In this article, a stochastic SIR epidemic model with saturated treatment function, non-monotone incidence rate and logistic growth is studied. Firstly, we prove that the stochastic model has a unique global positive solution. Next, by constructing a suitable Lyapunov function, we can show that the random SIR model exists an ergodic stationary distribution. Then, we show that a sufficient condition can make the disease tend to extinction. Finally, some numerical simulations were used to prove our analytical result.