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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 R0 is a threshold that completely determines the persistence or extinction of the disease. By using Lyapunov function techniques, we show that if R0>1, then the disease will prevail, the infective condition persists and the endemic state is asymptotically stable in a feasible region. If R0⩽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 R0, when R0>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.

Dynamics of deterministic and stochastic multi-group MSIRS epidemic models with varying total population size