Abstract
In this paper, to understand the impact of spatial heterogeneity of treatment and movement of individuals on the persistence and extinction of heroin spread, we propose a new diffusive heroin transmission model with treatment-dependent age-structure. The basic reproduction number in heterogenous environment R0 of the system is defined, which is consistent with the one deduced from the next generation operator approach R(x). The threshold dynamics in terms of the basic reproduction number is established: if R0≤1, the drug-free steady state is globally asymptotically stable, if R0>1, heroin transmission is uniformly persistent if it is present initially. In particular, when the environment is homogeneous and R0>1, our system has a unique space-independent drug spread steady state and it is globally asymptotically stable. Finally, some numerical simulations are carried out to illustrate the main results.
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