Dynamic Behavior Of Three Classes Of Epidemic Dynamical Models

Since ancient times,infectious diseases are a major problem endangering human health and life.Quantitative research on the epidemic law of diseases is helpful to the prevention and control of infectious diseases.Different types of infectious diseases should establish mathematical models that can reveal the changing law of infect,ious diseases,and then study the corresponding mathematical models.In this dissertation,the stochastic epidemic model with nonlinear incidence,the periodic three-patch Rift-Valley fever virus transmission model and the discrete coupled within-host and between-host epidemic model with two time scales are investigated,respectively.Specific research work is as follows:In chapter two,we study the stochastic epidemic model with nonlinear incidence.This chapter is divided into t,wo parts:In the first part,we study a stochastic peri=odic SIVS epidemic model with nonlinear incidence and vaccination.The existence of stochastic positive ?-periodic solutions of disease with probability one are established by defining the expression of the basic reproduction number of the model,establishing the appropriate Lyapunov function and using the existence criterion of periodic solut,ions of the periodic stochastic differential equation;The threshold conditions on extinction and permanence of disease with probability one are established by constructing a new stochastic Lyapunov function,using B-D-G inequality,Doob martingale inequality,B-C lemma and strong large number t,heorem to deal with the nonlinear incidence and vaccination for the stochastic epidemic model.The numerical simula.tions a.re given to illustrate the main theoretical results and present some new interesting conjectures.In the second part,we study a stochastic SIR epidemic model with nonlinear incidence.Based on the theory of integrable Markov semigroups,a threshold criterion is estab-lished to ensure the unique stationary distribution of the model.Finally,the numerical examples are carried out to illustrate our theoretical results.In chapter three,we investigate a three-patch Rift Valley fever virus transmission model wit,h periodic coefficients.Firstly,we discuss the first patch model and get the positivity and boundedness of the solution of the first patch model.By using the next generation matrix method,dynamic system persistence theory and periodic linear system properties,the basic regeneration number of the first patch model R10,according to the basic regeneration number,we get the threshold conditions of the persistence and extinction of the first patch model.Then,according to the conclusion of the first pat,ch model,the second patch model is discussed by applying the principle of differential equation comparison and the method of limit system.This part is divided into two cases,that is,The persistence and extinction of the second patch model are discussed when the disease is persistent and extinct in the first patch,respectively.Finally,the discussion of the third patch model is based on the conclusions of the first a.nd second patch models,and the thir-d patch model is discussed by using the method of the second patch.Based on the above three patches,the threshold conditions of RVF model persistence and extinction are obtained.Finally,the numerical simulations are given to confirm the theoretical results.In chapter four,we discuss the discrete epidemic model with two time scales.The system is divided into a fast time system and a slow time system by using the idea of limit equations.We discuss the positive,boundedness and existence of equilibrium for fast and slow systems,respectively.The stability of equilibrium points is a.nalyzed by constructing discrete Lyapunov functions and linearization methods.Since there are two positive equilibrium points in slow systems,we also discuss the backward bifurcat,ion of slow systems.Finally,we obtain the existence of equilibr-ium in coupled systems by discussing fast systems and slow systems,and study the loca.l asymptotic stability of equilibrium points by using linearization method.Finally,the correctness of the theorem and the rationality of the conjecture are verified by numerical simulation.