A stochastic SIRS epidemic model with nonlinear incidence rate

A stochastic SIRS model with ratio-dependent incidence rate is developed.The global dynamics of the deterministic model is shown.The stochastic dynamics of the SDE model is given.The existence of a unique stationary distribution of the SDE model is displayed. In this paper, we investigate the global dynamics of a general SIRS epidemic model with a ratio-dependent incidence rate and its corresponding stochastic differential equation version. For the deterministic model, we show that the basic reproduction number R0 determines whether there is an endemic outbreak or not: if R01, the disease persists. For the stochastic model, we show that its related reproduction number R0S can determine whether there is a unique disease-free stationary distribution or a unique endemic stationary distribution. In addition, we provide analytic results regarding the stochastic boundedness and permanence/extinction. One of the most interesting findings is that random fluctuations introduced in our stochastic model can suppress disease outbreak, which can provide us some useful control strategies to regulate disease dynamics.