Modeling the impact of rehabilitation, amelioration and relapse on the prevalence of drug epidemics

Substance abuse remains a global menace in spite of recurrent warnings, seizures, social and pharmacological effects associated with addiction to drugs. In this paper, we use a mathematical model which is a combination of the classical SIS and SIR models to investigate the dynamics of substance abuse. Initiation into drug use is based on contact of those at risk (the susceptible population) with drug users at different levels of drug use. We evaluate the threshold number and use it to analyze the model. We show that when this threshold number is less than unity, the drug-free steady state is globally asymptotically stable and when this threshold number is greater than unity the drug-persistent steady state is also globally stable. The impact of amelioration, rehabilitation and re-initiation on drug epidemics is investigated. Amelioration in presence of quitting for light users is observed to reduce the prevalence of substance abuse and this is supported by numerical simulations. The results show that both prevention and treatment/rehabilitation are necessary strategies for reduction of drug epidemics. Our recommendation is that preventive strategies should be directed toward reducing the contact rate and treatment should be combined with psychotherapy to accelerate quitting and reduce re-initiation.