Survival and stationary distribution of a SIR epidemic model with stochastic perturbations

In this paper, the dynamics of a SIR epidemic model is investigated. First, we show that the system admits a unique positive global solution starting from the positive initial value. Then, when R0>1R0>1, we show that the stochastic model has a stationary distribution under certain parametric restrictions. In particular, we show that random effects may lead the disease to extinction in scenarios where the deterministic model predicts persistence. When R0⩽1R0⩽1, a result on fluctuation of the solution around the disease-free equilibrium of deterministic system is established under suitable conditions. Finally, numerical simulations are carried out to illustrate the theoretical results.