Abstract: In this paper, a new model for the dynamics of heroin is formulated that incorporates both the infection age of drug users not in treatment and the treat age of drug users in treatment. We derive the basic reproduction number. By using the fluctuation lemma and Lyapunov functional, we show that if the basic reproduction number is less than unity, the disease-free steady state is globally asymptotically stable, whereas if the basic reproduction number is greater than unity, the endemic steady state is globally asymptotically stable.

Key words: infection age, treat age, basic reproduction number, Lyapunov functional, global stability