Backward, Hopf bifurcation in a heroin epidemic model with treat age

Heroin is considered as damage on the health of the individuals. In this paper, we consider a heroin epidemic model with treat age, where the inner rivalry in between the drug users for a small dose of drugs is investigated. The influence of treatment period for a drug consumer before quitting treatment is investigated, where it is obtained that the considered model can undergo backward bifurcation, which shows the possibility of having two endemic equilibriums, this type of bifurcation is discussed in terms of the basic reproduction number R-0. The bifurcation diagram is drown in the case of an age structured model. Also, we show the existence of Hopf bifurcation is shown under suitable conditions on the model parameters. The obtained mathematical results are confirmed numerically.