Curtailing the spread of drug-abuse and violence co-menace: An optimal control approach

Abstract

This work presents an optimal control of intervention strategies for a drug abuse and violence co-menace model considering a mass campaign against drug abuse, a mass campaign against violence, effective policing and rehabilitation of drug-abusing and violent individuals as control strategies. We obtained the drug abuse only free equilibrium, drug abuse only present equilibrium, violence only free equilibrium, and violence only present equilibrium. The stability analysis results suggest that the model drug abuse free equilibrium and violence-free equilibrium are both locally asymptotically stable when ROD<1 and ROV<1 and otherwise when ROD>1 and ROV>1. When ROD>1 and ROV>1 respectively, the model has a unique positive drug abuse and violence present equilibrium. The findings of the impact assessment carried out indicate that violence enhances drug abuse and drug abuse in a community often increases violent activities. The drug abuse and violence model is globally unstable. The result of the bifurcation analysis indicates that the model exhibits backward bifurcation. We updated the drug abuse and violence co-menace model by incorporating control measures namely: a mass campaign against drug abuse, a mass campaign against violence, effective policing, and rehabilitation of drug-abusing and violent individuals. Using Pontryagin’s Maximum Principle, the optimal control problem was solved. We observed from the numerical simulation that when the control measures are applied relative to the case of no control, there is a substantial decrease in the population of low drug-abusing individuals, high drug-abusing individuals, low drug-abusing and violent individuals, high drug-abusing and violent individuals, and violence victims. We conclude that the government should invest in a mass campaign against drug abuse, a mass campaign against violence, effective policing and rehabilitation of drug abuse and violent criminals in order to minimize or eradicate drug abuse and violence co-menace in the community.