Dynamic analysis based on the relapse rate of drug epidemic model

Abstract

In this paper, we study a drug epidemic model based on epidemiology by dividing the human population into four classes at time [Formula: see text]: susceptibles (S), drug users (I), drug users who are treated in isolation and temporarily quit drugs[Formula: see text] and drug users who are treated in isolation and permanently quit drugs [Formula: see text]. We obtain the basic reproduction number [Formula: see text] of the model and perform its sensitivity analysis. We show that if [Formula: see text], then the drug-free equilibrium is globally asymptotically stable, and if [Formula: see text], there exists an drug-abuse equilibrium and it is locally asymptotically stable. The proposed model may possess forward and backward bifurcations. Moreover, three different control strategies and numerical results are presented. Through different adjustments to obtain graphical results, we obtain the best strategy to control the drug epidemic.