Stationary Distribution And Extinction Of Epidemic Model With Nonlinear Incident Under Markov Switching

It is a more effective way that by establishing mathematical model to study the human's transmission mechanism of infectious diseases and putting forward reasonable prevention strategies.However,the spread of disease will inevitably be affected by environmental noise,In order to characterize this effect,scholars have established stochastic differential equations to simulate the epidemic model under the disturbance of white noise,These models clarified the transmission mechanism of infectious diseases from different perspectives.However,the spread of infectious disease is also affected by the noise of large disturbance.Hence,introducing the strong environmental noise into mathematical model can reflect the reality more.Although there are many researches have studied the dynamic model of stochastic infectious diseases,few researches study the stochastic epidemic model with general incidence rate based on Markovian Switching.In view of this,two kinds of stochastic infectious diseases models with Markovian Switching are considered in this paper.In Chapter three,the stochastic SIRS model with double saturated rates based on Markovian Switching is studied.Firstly,the existence and uniqueness of the global positive solution of the model is proved by constructing Lyapunov functions.Then,the sufficient condition for the extinction of the disease are given by using the generalized It(?) formula;Thirdly,by using Khasminskii's theory and constructing Lyapunov functions,it is proved that the model has ergodicity and stationary distribution for any initial value solution;Finally,the correctness of the theoretical analysis results is verified by numerical simulation.In the fourth chapter,the stochastic SIRS model of Beddington-De Angelis incidence rates based on Markovian Switching is studied.Similarly,using the Lyapunov function and stopping time theory to prove that the solution of the model is globally positive.Then,the sufficient conditions for disease extinction are given by using the generalized It(?) formula.Furthermore,the appropriate Lyapunov function is constructed and using Khasminskii's theorem,proving that the model has ergodicity and stationary distribution for any initial value solution;Finally,the results of theoretical analysis are verified by numerical simulation.