Modelling multiple relapses in drug epidemics

TLDR: A mathematical model with n-alternate stages of rehabilitation and relapsing, which has two equilibria, the drug free equilibrium and the drug persistent equilibrium, that are both globally stable when the threshold of R0<1 and R0>1 is reached.

Abstract: Drug dependence is a ‘chronic disease’ treatable through rehabilitation. Many drug addicts progress through a series of rehabilitation and relapsing episodes. In this paper, we formulate a mathematical model with n-alternate stages of rehabilitation and relapsing. The dynamics of drug abuse are treated as an infectious disease that spreads through a population. The model analysis shows that the model has two equilibria, the drug free equilibrium and the drug persistent equilibrium, that are both globally stable when the threshold $${\mathcal {R}}_{0}<1$$ and $${\mathcal {R}}_{0}>1$$ respectively. The model is fitted to data on individuals under repeated rehabilitation and parameter values that give the best fit chosen. The projections carried out the long term trends of proportions for repeated rehabilitants. The relative impact for each subgroup is determined to find out which population subgroup is responsible for a disproportionate number of initiations. The results have huge implications to designing policies aligned to rehabilitation processes.