Two-group SIR epidemic model with stochastic perturbation

A stochastic two-group SIR model is presented in this paper. The existence and uniqueness of its nonnegative solution is obtained, and the solution belongs to a positively invariant set. Furthermore, the globally asymptotical stability of the disease-free equilibrium is deduced by the stochastic Lyapunov functional method if R ₀ ≤ 1, which means the disease will die out. While if R ₀ > 1, we show that the solution is fluctuating around a point which is the endemic equilibrium of the deterministic model in time average. In addition, the intensity of the fluctuation is proportional to the intensity of the white noise. When the white noise is small, we consider the disease will prevail. At last, we illustrate the dynamic behavior of the model and their approximations via a range of numerical experiments.