The threshold of a stochastic delayed SIR epidemic model with vaccination

In this paper, we study the threshold dynamics of a stochastic delayed SIR epidemic model with vaccination. We obtain sufficient conditions for extinction and persistence in the mean of the epidemic. The threshold between persistence in the mean and extinction of the stochastic system is also obtained. Compared with the corresponding deterministic model, the threshold affected by the white noise is smaller than the basic reproduction number R¯0 of the deterministic system. Results show that time delay has important effects on the persistence and extinction of the epidemic.