Dynamical Analysis Of Two Classes Of Discrete Epidemic Models

In this thesis,combined with statistical software(partial rank coefficients analy-sis PRCC),the stability theory and bifurcation theory of discrete dynamical systems are used to analyze the dynamics of two classes of discrete infectious disease models.We assume that the incidance rate is not a constant but a general function.We first-ly study a discrete SIRS epidemic model with vaccination,The basic reproductive number R0 of the model is defined,the existence of equilibria is discussed,and the stability analysis of the model is given.It is proven that the disease-free equilibrium is locally asymptotically stable if R0<1 and globally asymptotically stable for some additional conditions,and the endemic equilibrium is locally asymptotically stable if R0>1.Numerical simulations are presented to illustrate the stability results,the time series diagram shows the global asymptotic stability of the disease-free equi-librium when it is locally asymptotically stable.Partial rank correlation coefficients(PRCC)is used to perform the sensitivity analysis of the basic reproduction number R0 in terms of different parameters on the disease.Therefore,measures to control the spread of diseases can be proposed.Finally,the influenza data of China from 2004 to 2018 are fitted by this model,the trend of influenza prevalence is predicted.Our numerical simulations indicate that the number of influenza infected cases will be increasing in the next few years in China.In the third part of the thesis,we study a discrete SIS epidemic model with disease induced death and Ricker-type recruit-ment.The basic reproductive number R0 of the model is defined,the existence of the equilibria is proved,and the local asymptotically stability of the disease-free equi-librium is given.We give conditions on the existence of transcritical bifurcation,flip bifurcation and Neimark-Sacker bifurcation by using center manifold theorem and bifurcation theory.Numerical simulations,including bifurcation diagrams,phase portraits,maximum Lyapunov exponents and feasible sets,are presented which not only illustrate theoretical results,but also indicate the existence of more complex dynamical behaviors,such as complex periodic windows,the coexistence of periodic points and invariant tori,chaotic attractors.The thesis consists of four chapters as the following.Chapter 1 describles the background knowledge on modeling infectious diseases and the preparation knowledge.A brief review of local bifurcation theory,central manifold theorem,Jury condition for discrete dynamical systems is presented.In chapter 2,a discrete SIRS epidemic model with vaccination and a general probability function is investigated.The basic reproductive number R0 of the model is defined.It is proven that the disease-free equilibrium is locally asymptotically stable if R0<1,and the endemic equilibrium is locally asymptotically stable if R0>1.Numerical simulations are presented to illustrate the stability results,partial rank correlation coefficients(PRCC)is used to perform the sensitivity analysis of the basic reproduction number R0 to study the impact of different parameters on the disease,the result shows that the proportionality coefficient of unvaccinated susceptible,the probability that infected individuals do not recover,and the probability of survival have much more effect on R0.In chapter 3,we investigate the complex dynamics in a discrete SIS epidem-ic model with disease induced death.It is shown that the model has a unique disease-free equilibrium if the basic reproduction number R0<1 and a unique en-demic equilibrium if R0>1.Sufficient conditions for the local asymptotic stability of the equilibria are obtained.A detailed bifurcation analysis at the endemic e-quilibrium reveals a rich bifurcation structure,including transcritical bifurcation,flip bifurcation and Neimark-Sacker bifurcation,as the parameters vary.Numerical simulations,including bifurcation diagrams,phase portraits,maximum Lyapunov exponents and feasible sets are presented.In chapter 4,we summarize the current work and address the future research goals.