Partial Differential Equations of an Epidemic Model with Spatial Diffusion

The aimof this paper is to study the dynamics of a reaction-diffusion SIR epidemicmodel 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 diseasefree 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.


Introduction
In this paper, we consider the following SIR epidemic model with a specific nonlinear incidence rate described by = Λ − − 1 + 1 + 2 + 3 , where , , and are susceptible, infectious, and recovered classes, respectively. Λ is the recruitment rate of the population, is the natural death rate of the population, is the death rate due to disease, is the recovery rate of the infective individuals, is the infection coefficient, and /(1 + 1 + 2 + 3 ) is the incidence rate, where 1 , 2 , 3 ≥ 0 are constants. It is very important to note that this incidence rate becomes the bilinear incidence rate if 1 = 2 = 3 = 0, the saturated incidence rate if 1 = 3 = 0 or 2 = 3 = 0, the Beddington-DeAngelis functional response introduced in [1,2] and used in [3] when 3 = 0, and Crowley-Martin functional response presented in [4][5][6] if 3 = 1 2 . Moreover, the function /(1 + 1 + 2 + 3 ) satisfies the hypotheses ( 1), ( 2), and ( 3) 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 /(1 + 1 + 2 + 3 ) 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: where ( , ), ( , ), and ( , ) represent the numbers of susceptible, infected, and removed individuals at location and time , respectively. The positive constants , , and 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 does not appear in the first two equations; this allows us to study the system with homogeneous Neumann boundary conditions and initial conditions Here, Ω is a bounded domain in R with smooth boundary Ω. / ] and / ] are, respectively, the normal derivatives of and 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.

Global Existence, Positivity, and Boundedness of Solutions
In this section, we establish the global existence, positivity, and boundedness of solutions of problem (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 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 (Λ/ , 0). Further, if 0 > 1, the system (3) has an endemic stationary state * ( * , * ) where * = 2 ( + 2 Λ) with = + + and = ( − 1 + 2 − 3 Λ) 2 + 4 3 ( + 2 Λ).
The objective of this section is to discuss the local and global stability of the equilibria.

Local Stability of the Equilibria.
First, we linearize the dynamical system (3) around arbitrary spatially homogeneous fixed point ( , ) for small space-and time-dependent fluctuations and expand them in Fourier space. For this, let where ⃗ = ( , ) and ⃗ ⋅ ⃗ := ⟨ ⃗ , ⃗ ⟩ := 2 ; ⃗ and are the wavenumber vector and frequency, respectively. Then we can obtain the corresponding characteristic equation as follows: where 2 is the identity matrix, = diag( , ) is the diffusion matrix, and is the Jacobian matrix of (3) without diffusion ( = = 0) at which is given by The characterization of the local stability of disease-free equilibrium is given by the following result.

Theorem 2.
The disease-free equilibrium is locally asymptotically stable if 0 < 1 and it is unstable if 0 > 1.
Next, we focus on the local stability of the endemic equilibrium * . Theorem 3. The endemic equilibrium * is locally asymptotically stable if 0 > 1.

Global
Stability of the Equilibria. The purpose of this subsection is to determine the global stability for reactiondiffusion 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].
By the above discussion, we deduce that the largest compact invariant set in Γ = {( , ) | ( 1 )/ = 0} is just the singleton . From LaSalle invariance principle [15], we conclude that is globally asymptotically stable. Using same technique, we construct a Lyapunov functional 2 for system (3)-(5) at * from the Lyapunov functional 2 defined by Hattaf et al. in [7]. It is easy to show that 2 verifies the condition (15) given in [14]. Hence, it follows from [14, Proposition 2.1] that 2 is a Lyapunov functional for the reaction-diffusion system (3)-(5) at * when 0 > 1. We summarize the above in the following result. (ii) If 0 > 1, the endemic equilibrium * of (3)-(5) is globally asymptotically stable for all diffusion coefficients.

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 International Journal of Partial Differential Equations    By calculation, we have 0 = 0.9524. In this case, system (3) has a disease-free equilibrium (5, 0). Hence, by Theorem 4(i), is globally asymptotically stable. Numerical simulation illustrates our result (see Figure 2).
In Figure 3, we choose = 0.6 and do not change the other parameter values. By calculation, we have 0 = 2.8571 which satisfy Theorem 4(ii); then the disease-free equilibrium is still present and the system (2) has a unique endemic equilibrium * (1.3625, 0.5196). Therefore, by Theorem 2 and Theorem 4(ii), is unstable, while * is globally asymptotically stable. Numerical simulation illustrates our result (see Figure 3).

Conclusion
In this paper, we investigated the dynamics of a reactiondiffusion 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 number 0 . We proved that the disease-free equilibrium is globally asymptotically stable if 0 ≤ 1,, which leads to the eradication of disease from population. When 0 > 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