Dynamic Behaviors Of Two Classes Of Stochastic Epidemic Models With Different Incidence

The application of stochastic differential equations is very broad.It plays an important role in many fields,especially in the fields of biology and medicine,and provides some theoretical guidance for the development of these fields.At the same time,in real life,random disturbances in the environment will inevitably have a certain impact on the infectious disease system.Therefore,it is crucial and realistic to study how environmental noise affects the development of infectious diseases.This paper studies the deterministic SIQR and SIRS infectious disease models under the disturbance of Brownian motion as noise form,and obtain corresponding stochastic models of SIQR and SIRS.For the SIQR model with non-monotonic rate.Firstly,the use of the counterevidence method proves that there is a unique global positive solution for the SIQR infectious disease system.Secondly,by constructing the Lyapunov function,it is proved that when the parameters in the model satisfy certain conditions,the stochastic model is respectively asymptotic stability near the disease-free equilibrium and the endemic equilibrium,and using numerical simulation to verify the conclusion.For the SIRS model with non-monotonic incidence and vaccination,this paper uses the method of random perturbation to the whole model based on the deterministic model.In this paper,four aspects of stochastic SIRS infectious disease model are studied,which proves the existence and uniqueness of the global positive solution of the model,the asymptotic behavior of the solution near the disease-free equilibrium,the asymptotic behavior of the endemic equilibrium,and the numerical simulation.In the course of the research,the corresponding proofs were obtained by using the methods of counter-evidence,It^ formula and constructing Lyapunov functions.For the SIQR model with standard incidence.First,it is proved that the SIQR dynamic model has a unique global positive solution.Then,by constructing the appropriate Lyapunov function,the It^o formula is used to discuss the sufficient conditions to ensure the extinction and persistence in mean of the disease.The threshold for the extinction and persistence of the disease in a stochastic SIQR system under the influence of white noise.