Dynamics Research Of Two Stochastic SIVS Epidemic Models With Vaccination Control

In the field of biomathematics,the problem of infectious diseases has always been one of the research hotspots,and mathematical models with differential equations are of great significance to analyze the dynamic behavior of infectious diseases,predict the trend of disease development and prevent and control the occurrence and spread of infectious diseases.The study of dynamics behavior of infectious diseases enables people to understand the epidemic mechanism of infectious diseases in a more comprehensive way and to adopt the optimal control strategy.At present,many experts in biomathematics have studied the dynamics of infectious disease systems by establishing deterministic mathematical models.However,there always exists random factors such as white noise and Levy noise in the environment,stochastic differential equation model can more accurately reflect the reality than deterministic model.Therefore,by using stochastic differential equation theory and some important inequalities.This paper mainly studies the two kinds of stochastic epidemic models and their dynamic behavior,one is a stochastic SIVS epidemic model with nonlinear saturated infectious rate under vaccination control,the other is a stochastic SIVS epidemic model with double infectious diseases and Levy jumps.This paper includes the following four sections.In chapter 1,we introduce the research background and development status of the subject,then present some basic knowledge and theorems related to stochastic differential equations,and prepare some important inequalities.In chapter 2,we consider a non-linear stochastic SIVS epidemic system under vaccination control and analyze the dynamics of the model.Firstly,we verify there exists a unique global positive solution of this infectious disease systems by using Ito formulas and Lyapunov function method.Which is significant to investigate the dynamics of stochastic system.Secondly,we study the stochastic dynamical behavior of the system and obtain the conditions for controlling the extinction and persistence of infectious diseases.Finally,we use a series of numerical simulations to verify the results,which show that large random noise can lead to the extinction of infectious diseases,in other words,random interference can control the outbreak of infectious diseases,which provides a theoretical basis for the control of infectious diseases.In Chapter 3,we propose a class of stochastic SIVS epidemic models with double infectious diseases and Levy jumps.Firstly,when the SIVS model with two diseases is disturbed by both white noise and Levy noise,we study its stochastic disease-free dynamics and stochastic endemic dynamics.Then,by using a series of stochastic inequality techniques,discuss the persistence in the mean and extinction of two diseases on a case-by-case basis and obtain sufficient conditions of the persistence in mean and extinction of stochastic systems and the threshold for controlling the extinction and diffusion of infectious diseases.Finally,we carry out the numerical simulation results of the dynamic effects of stochastic disturbances to illustrate the theoretical findings.In Chapter 4,we make a summary and prospect of the whole paper,and summarize the main contents and biological implications of this paper,and then we make a forward-looking vision of the next step of work.