Analysis Of Dynamics Of Stochastic Epidemic Models With Delays

Epidemic dynamics studies the epidemic law of infectious disease quantitative-ly by studying the dynamic behavior of the model and analyzing the development process of the disease.However,on the one hand,the transmission of diseases in nature is inevitably influenced by random environmental factors.In addition,infec-tious disease systems are often subject to some severe shocks,leading to a sudden jump in the mode and regularity of transmission of infectious diseases.Therefore,it is necessary to study epidemic systems driven by white noise and jump noise.On the other hand,the law of transmission of infectious diseases is not only re-lated to the current state,but also to the historical state.At this time,using the original differential equation model can not well describe the internal law of the development of infectious diseases.Therefore,it is more practical to describe the transmission mechanism of infectious diseases by stochastic differential equation model with time delay.In this paper,we mainly consider the dynamics of epidem-ic models with time delay affected by environmental noise.The main contents are as follows:1.Asymptotic behavior of stochastic SIR and SEIR epidemic models with nonlinear incidence rates and delay.By constructing appropriate Lyapunov func-tionals,we study the asymptotic behavior of the disease-free equilibrium and the endemic equilibrium of stochastic delayed SIR and SEIR epidemic models.2.Dynamical behavior of stochastic SIR and SVEIR epidemic models with distributed delay subject to system perturbation.For the SIR model with degen-erate diffusion,by using Markov semigroup theory,we obtain that the solution of the stochastic system with infinitely distributed delay can converge in~1to an ergodic stationary distribution.For the stochastic SVEIR model with non degen-erate diffusion and distributed delay,applying Khasminskii's ergodicity theory and stochastic Lyapunov function method,we obtain the existence and uniqueness of an ergodic stationary distribution.3.Dynamic behavior of stochastic SIR epidemic models with L?evy noise,vacci-nation and delay.For the stochastic SIR model with disease transmission coefficient affected by L?evy noise and white noise and delay,we study the persistence in the mean and extinction of the model.The results show that the persistence and ex-tinction of the disease have a very closed relationship with the intensity of L?evy noise and the validity period of the vaccination.For the stochastic SIR model with double diseases and delay for system affected by L?evy noise and white noise,we establish sufficient conditions for extinction and persistence in the mean of two dis-eases.The results show that:(i)delay and L?evy noise have very important effects on the persistence and extinction of the diseases;(ii)the two diseases can coexist under certain conditions.The results obtained in this paper greatly extend the research results on the dynamic behavior of deterministic epidemic models with delays,which enables us to have a deeper understanding of the dynamic behavior of stochastic epidemic models with delays.From this point of view,the conclusions obtained in this paper are very meaningful.