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. 1. 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 is the incidence rate, where are constants. It is very important to note that this incidence rate becomes the bilinear incidence rate if , the saturated incidence rate if or , the modified saturated incidence rate proposed in [2, 3] when , and Crowley-Martin functional response presented in [4–6] 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), are independent standard Brownian motions, and 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. 2. Stability Analysis of Stochastic Model Clearly, the system (1) has a