Abstract: In this paper, an antibody immune HIV model with a saturated proliferation rate is established, and some conditions of local stability and global stability for the disease-free equilibrium, the no-immune endemic equilibrium and the immune endemic equilibrium are obtained by the linearization method and the Lyapunov function method, respectively. Moreover, motivated by the success of the latest anti-AIDS vaccine in some animal experiments, the dual action of antibody immunity and drug therapy is builded into the above kinetic model, which forms a optimal control problem to minimize the concentration of infected cells and virus and the cost of control. Using the Pontryagin maximum principle, the optimality conditions for the optimal control problem are gotten. After obtaining the parameters of the model , numerical simulation tests are carried out for the double control problem and the single control problem respectively. Experimental results show that the concentration of the infected cells and viruses can quickly reduce under the condition of the effective vaccine, and the immune control effect is almost as well as the effect of the double control by comparing with double control effect, which illustrates the vaccine is fairly effective in controlling AIDS. It can be expected that vaccine will greatly change the current situation of AIDS spread and may even eliminate AIDS eventually, after the vaccine is put into clinical practice.

Key words: HIV model ,  Lyapunov function ,  Stability ,  Vaccination ,  Optimal control