In this paper, we study stochastic SIR and SIRS epidemic models with delay. A nonlinear incidence function that includes some special incidence rates is also considered. Two thresholds RS0 and R ̃S0 of the two models are derived by using the nonnegative semimartingale convergence theorem. The disease goes extinct when the value of RS0 is below 1 and it prevails when R ̃S0 value is above 1 for any size of the white noise. The comparison between the two thresholds is made.

delays, stochastic SIR model, nonlinear incidence, extinction, persistence in mean