Global behaviour of a heroin epidemic model with distributed delays

In this paper, we study a heroin epidemic model with distributed time delays. The basic reproduction number R 0 for the model is identified and the threshold property of R 0 is established. It is shown that drug-free equilibrium is globally asymptotically stable if R 0 < 1 . When R 0 > 1 , there is a disease endemic equilibrium which is locally asymptotically stable, it is proved that the disease is uniformly persistent in the population, and explicit formulae are obtained by which the eventual lower bound of the drug user individuals can be computed.