero at the boundaries x=0 and x=1.Figure 1
The initial distributions of the numbers of susceptibles and infectious individuals for Figures2 and 3.Now, we choose the following data set of system (3): dS=0.1, dI=0.5, Λ=0.5, β=0.2, μ=0.1, α1=0.1, α2=0.02, α3=0.03, d=0.1 and r=0.5. By calculation, we have R0=0.9524. In this case, system (3) has a disease-free equilibrium Ef(5,0). Hence, by Theorem 4(i), Ef is globally asymptotically stable. Numerical simulation illustrates our result (see Figure 2).Figure 2
The temporal solution found by numerical of problem (3) with the Neumann boundary conditions (27) and initial conditions (28).In Figure3, we choose β=0.6 and do not change the other parameter values. By calculation, we have R0=2.8571 which satisfy Theorem 4(ii); then the disease-free equilibrium is still present and the system (2) has a unique endemic equilibrium E*(1.3625,0.5196). Therefore, by Theorem 2 and Theorem 4(ii), Ef is unstable, while E* is globally asymptotically stable. Numerical simulation illustrates our result (see Figure 3).Figure 3
The temporal solution found by numerical of problem (3) with the Neumann boundary conditions (27) and initial conditions (28).


## 5. Conclusion

In this paper, we investigated the dynamics of a reaction-diffusion epidemic model with specific nonlinear incidence rate. This specific nonlinear incidence rate includes the traditional bilinear incidence rate, the saturated incidence rate, the Beddington-DeAngelis functional response, and Crowley-Martin functional response. The global dynamics of the model are completely determined by the basic reproduction numberR0. We proved that the disease-free equilibrium is globally asymptotically stable if R0≤1,, which leads to the eradication of disease from population. When R0>1 then disease-free equilibrium becomes unstable and a unique endemic equilibrium exists and is globally asymptotically stable, which means that the disease persists in the population.From our theoretical and numerical results, we conclude that the spatial diffusion has no effect on the stability behavior of equilibria in the case of Neumann conditions and spatially constant coefficients.



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