Abstract
In this paper, we consider a stochastic SIR epidemic model with Logistic birth. By using the stochastic Lyapunov function method, we show that the stochastic basic reproduction number R0S can be used to determine the threshold dynamics of the stochastic system. If R0S>1, we establish sufficient conditions for the existence of a stationary distribution of the positive solutions to the model. While if R0S<1, under some extra conditions, we obtain sufficient conditions for extinction of the disease. Finally, some examples and numerical simulations are provided to illustrate the theoretical results.
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