Stability Analysis of a Stochastic SIR Epidemic Model with Specific Nonlinear Incidence Rate

We investigate a stochastic SIR epidemic model with specific nonlinear incidence rate. The stochastic model is derived from the deterministic epidemic model by introducing random perturbations around the endemic equilibrium state. The effect of random perturbations on the stability behavior of endemic equilibrium is discussed. Finally, numerical simulations are presented to illustrate our theoretical results.


Introduction
Many mathematical models have been developed in order to understand disease transmissions and behavior of epidemics.One of the earliest of these models was used by Kermack and Mckendrick [1], by considering the total population into three classes, namely, susceptible () individuals, infected () individuals, and recovered () individuals which is known to us as SIR epidemic model.This SIR epidemic model is very important in today's analysis of diseases.
The disease transmission process is unknown in detail.However, several authors proposed different forms of incidences rate in order to model this disease transmission process.In this paper, we consider the following model with specific nonlinear incidence rate: where  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 modified saturated incidence rate proposed in [2,3] when  3 = 0, and Crowley-Martin functional response presented in [4][5][6] if On the other hand, environmental fluctuations have great influence on all aspects of real life.The aim of this work is to study the effect of these environmental fluctuations on the model (1).We assume that the stochastic perturbations are of white noise type and that they are proportional to the distances of  and , respectively.Then, the system (1) will be extended to the following system of stochastic differential equation: where  * ,  * are the positive points of equilibrium for the corresponding deterministic system (1),   ( = 1, 2) are International Journal of Stochastic Analysis independent standard Brownian motions, and  2  ( = 1, 2) represent the intensities of   , respectively.
The rest of paper is organized as follows.In the next section, we present the stability analysis of our stochastic model (2).In Section 3, we present the numerical simulation to illustrate our result.The conclusion of our paper is in Section 4.

Stability Analysis of Stochastic Model
Clearly, the system (1) has a basic reproduction number given by Using the results presented by Hattaf et al. in [7], it is easy to show that if  0 ≤ 1, the system (1) has just one diseasefree equilibrium   (/, 0) which is globally asymptotically stable; otherwise, if  0 > 1, the disease-free equilibrium   is still present and is unstable, but there is also a unique positive endemic equilibrium  * ( * ,  * ), where This endemic equilibrium is globally asymptotically stable.
The system (2) has the same equilibria as the system (1).We assume that  0 ≤ 1, and we discuss the stability of the endemic equilibrium  * of (2).The stochastic system (2) can be centered at its interior endemic equilibrium  * by the changes of the variables as follows: Hence, the linearized version corresponding to the stochastic model ( 2) around  * is given by the following form: where and the superscript "" represents transposition.
According to [8], we have the following theorem.
From Theorem 1, we get the conditions for stochastic asymptotic stability of trivial solution of (5) which are given by the following theorem.
Proof.We consider the following Lyapunov function: where  1 and  2 are nonnegative constants that will be chosen later.It is easy to verify that inequality (8) holds true with  = 2.
By applying the operator  on (, ), we get where From the assumptions of the theorem, we deduce that   > 0,  = 1, 2 and || > 0. Hence,  is a symmetric positive definite matrix.Let   denote the minimum of its two positive eigenvalues  1 and  2 ; then, we can easily get According to Theorem 1, we conclude that the trivial solution of system ( 5) is globally asymptotically stable.

Conclusion
The purpose of this work is to study the effects of the environmental fluctuations on dynamical behavior of a deterministic SIR epidemic model with specific nonlinear incidence rate by considering the white noise perturbation around the endemic equilibrium state.We have shown that our stochastic model is globally asymptotically stable in probability when the intensities of white noise are less than certain threshold of parameters.However, if these intensities of white noise are zero, which meant that there is no environmental stochastic perturbation, then the conditions of Theorem 2 are reduced to the condition  0 > 1, which gives a nonlinear stability condition for the deterministic model (1).
From our analytical and numerical results, we conclude that the main factor that affects the stability of the stochastic model is the intensities of white noise.In addition, our main results extend the corresponding results in paper [3] and those in [9] when the value of the parameter  ℎ  is equal to one into the stochastic model [9].