Dynamics Of A Stochastic SIRI Epidemic Model With Nonmonotone Incidence Rate

In this paper,we discuss the dynamics of a stochastic susceptible-infective-removed-infective(SIRI)epidemic model with nonmonotone incidence rate.The specific content is mainly introduced through the following two aspects.In the first part,a stochastic SIRI epidemic model affected by white noise is considered.Firstly,we derive the existence and uniqueness of the global positive solution.Then,we give sufficient conditions of extinction of the disease by constructing a suitable stochastic Lyapunov function.We also investigate the dynamical properties of the solution around both disease-free and endemic equilibria of the deterministic model.In addition,we study the existence of a stationary distribution.Finally,numerical simulations are given to verify the correctness of our theoretical results.In the second part,we consider a stochastic SIRI epidemic model with white noise and Lévy noise.To begin with,we show the existence and uniqueness of the global positive solution,and get sufficient conditions for extinction of the disease.We also study the dynamic properties of the solution around both disease-free and endemic equilibria of the deterministic model.At last,we carry out numerical simulations to illustrate our theoretical results.