Abstract

The synthetic drugs that are becoming increasingly popular in drug-markets are some of the most destructive drugs which have made headlines causing serious social and health care issues in the past few years. In this work, a synthetic drug transmission model with general contact rate and Holling Type-II functional responses among susceptible and drug addicts (both psychological and physiological) is proposed. Sensitivity analysis provides that controlling the contact rate among the drug addicts and the susceptible is better than the treatment. Further, an optimal control problem has been formulated to minimize the cost and drug addiction by choosing the counselling treatment (including awareness programmes) as a control variable. Numerical analysis has indicated that if the control policy is implemented, then it will be economically viable for long term. In the proposed model system, it is observed how counselling can prevent the psychologically addicted persons from taking more drugs.