Threshold dynamics for an age-structured heroin epidemic model with distributed delays

We introduce, in this paper, a delayed hybrid epidemic model describing the evolution of heroin addiction in a given population. The main objective of our mathematical analysis will be to provide the global behavior of the solutions. To the best of our knowledge, there are no results about the global dynamics of such a model containing both age structure and distributed time delay. We will prove that our model has threshold dynamics in terms of the basic reproduction number R-0. It will be established that for R-0 = 1, the addiction-free equilibrium is globally stable and for R-0 > 1, the endemic equilibrium is globally stable. The theoretical results are checked numerically. Especially, we focus on the influence of the addiction rate on the population dynamic.