Abstract
We aim in this research to show the global behavior of each equilibrium of a heroin epidemic model with distributed delays. We obtained that the global behavior of the considered system is completely governed by the value of the basic reproduction number [Formula: see text]. For the case when [Formula: see text] is less than one, we established the global stability of the drug-free equilibrium using the Lyapunov direct method. However, for [Formula: see text] greater than one, the addiction persists and the drug equilibrium is globally asymptotically stable, which is shown also by the Lyapunov direct method. Moreover, we obtained that by controlling the density of occasional heroin users we can control the heroin epidemic spread, and this measure is more effective than the treatment measure.
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