Backward bifurcation in a two strain model of heroin addiction

Among the various causes of heroin addiction, the use of ‎prescription ‎opioids‎ is one of the main reasons. In this article, we introduce and analyze a two ‎strain‎ epidemic model with super infection for modeling the effect of ‎prescrib‎ed opioids abuse on heroin ‎addiction.‎ ‎Our ‎model ‎contains ‎the ‎effect ‎of ‎relapse ‎of ‎individuals ‎under ‎treatment/rehabilitation‎ ‎to drug abuse in each ‎strain.‎ ‎We ‎extract‎ the basic reproductive ‎ratio, ‎the‎ invasion numbers‎, ‎and study the occurrence of backward bifurcation in strain ‎domi‎nance equilibria, i.e., boundary ‎equilibria. ‎Also, ‎we ‎study ‎both‎ ‎‎local and global stability of DFE and boundary equilibria ‎under suitable conditions‎.‎ ‎Furthermore, we study the ‎existence of the coexistence equilibrium point‎. We prove that when ‎$‎R_0<1‎$‎, the coexistence equilibrium point can exist, i.e., backward bifurcation ‎occurs‎ in coexistence equilibria. ‎Finally, we use numerical simulation to describe the obtained analytical results.‎