Analysis Of Several HIV-1Dynamics Models With Immune Response

In this thesis, we investigate several HIV-1dynamics models with immune response by combining Hurwitz theorem, Lyapunov functionals, the normal form argument and center manifold theory. The results show that the basic reproduction number and the immune response reproduction number play important roles in the stability of the equilibria. The thesis is composed of four chapters.In Chapter One, the historical background of AIDS, signification and evolving of HIV-1are briefly reviewed. Furthermore, we simply introduce the main work in this paper.In Chapter Two, an HIV-1infection model with Beddington-DeAngelis inci-dence rate and CTL immune response is studied, we obtain the sufficient condi-tions of the globally asymptotically stable for the infection-free equilibrium, the immune-free equilibrium and the infected equilibrium with immune response. The model considers the feature that an eclipse stage for the infected cells is reverted to uninfected class. The numerical results show that the rate which the infected cells are reverted to uninfected class has no effect on the stability of the equilibria. On the other hand, our model can better describe the HIV-1infection in the real world because of the general incidence rate.In Chapter Three, several HIV-1infection model with different delays are studied. And, this chapter is composed of three sections. The first section investi-gates an HIV-1model which include the delay for the infected cells in eclipse stage is reverted to uninfected cells. The second section investigates an HIV-1model which include the time uninfected cells are contacted by the virus particles and the time the virons enter the cells. In these two sections, we all obtain that the infection-free equilibrium is locally asymptotically stable if the basic reproduction number is less than or equal to1and the delay number is greater than or equal to zero. We also obtain that the immune-free equilibrium and the infected equilib-rium with immune response are locally asymptotically stable if the delay number is equal to zero, and, the basic reproduction number and the immune response repro-duction number under some conditions. However, if the delay number is greater than a positive number, then the Hopf bifurcation can occur at the two equilibria under some conditions. In the third section, we consider an HIV-1model with delayed CTL immune response. We prove the positivity and the boundedness of the solution for system, furthermore, the uniform persistence of the system can be obtained if the immune response reproduction number is greater than1. In ad-dition, the infection-free equilibrium and the immune-free equilibrium are always globally asymptotically stable if the basic reproduction number and the immune response reproduction number under some conditions. On the other hand, the infected equilibrium with immune response is globally asymptotically stable if the delay number is equal to zero. However, the Hopf bifurcation can occur if the delay number is greater than a positive number, we further study the direction and stability of Hopf bifurcation. The models which include different delays in this chapter represent that the delay can change the stability of the models, that is, the infection of HIV-1in clinical is complex.Finally, in Chapter Four, we consider the HIV-1model including eclipse stage of infected cells and multi-target cells. Firstly, we investigate the globally asymp-totically stable for the infection-free equilibrium, the immune-free equilibrium and the infected equilibrium with immune response when n=2. Furthermore, we obtain the sufficient conditions for the globally asymptotically stable of the three equilibria when n≥2. By comparison with the results, we know that the suffi-cient conditions are strong if the multi-target cells are infected. By the biological explanations, it shows that the complexity and the variety after the virus entering into the body and it is accord with the unpredictability in the clinical.