Study On The Dynamic Behavior Of Three Kinds Of Stochastic Models

Nowadays,the spread and prevention and control of infectious diseases have always been the focus of the medical and health departments around the world.Stochastic differential equations play an important role in the research of infectious diseases.At the same time,since the 21st century,the network technology and digital information have been developed exponentially,and the world has entered the era of network.Under the background of the network era,the global spread of computer virus has a great negative impact on today's society.In this context,it is particularly important to study the dynamic behavior of infectious disease model and computer virus model by using the theory of differential equation.Considering the disturbance of environmental noise to the infectious disease model and computer virus model,the study of stochastic system is more practical.Therefore,the white noise disturbance is introduced into the classical deterministic SIQR and SIR mathematical models,and the corresponding stochastic SIQR and SIR mathematical models are obtained.In addition,considering the time delay of computer virus propagation in real environment,the stochastic SIQR computer virus model with time delay effect is considered,the main contents are as follows:Firstly,consider stochastic SIQR model with bilinear saturation rate and stochastic SIR models with nonmonotonic incidence.Based on the deterministic infectious disease model,the influence of random disturbance on the model was considered to analyze the infectious disease model with random mortality.Firstly,it is proved that the model has a unique positive solution,then we use the stochastic analysis theory of Lyapounov function,It(?) formula to discuss the sufficient condition of disease extinction and persistence,meanwhile,the disease extinction and persistence threshold of R_swere also obtained.Secondly,consider a random SIQR computer virus model with nonlinear contagion rate and time delay.Based on the deterministic SIQR model,considering the influence of stochastic disturbances on the model,a random SIQR computer virus model with nonlinear contagion rate and delay is established.Firstly,it is proved that the model has a unique global positive solution under certain conditions.Secondly,by constructing the appropriate Lyapounov function,the asymptotic behavior of the two equilibrium points is proved.Finally,the asymptotic behavior of the stochastic system solution is further analyzed and verified by numerical simulation.