Dynamics of a delayed nonlocal reaction–diffusion heroin epidemic model in a heterogenous environment

To study the consumption of heroin in a heterogeneous environment, we propose and analyze a spatiotemporal model with a distributed delay. Using the spectral theory, we determine the basic reproduction number R0$$ {R}_0 $$, which serves a threshold role. If R01$$ {R}_0>1 $$, then there is at least one addictive steady state. Moreover, when R0>1$$ {R}_0>1 $$, if one of the dispersal coefficients is zero, then there is only one addictive steady state, and it is globally asymptotically stable; if both diffusions of susceptible and addicted individuals are present, we cannot identify the temporal behavior of solutions, and hence, we study the asymptotic profile of addictive steady states when one of the dispersal coefficients tend to zero.