Global Analysis Of Several Epidemic Models

In this paper,we mainly study the dynamics of some classes of infectious disease systems.The article includes four chapters.The preface is in chapter 1,we introduce the research background of this article,the main task and some important preliminaries.In Chapter 2.we consider a.disease that spread by droplet infection and also through direct contact,establish a delayed epidemic model with pulse vaccination and nonlinear transmission nonlinear cure rate.Firstly,by using the comparison theorem of impulsive equations.we discuss the global attractor of the disease free periodic solution of the system.Then.the persistence of the system is proved with the.help of Lyapunov function.In the end.numerical simulations axe carried out.to explain the mathematical conclusions.In Chapter 3,based on the background of the second chapter we build a model with two delays and general nonlinear incidence rate.we use the f(P)and f(I)to express incidence of contacting with individuals infected droplets and direct contact.This makes the system more general.Firstly,We investigate the positivity.boundedness of the solutions and the existence of the equilibria of this system.Then,the global asymptotical stabilities of the disease-free equilibrium is given by constructing Lyapunov function.The local asymptotic stability and the existence of Hopf bifurcations at the epidemic equilibrium is also proved by the principle of switching.Numerical simulations were carried out to explain the mathematical conclusions.In Chapter 4,we propose a rumor transmission model with incubation period and constant recruitment in social networks.Firstly,this chapter proves the existence of rumor-free equilib-rium and rumor equilibrium.Then,the local stability of rumor-free equilibrium is proved by the method of characteristic.A transcritical bifurcation is analyzed by the method of left and right eigenvectors.In addition,we proved the local asymptotic stability of the rumor-free equi-librium with the help of the Routh-Hurwitz criterion,and we also showed the global stability of the rumor equilibrium by the M-matrix.When certain conditions are established.there is only a rumor equilibrium point in the positive invariant set of the system and the equilibrium is globally stable.Numerical simulations were carried out to illustrate the main theoretical results.