Dynamics of illicit drug use and banditry population with optimal control strategies and cost-effectiveness analysis

Illicit drug use and banditry (IDB) have become public health and social issues in the world. In recent years, IDB-related violence has led to the deaths of many people in different countries. If considerable control efforts are not taken, the epidemic of IDB will continue to ravaging many societies. Hence, this paper is largely concerned with the formulation and analysis of a nonlinear optimal control model for the population dynamics of IDB. A deterministic eight-dimensional autonomous system of ordinary differential equations incorporating four control parameters with constant rates that account for public enlightenment campaign, corrective strategy of individuals in detention, corrective strategy of individuals in prison, and rehabilitation control of individuals in rehabilitation center is formulated to describe the dynamics of IDB spread in a population. The IDB-free state of the autonomous system is obtained, and shown to be locally asymptotically stable whenever the effective reproduction number R c is below unity. To determine the optimal cost-effective control strategy in minimizing the spread of IDB, the autonomous system is extended to a non-autonomous version with associated four time-dependent control functions. By employing optimal control theory, the system is analysed qualitatively to derive the existence results and characterize the optimal controls. The effects of combining at least any three control functions on the IDB dynamics are evaluated numerically. Cost-effectiveness analysis is conducted to derive the most effective and least costly strategy in minimizing the crimes in a population. The results of the cost-effectiveness analysis reveal that a strategy which combines all the four control interventions is the most efficacious and cost-effective strategy.