On invariant measures and the asymptotic behavior of a stochastic delayed SIRS epidemic model

Abstract

In this paper, we consider a stochastic epidemic model with time delay and general incidence rate. We first prove the existence and uniqueness of the global positive solution. By using the Krylov–Bogoliubov method, we obtain the existence of invariant measures. Furthermore, we study a special case where the incidence rate is bilinear with distributed time delay. When the basic reproduction number R0<1, the analysis of the asymptotic behavior around the disease-free equilibrium E0 is provided while when R0>1, we prove that the invariant measure is unique and ergodic. The numerical simulations also validate our analytical results.