A class of stochastic delayed SIR epidemic models with generalized nonlinear incidence rate and temporary immunity

A class of SIR epidemic model with generalized nonlinear incidence rate is presented in this paper. Temporary immunity and stochastic perturbation are also considered. The existence and uniqueness of the global positive solution is achieved. Sufficient conditions guaranteeing the extinction and persistence of the epidemic disease are established. Moreover, the threshold behavior is discussed, and the threshold value R0 is obtained. We show that if R0<1, the disease eventually becomes extinct with probability one, whereas if R0>1, then the system remains permanent in the mean.