•A Stochastic SIRS epidemic model with logistic growth and nonlinear incidence is probed.•A critical parameter R0s for the ergodicity is presented.•A unique stationary distribution exists under certain criteria.•The existence of a unique stationary distribution implies stochastic weak stability.•Sufficient conditions for the extinction of the disease are obtained. A stochastic SIRS model with logistic growth and nonlinear incidence rate is probed in this paper. We exemplify that the proposed stochastic SIRS model reveals a global and positive solution. By applying suitable Lyapunov functions, the sufficient conditions for the existence of ergodic stationary distribution of the solution to the stochastic SIRS model are derived. Furthermore, we acquire the sufficient conditions for extinction of the infectious disease.