The Influence Of Environmental Noise On Epidemic Models

In the realistic ecological environment,the environmental noise exists everywhere.Therefore,introducing stochasticity into the corresponding deterministic models and studying the influence of random factor on epidemic models can be important for predicting the development of the infectious diseases.In this paper,we study an SIRS epidemic model with non-monotone incidence and a model of SEIS with varying population size under random perturbation.According to the stochastic Lyapunov functional method,we show that stochastic model possesses non-negative solutions.Then,we obtain that the extinction and persistence when subject to some simple conditions on some of parameters,respectively.We also give the asymptotic behavior of the stochastic model.Furthermore,based on the theory of Hasminskii,we show that there is a stationary distribution to stochastic model,and it is ergodic.The above research results suggest that,compared with the corresponding deterministic systems,the dynamical properties of the systems have some changes.When the noise is small,the systems still maintain the similar properties of the corresponding deterministic models.While when noise is large,the dynamical characteristics of stochastic models are different from the corresponding deterministic models.