A remark on stationary distribution of a stochastic SIR epidemic model with double saturated rates

In this paper, we propose and discuss a stochastic SIR epidemic model with saturated treatment and incidence rates. The existence and uniqueness of the global positive solution are achieved. By constructing suitable Lyapunov functions and using Khasminskii's theory, we establish appropriate conditions to prove that the stochastic model has a unique stationary distribution and the ergodicity holds. Moreover, sufficient conditions that guarantee the epidemic disease's extinction are given.