Modelling and stability of a synthetic drugs transmission model with relapse and treatment

Abstract

In this paper, a synthetic drugs transmission model with treatment is formulated based on the principles of mathematical epidemiology. The model considers that relapse can occur among those individuals who have a history of drug abuse and we distinguish the addiction rates of susceptible individuals who have a history of drug abuse and those who have not. The global dynamics of this model are determined by the basic reproduction number, $$R_{0}$$ , under certain conditions. If $$R_{0}<1$$ , the drug-free equilibrium is globally exponentially stable for a special case and the exponential convergence rate can be unveiled, and if $$R_{0}>1$$ , the drug-addiction equilibrium is globally asymptotically stable under certain conditions. Sensitivity analysis is performed to seek for effective control measures for drug abuse. Numerical simulations are also carried out to confirm the analytical results.