Long-time behavior of a stochastic epidemic model with varying population size

In this paper we investigate the persistence and extinction of a stochastic epidemic model with a varying population environment in the long-term behavior. Our model consists of two stochastic differential equations: one for the susceptible individuals in which the transmission rate is disturbed by white noise, one for the exposed individuals in which the same perturbation occurs, and one ordinary differential equation in which describes the infective individuals in a varying population environment. We derive sufficient conditions for the extinction and persistence of the epidemic model depending on the constant contact rate. Moreover, we carry out several numerical simulations to illustrate the main results of this contribution.