Dynamic Behavior Of Two Stochastic SIR And SIQS Epidemic Models With Lévy Jumps

Stochastic differential equation plays a vital role in many fields,such as medicine,biology and other fields.It provides certain theoretical basis for these fields,especially in the study of the spread and prevention of infectious diseases.The stochastic differential equation established by scholars has made great contributions to the control and prevention of infectious diseases In addition,isolation is an important means to control the spread of the disease.In the early stage of the outbreak of the novel corona virus,isolation of infected individuals can better control the further spread of the disease.So this paper studied two types of noise with white noise and Levy noise.In this paper,we consider two stochastic SIR and SIQS epidemic models with Levy jumps.In the first part,we expatiate the background and significance of the research of infectious disease dynamics,the research status of infectious disease dynamics and the definitions and theorems that will be used in this paper.In the second part,we study a kind of stochastic SIR model with saturation incidence.Due to the influence of psychological and social factors,the effective contact rate between susceptible and infected persons tends to saturation state.By constructing a suitable Lyapunov function and using Ito theorem and the concept of stop-time,we prove the global nature of the infectious disease model The existence and uniqueness of the positive solution are obtained,and then we obtain sufficient conditions for the extinction and persistence of the disease.In the third part,we study a class of stochastic sloR infectious disease model with isolation strategy.For this model,we use taylor formula and Ito theorem to prove the existence and uniqueness of global positive solutions,and obtain the threshold of disease extinction and disease persistence.At the end of the second and third parts of the article,for the above two models,we respectively use MATLAB numerical simulation to verify the correctness of the theoretical results.