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

Abstract

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 , which serves a threshold role. If , then the addiction‐free steady state is globally asymptotically stable while if , then there is at least one addictive steady state. Moreover, when , 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.