A graph theoretical perspective of a drug abuse epidemic model

A drug use epidemic can be represented by a finite number of states and transition rules that govern the dynamics of drug use in each discrete time step. This paper investigates the spread of drug use in a community where some users are in treatment and others are not in treatment, citing South Africa as an example. In our analysis, we consider the neighbourhood prevalence of each individual, i.e., the proportion of the individual's drug user contacts who are not in treatment amongst all of his or her contacts. We introduce parameters α*,β* and γ*, depending on the neighbourhood prevalence, which govern the spread of drug use. We examine how changes in α*, β* and γ* affect the system dynamics. Simulations presented support the theoretical results.