A Discrete Mathematical Modeling of the Influence of Alcohol Treatment Centers on the Drinking Dynamics Using Optimal Control

Abstract

In this paper, we propose a discrete mathematical model that describes the interaction between the classes of drinkers, namely, potential drinkers $\left(P\right)$, moderate drinkers $\left(M\right)$, heavy drinkers $\left(H\right)$, poor heavy drinkers $\left(T^p\right)$, rich heavy drinkers $\left(T^r\right)$, and quitters of drinking $\left(Q\right)$. We also focus on the importance of treatment within addiction treatment centers aiming to find the optimal strategies to minimize the number of drinkers and maximize the number of heavy drinkers who join addiction treatment centers. We use three controls which represent awareness programs through media and education for the potential drinkers, efforts to encourage the heavy drinkers to join addiction treatment centers, and psychological support with follow-up for the individuals who quit drinking. We use Pontryagin’s maximum principle in discrete time to characterize these optimal controls. The resulting optimality system is solved numerically by Matlab. Consequently, the obtained results confirm the performance of the optimization strategy.