Dynamics Of An HIV-1 Model With Two Delays

In this paper,the HIV-1 model and smoking model are established and studied by applying relevant theories of differential equations.The whole paper is divided into four chapters:In Chapter 1.we introduce the research background of this article,the main task and some important preliminariesIn Chapter 2,we get the immune-inactivated reproduction rate,we establish and inves-tigate an HIV-1 virus model with intracellular delay and humoral immunity delay.The local stability of feasible equilibria is established by analyzing the characteristic equations.The glob-ally stability of infection-free equilibrium and immunity-inactivated equilibrium are obtained by using Lyapunov functional and LaSalle's invariance principle.And the global stability of the infection-free equilibrium and the immunity-inactivated equilibrium are independent of humoral immune delay.Then,it is proved that humoral immune delay affects the stability of the positive equilibrium.We prove that Hopf bifurcation will occur when the humoral immune delay exceeds the critical value.The direction of Hopf bifurcation and the stability of bifurcation periodic so-lutions are studied by using the normal form method and the center manifold theorem.Finally,in order to verify our theoretical results,some numerical simulations are included.In Chapter 3,On the basis of the previous chapter,Logistic growth was introduced.Through the analysis of the characteristic equation,the local stability of the feasible equi-librium was established.Using Lyapunov functional and LaSalle's invariance principle,we can obtain that for any T1,T2,the infection-free equilibrium E0 is globally asymptotically stable when R0<1.For any T1,T2,the immunity-inactivated equilibrium E1 is globally asymptoti-cally stable if R1<1<R0.Besides,we proved that humoral immune delay affects the stability of the positive equilibrium,and we prove that Hopf bifurcation will occur when the humoral immune delay exceeds the critical value.The expressions determining the bifurcation direction and stability of bifurcation periodic solutions are obtained by using the normal form method and the center manifold theorem.Finally,in order to verify our theoretical results,some numerical simulations are included.In Chapter 4,we formulate a smoking model with delay and immigration,The introduction of two control functions,u1 and u2,represent the ratio of educational activities and media campaigns for the susceptible and heavy smokers,respectively.The existence of the optimal control pair is also proved.The optimal control solution is obtained by constructing the Hamil-tonian and using the Pontryagin maximum principle.Finally,numerical simulation is carried out to verify the main results...