A non-linear optimal control model for illicit drug use and terrorism dynamics in developing countries with time-dependent control variables

The burden of illicit drug use and terrorism has become a significant health and social problem in many developing countries. Illicit drug use and terrorism have negatively affected the economy and development of some nations because the death rate due to these two phenomena is increasing daily, particularly among the young generation. Implementing preventive and control measures for illicit drug use and terrorism requires considerable government spending. This study aims to formulate and critically examine a non-linear optimal control problem based on an existing non-linear compartmental model of illicit drug use and terrorism dynamics. The existing seven-dimensional deterministic model is modified to capture four time-dependent control variables denoting public awareness, empowerment rate, detention rate of illicit drug users, and rehabilitation rate of illicit drug users. The non-linear optimal control model is rigorously analyzed to explicitly prove its existence and characterize the optimal control quadruple using optimal control theory. Effects of different combinations of at least two control variables in minimizing illicit drug use and terrorism are evaluated through simulations. Moreover, a cost-effectiveness analysis is conducted to ascertain the most effective and least expensive strategy required for preventing and controlling the burden of illicit drug use and terrorism with limited resources in the population.