The Behavior of an SIR Epidemic Model with Stochastic Perturbation

In this article, we discuss an SIR model with stochastic perturbation. We show that there is a nonnegative solution that belongs to a positively invariant set. Then, by stochastic Lyapunov functional methods, we deduce the globally asymptotical stability and exponential meansquare stability of the disease-free equilibrium under some conditions, which means the disease will die out. Comparing with the deterministic model, there is no endemic equilibrium. To show when the disease will prevail, we investigate the asymptotic behavior of the solution around the endemic equilibrium of the deterministic model. Last, we illustrate the dynamic behavior of the model and their approximations via a range of numerical experiments.