Stability Analysis Of An HIV/AIDS Model With Vertical Transmission

Acquired Immune Deficiency Syndrome (AIDS) is a infectious disease transmitted by Human Immune deficiency Virus(H IV) through sexual contact, blood and mother to child transmission. The virus invades the human body, destroys the body's immune system and causes a series of serious diseases. There is no specific therapy for this disease, so it's fatality rate is high.There are three main routes of AIDS transmission, the vertical transmission (moth-er to child transmission) ratio is increasing rapidly, therefore this paper mainly study the HIV/AIDS vertical transmission. The main contents can be summarized as follows:The first section is introduction, in which we present research background, purpose and significance of vertical transmission.The second section is preliminary knowledge, in which we present some theorems, lem-mas that we will use in our paper.In the third section, we developed a SIJA model and established an HIV/AIDS model with vertical transmission. We used basic reproduction number to determine the stability of the disease-free and endemic equilibrium, we also proved the global stability of equilibria by using Lyapunov function. Furthermore, we investigate the effect of the vertical transmission on HIV/AIDS prevention.In the forth section, we considered a delayed HIV/AIDS transmission model with vertical transmission. The basic reproduction number Ro which determines whether the disease goes to extinction or not is obtained. Throughout the paper, we investigated the global stability of the disease-free equilibrium and the endemic equilibrium. Furthermore, we also investigated the effect of the vertical transmission on the prevention of HIV/AIDS.In the fifth section, we considered a delayed HIV/AIDS model with vertical transmis-sion, but the model differs from the model in the forth section. The conditions and thresholds for the existence of two equilibria are obtained, we proved that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number is less than 1. The global stability for the endemic equilibrium we only considered the special case of r= 0, and proved the global stability.