Optimal control of a heroin epidemic mathematical model

TL;DR

This paper develops and analyzes an optimal control model for a heroin epidemic, incorporating prevention and treatment strategies influenced by information dissemination, and identifies the most effective combined intervention approach.

Contribution

It introduces an optimal control framework for heroin epidemic management that accounts for behavioral responses to information and derives optimal strategies using Pontryagin's maximum principle.

Findings

Combined prevention and treatment strategies are most effective and cost-efficient.

The basic reproduction number determines stability of drug-free and endemic states.

Optimal controls can be characterized and numerically computed.

Abstract

A heroin epidemic mathematical model with prevention information and treatment, as control interventions, is analyzed, assuming that an individual's behavioral response depends on the spreading of information about the effects of heroin. Such information creates awareness, which helps individuals to participate in preventive education and self-protective schemes with additional efforts. We prove that the basic reproduction number is the threshold of local stability of a drug-free and endemic equilibrium. Then, we formulate an optimal control problem to minimize the total number of drug users and the cost associated with prevention education measures and treatment. We prove existence of an optimal control and derive its characterization through Pontryagin's maximum principle. The resulting optimality system is solved numerically. We observe that among all possible strategies, the most effective and cost-less is to implement both control policies.