Dynamics of an age-structured heroin transmission model with vaccination and treatment

Based on the development of heroin vaccine, in this paper, we propose an age structured heroin transmission model with treatment and vaccination. The model allows the drug reuse rate of the individuals in treatment to depend on a treatment-age and the vaccine waning rate of the vaccinated to depend on a vaccination age. Meanwhile, the model allows that the heroin vaccine provides an imperfect protection (i.e., the vaccinated individuals can also become drug addicted). We derive the basic reproduction number which dependents on vaccination. The basic reproduction number completely determines the persistence and extinction of heroin spread, i.e., if the basic reproduction number is less than one the drug-free steady state is globally asymptotically stable (i.e., the heroin spread dies out), if the basic reproduction number is larger than one, there exists an unique positive steady state and it is locally and globally stable in some special cases. Finally, some numerical simulations are carried out to illustrate the stability of the positive steady state.