Modelling and qualitative analysis of an illicit drugs model with saturated incidence rate and relapse

Abstract

Drug abuse is now regarded as a global issue that brings severe consequences on the health, social well-being, and economy. In this study, an illicit drugs model with saturated incidence rate and relapse of individuals who quit using drugs is proposed and analyzed qualitatively. This study aims to determine the behavior of a drug epidemic when the psychological or inhibitory effect and relapse are being considered as well as assist the policymakers in devising effective control measures. The basic reproduction number \(R_0\) is derived and used as a threshold parameter in the global stability analysis. It is found that the drug-free equilibrium is globally asymptotically stable when \(R_0\le 1\). This implies that we can eradicate a drug epidemic when the threshold \(R_0\) is less than or equal to one, irrespective of the initial population size of drug users. On the other hand, the drug persistent equilibrium is globally asymptotically stable when \(R_0>1\). This indicates that the phenomenon of drug use will remain in a community when the threshold \(R_0\) is greater than one, irrespective of the initial population size of drug users. Next, the sensitivity analysis is performed and the results show that the effective contact rate should be targeted to reduce its value. The numerical simulations are also carried out to illustrate the analytical results and investigate the relationship between the measure of psychological or inhibitory effect and the number of drug users.