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Dynamics of a Stochastic SIR Epidemic Model with Logistic Growth
  
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KeyWord:Logistic growth  Saturated treatment  Stationary distribution and ergodicity  Non-monotone incidence  Extinction
Author NameAffiliation
Yubo Liu College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China 
Jianli Li College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China 
Daipeng Kuang College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China 
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Abstract:
      In this paper, a stochastic SIR epidemic model with saturated treatment function, non-monotone incidence rate and logistic growth is studied. First, we prove that the stochastic model has a unique global positive solution. Next, by constructing a suitable Lyapunov function, we can show that there exists an ergodic stationary distribution in the random SIR model. Then, we show that a sufficient condition can make the disease tend to extinction. Finally, some numerical simulations are used to prove our analytical result.