A stochastic SIRS epidemic model with nonlinear incidence rate

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 R₀<1, the disease dies out; while if R₀>1, the disease persists. For the stochastic model, we show that its related reproduction number R₀^S 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.