Dynamical behaviors of a stochastic SIRS epidemic model

Abstract

In this paper, we study the dynamical behavior of a stochastic SIRS epidemic model with specific nonlinear incidence rate and vaccination. We show the existence and positivity of the solution of the SIRS stochastic differential equation. We defined a number $\mathcal{R}$ and we prove the disease free equilibrium is almost sure exponentially stable if $\mathcal{R}<1$. We studying the behavior around the endemic equilibrium E*. Numerical simulations presented our theoretical results.