Global asymptotic properties of a heroin epidemic model with treat-age

In this paper, a model for the use of heroin with treat-age is formulated based on the principles of mathematical epidemiology. The model accounts for relapse rate that depends on how long the host has been in treatment for heroin addiction. An explicit formula for the reproductive number of the heroin spread is obtained. By using the method of Lyapunov functional, we established the dynamical properties of the heroin epidemic model, and the results show that the global dynamics of the model is completely determined by the basic reproduction number. It is shown that the drug-free equilibrium is locally and globally asymptotically stable if the basic reproduction number is less than one. In addition, the heroin spread system is uniform persistence and the unique drug spread equilibrium is locally and globally asymptotically stable if the basic reproduction number is greater than one.