Persistence and distribution of a stochastic susceptible-infected-removed epidemic model with varying population size

In this paper, the dynamics of a stochastic susceptible-infected-removed model in a population with varying size is investigated. We firstly show that the stochastic epidemic model has a unique global positive solution with any positive initial value. Then we verify that random perturbations lead to extinction when some conditions are being valid. Moreover, we prove that the solution of the stochastic epidemic model is persistent in the mean by building up a suitable Lyapunov function and using generalized Itô's formula. Further, the stochastic epidemic model admits a stationary distribution around the endemic equilibrium when parameters satisfy some sufficient conditions. To end this contribution and to check the validity of the main results, numerical simulations are separately carried out to illustrate these results.