Backward bifurcation in a two strain model of heroin addiction

Document Type : Research Paper


1 Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran.

2 Esfarayen University of Technology, Esfarayen, North Khorasan, Iran.


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 the superinfection for modeling the effect of prescribed opioids abuse on heroin addiction. Our model contains the impact 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 explore both the 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 R0 < 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. 


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Volume 10, Issue 3
July 2022
Pages 656-673
  • Receive Date: 17 February 2021
  • Revise Date: 28 July 2021
  • Accept Date: 30 July 2021
  • First Publish Date: 08 August 2021