The Dynamics Of Stochastic Epidemic Models With Impulsive Effect

Recently,deterministic impulsive epidemic models have been widely studied and excellent works have been obtained.The study of impulsive epidemic models and impulsive eco-epidemic models can provide deep insight for people to understand the dynamic behavior of diseases and help to make and assess the prevention and control strategies of infectious diseases.However,in the real world,the spread of infectious diseases are inevitably affected by random noises.Therefore,it is important to study the dynamic behavior of infectious disease model and eco-epidemic model under the disturbances of environmental noise and pulses.1.Dynamic behavior of stochastic SIR epidemic models with pulse vaccination.We consider the stochastic models is influenced by linear perturbation,or with incidence rate perturbed by white noise.First,we prove the existence and uniqueness of the global positive solution by constructing the equivalent system without impulse.Second,based on the equivalent system,we obtain a sufficient condition which determines epidemic to go extinct or persist.Then we demonstrate the existence and global attraction of the boundary periodic solution under the certain condition.Finally,we present a sufficient condition which allows the existence of a positive periodic solution.2.Dynamic behavior of stochastic SEIR epidemic models with pulse vaccination.We consider a general nonlinear incidence rate and a specific nonlinear incidence rate respectively.First,we prove the existence and uniqueness of the global positive solution by constructing the equivalent system without impulse.Then we obtain a sufficient condition for the extinction of disease,which means large white noises help wipe out disease.Finally,we establish a sufficient condition which allows the existence of a positive periodic solution by Khasminskii's theory for periodic Markov processes.3.Dynamic behavior of impulsive stochastic eco-epidemic models.We consider two stochastic eco-epidemic models.For the first model,we obtain sufficient conditions which determine the species to go extinct or persist and derive the existence of a positive periodic solution.For the second model,we demonstrate sufficient conditions of the extinction and persistence in the mean for the species.Above studies show that white noises make an impact on the dynamics of impulsive stochastic epidemic models and impulsive stochastic eco-epidemic models.When the white noise is small,the stochastic systems will have the properties similar to the corresponding deterministic systems,such as periodicity;when the white noise is large,it will lead to the extinction of disease or species.The results of these studies can provide an insight into the dynamics of impulsive stochastic epidemic models.In this sense,our results are interesting.