The threshold behavior and periodic solution of stochastic SIR epidemic model with saturated incidence

We investigate degenerate stochastic SIR epidemic model with saturated incidence. For the constant coefficients case, we achieve a threshold which determines the extinction and persistence of the epidemic by utilizing Markov semigroup theory. Furthermore, we conclude that environmental white noise plays a positive effect in the control of infectious disease in some sense comparing to the corresponding deterministic system. For the stochastic nonautonomous system, we prove the existence of periodic solution.