Optimal control of a heroin epidemic mathematical model

Abstract

A heroin epidemic mathematical model with prevention information and treatment, as control interventions, is analysed, assuming that an individual's behavioural 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 the 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.

Keywords:

• Heroin epidemic mathematical model
• stability analysis
• behavioural change
• optimal control

2020 Mathematics Subject Classifications:

• 34D20
• 49J15
• 92D30

Acknowledgements

The authors are grateful to two anonymous reviewers for several constructive comments that really helped to improve the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Sowndarrajan is thankful to the Ministry of Human Resources Development (MHRD) and National Institute of Technology Goa, India, for awarding him a Senior Research Fellowship to Sowndarrajan. Debbouche and Torres are grateful to the Portuguese Foundation for Science and Technology (FCT) (Fundação para a Ciência e a Tecnologia), project UIDB/04106/2020 (CIDMA).