Dynamic Behavior Of Two Classes Of Stochastic Epidemic Models With Markov Switching

In recent years,mathematical models play a key role in the study of infectious diseases,and they can provide theoretical basis for practical problems such as the control and prevention of infectious diseases.In real life,the spread of infectious diseases will be subject to different forms of random interference.In order to describe the system more accurately,our research has more practical significance.Based on the deterministic model,this paper establishes two kinds of stochastic infectious disease models with Markov switching one is the stochastic SIRS infectious disease model with nonmonotonic infectious rate and Markov switching,the other is the stochastic SIQR infectious disease model with saturated infectious rate and Markov switching.Based on the theory of stochastic differential equation,the existence and uniqueness of global positive solution of stochastic infectious disease model are discussed.The extinction and persistence of the model are studied,and existence of stationary distribution of the model are analyzed.Finally,the correctness of the conclusion of the theorem is proved by numerical simulation.In Chapter 1,we mainly expound the research background?motivation and current situation of infectious disease model and show the definitions,theorems and lemmas related to the extinction and persistence of infectious diseases and ergodicity of the system.In Chapter 2,we study the stochastic SIRS epidemic model with nonmonotonic infection rate and Markov switching.Firstly,based on the deterministic model and considering the influence of white noise and color noise on the system,the stochastic SIRS epidemic model is established.Secondly,the appropriate Lyapunov function is constructed,and the ?'s formula is used for reasonable analysis,the existence and uniqueness of the global positive solution is obtained.Then,using the strong law of large numbers,the sufficient conditions for the extinction of the disease are given.The ergodicity and stationary distribution of the model by using a Lyapunov function with Markov switching is proved.Finally,through a specific example,the correctness of the conclusion is verified by numerical simulation.In Chapter 3,the stochastic SIQR epidemic model with saturated infection rate and Markov switching is studied.Firstly,based on the deterministic model and considering the influence of white noise and color noise on the system,the stochastic SIQR epidemic model is established.Secondly,the appropriate Lyapunov function is constructed,and the ?'s formula is used for reasonable research,the existence and uniqueness of the global positive solution is obtained.Then,using the strong law of large numbers,the sufficient conditions for the extinction and persistence of the disease are given.Further,the conditions for the existence of the stationary distribution of the model are analyzed.Finally,through a specific example,the correctness of the conclusion is verified by numerical simulation.In Chapter 4,we summarize the full paper and give further prospects.