Dynamical behaviors of a stochastic periodic SIRS epidemic model with time delay are investigated. By constructing suitable Lyapunov functions and applying Itô’s formula, the existence of the global positive solution and the property of stochastically ultimate boundedness of model (1.1) are proved. Moreover, the extinction and the persistence of the disease are established. The results are verified by numerical simulations.

In this paper, we consider a stochastic delayed SIRS epidemic model with seasonal variation. Firstly, we prove that the system is mathematically and biologically well-posed by showing the global existence, positivity and stochastically ultimate boundneness of the solution. Secondly, some sufficient conditions on the permanence and extinction of the positive solutions with probability one are presented. Thirdly, we show that the solution of the system is asymptotical around of the disease-free periodic solution and the intensity of the oscillation depends of the intensity of the noise. Lastly, the existence of stochastic nontrivial periodic solution for the system is obtained.