High and Low Frequency Oscillations in Drug Epidemics

We extend the two-dimensional model of drug use introduced in Behrens et al. [1999, 2000, 2002] by introducing two additional states that model in more detail newly initiated (“light”) users’ response to the drug experience. Those who dislike the drug quickly “quit” and briefly suppress initiation by others. Those who like the drug progress to ongoing (“moderate”) use, from which they may or may not escalate to “heavy” or dependent use. Initiation is spread contagiously by light and moderate users, but is moderated by the drug’s reputation, which is a function of the number of unhappy users (recent quitters + heavy users). The model reproduces recent prevalence data from the U.S. cocaine epidemic reasonably well, with one pronounced peak followed by decay toward a steady state. However, minor variation in parameter values yields both long-run periodicity with a period akin to the gap between the first U.S. cocaine epidemic (peak ~1910) and the current one (peak ~1980), as well as short-run periodicity akin to that observed in data on youthful use for a variety of substances. The combination of short- and long-run periodicity is reminiscent of the elliptical burstors described by Rubin and Terman [2002]. The existence of such complex behavior including cycles, quasi periodic solutions, and chaos is proven by means of bifurcation analysis.