On the dynamic and application of a mathematical model of the spread of HIV amongst drug users who inject

Kaplan developed the early deterministic mathematical model of the spread of HIV spread amongst IDUs (injecting drug users). This was followed by Greenhalgh and Hay, who extended the Kaplan model by considering some realistic assumptions. The models detailed the dynamic of probability of exposure to HIV of an IDU after they had used contaminated needles, and the dynamic of the IDU fraction subject to HIV infection. The model from Greenhalgh and Hay has two equilibria (fixed points), namely the HIV-free equilibrium and the HIV-endemic equilibrium. Greenhalgh and Hay demonstrated the global stability of the fixed points for some specific conditions. If the specific condition was not satisfied, Greenhalgh and Hay left it as an open problem. In this paper, we show that the dynamics of HIV spread among injecting drug users completely results from the basic reproduction number by constructing suitable Lyapunov functions, which resolves the open problem. We also apply the model to describe HIV/AIDS spread in a real case. The predicted result agrees with the data.