Mathematical analysis of an age structured heroin-cocaine epidemic model
Abdennasser Chekroun, Mohammed Nor Frioui, Toshikazu Kuniya, Tarik Mohammed Touaoula
Abstract
This paper is devoted to studying the dynamics of a certain age structured heroin-cocaine epidemic model. More precisely, this model takes into account the following unknown variables: susceptible individuals, heroin users, cocaine users and recovered individuals. Each one of these classes can change or interact with others. In this paper, firstly, we give some results on the existence, uniqueness and positivity of solutions. Next, we obtain a threshold value $ r(\Psi'[0]) $ such that an endemic equilibrium exists if $ r(\Psi'[0]) > 1 $. We then show that if $ r(\Psi'[0]) < 1 $, then the disease-free equilibrium is globally asymptotically stable, whereas if $ r(\Psi'[0]) > 1 $, then the system is uniformly persistent. Moreover, for $ r(\Psi'[0]) > 1 $, we show that the endemic equilibrium is globally asymptotically stable under an additional assumption that epidemic parameters for heroin users and cocaine users are same. Finally, some numerical simulations are presented to illustrate our theoretical results.