We discuss in this paper a deterministic multi-group MSIR epidemic model with a vaccination rate, the basic reproduction number Ro, a key parameter in epidemiology, is a threshold which determines the persistence or extinction of the disease. By using Lyapunov function techniques, we show if Ro is greater than 1 and the deterministic model obeys some conditions, then the disease will prevail, the infective persists and the endemic state is asymptotically stable in a feasible region. If Ro is less than or equal to 1, then the infective disappear so the disease dies out. In addition, stochastic noises around the endemic equilibrium will be added to the deterministic MSIR model in order that the deterministic model is extended to a system of stochastic ordinary differential equations. In the stochastic version, we carry out a detailed analysis on the asymptotic behavior of the stochastic model. In addition, regarding the value of Ro, when the stochastic system obeys some conditions and Ro is greater than 1, we deduce the stochastic system is stochastically asymptotically stable. Finally, the deterministic and stochastic model dynamics are illustrated through computer simulations.
