Asymptotic Dynamics of a Stochastic SIR Epidemic System Affected by Mixed Nonlinear Incidence Rates

This paper considers a stochastic SIR epidemic system affected by mixed nonlinear incidence rates. Using Markov semigroup theory and the Fokker–Planck equation, we explore the asymptotic dynamics of the stochastic system. We first investigate the existence of a positive solution and its uniqueness. Furthermore, we prove that the stochastic system has an asymptotically stable stationary distribution. In addition, the sufficient conditions for disease extinction are also obtained, which imply that the white noise can suppress and control the spread of infectious diseases. Finally, in order to illustrate the analytical results, we give some numerical simulations.