Modeling and mathematical analysis of drug addiction with the study of the effect of psychological and biological treatment

In this article, we propose a discrete mathematical model which describes the propagation of the drug phenomenon in a human population. The population is unscrewed in five compartments: "S" People likely to become drug addicts, "M" Moderate drug addicts, "H" Heavy drug addicts, "T" People receiving drug addiction treatment, "R" The recovered people who have completely abstained from drug addiction. Our goal is to find a better strategy to reduce the number of heavy addicts and to maximize the number of people receiving full treatment. The tools of optimal control theory were used in this study, in particular the Pontryagin maximum principle.