摘要：In this paper, we extend the deterministic single-group MSIRS epidemic model to a multi-group model, and we also extend the deterministic multi-group framework to a stochastic one and formulate it as a stochastic differential equation. In the deterministic multi-group model, the basic reproduction number R 0 is a threshold that completely determines the persistence or extinction of the disease. By using Lyapunov function techniques, we show that if R 0 > 1 , then the disease will prevail, the infective condition persists and the endemic state is asymptotically stable in a feasible region. If R 0 ⩽ 1 , then the infective condition disappears and the disease dies out. For the stochastic version, we perform a detailed analysis on the asymptotic behavior of the stochastic model, which also depends on the value of R 0 , when R 0 > 1 , we determine the asymptotic stability of the endemic equilibrium by measuring the difference between the solution and the endemic equilibrium of the deterministic model in time-averaged data. Numerical methods are used to illustrate the dynamic behavior of the model and to solve the systems.
关键词：multi-group MSIRS model ; stochastic perturbation ; graph theory ; Brownian motion
