The Study Of Some Classes Of Epidemic Models With Cure Rate

In this paper, we mainly study the dynamics of some classes of epidemic model with cure rate in population. 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, the stability of the SIR epidemic model with information variable, saturated incidence rate and graded cure rate was studied. We obtain the basic reproduction number R₀, The local asymptotical stability of equilibria is verified by analyzing the eigenvalues and using the Routh-Hurwitz criterion, this model exhibits two bifurcations, one is transcritical bifurcation and the other is Hopf bifurcation. We also discuss the global asymptotical stability of the disease free equilibrium by constructing a Lyapunov function and the endemic equilibrium by autonomous convergence theorem. A numerical analysis is given to show the. effectiveness of the main results.In Chapter 3. the dynamics behavior of a delayed viral infection model with immune im-pairment and cure rate is studied. It is shown that there exists three equilibria. By analyzing the characteristic equations, the local stability of the infection-free equilibrium and the immune-exhausted equilibrium of the model are established. In the following, the stability of the positive equilibrium is studied. Furthermore, we investigate the existence of Hopf bifurcation by using a delay as a bifurcation parameter. Finally, numerical simulations are carried out to explain the mathematical conclusions.In Chapter 4. we consider an HIV pathogenesis model including cure rate and logistic proliferation term of CD4⁺T cells in healthy populations. Let N be the number of virus released by each productive infected CD4⁺ T cell. The critical number Ncrit that ensures the existence of the positive equilibrium is obtained. We further show that if N?N_crit. then there exists a unique uninfected equilibrium point E₀ that is locally asymptotically stable. If N>N_Crit, then the system is persistent and the only infected steady state E* is globally asvmptoticallv stable. Numerical simulations are presented to illustrate the results.