Asymptotic behavior of stochastic multi-group epidemic models with distributed delays

In this paper, we introduce stochasticity into multi-group epidemic models with distributed delays and general kernel functions. The stochasticity in the model is a standard technique in stochastic population modeling. When the perturbations are small, by using the method of stochastic Lyapunov functions, we carry out a detailed analysis on the asymptotic behavior of the stochastic model regarding of the basic reproduction number R0. If R0≤1, the solution of the stochastic system oscillates around the disease-free equilibrium E0, while if R0>1, the solution of the stochastic model fluctuates around the endemic equilibrium E∗. Moreover, we also establish sufficient conditions of these results.