Abstract
In this paper, we propose a discrete mathematical model that describes the interaction between the classes of drinkers, namely, potential drinkers P, moderate drinkers M, heavy drinkers H, poor heavy drinkers Tp, rich heavy drinkers Tr, and quitters of drinking Q. 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.
Highlights
Addiction is a social phenomenon and a psychological scourge that all societies suffer from
In order to achieve this purpose, we use optimal control strategies associated with three types of controls: the first represents awareness programs for potential drinkers, the second is the effort to encourage the rich to go to private treatment centers and the poor heavy drinkers to go to public treatment centers, and the third represents follow-up and the psychological support for temporary quitters of drinking
In order to achieve these objectives, we introduce three control variables. e first control u1 represents the effort of the awareness programs to protect the potential drinkers not to be drinkers. e second control u2 measures the effort made to make the heavy drinkers know private and public treatment centers of alcohol addiction and to encourage them to join those centers
Summary
Addiction is a social phenomenon and a psychological scourge that all societies suffer from. Adu et al [8] used a nonlinear SHTR mathematical model to study the dynamics of drinking epidemic; they divided their population into four classes: nondrinkers (S), heavy drinkers (H), drinkers receiving treatment (T), and recovered drinkers (R). E numerical exploration of discrete-time models is rather straightforward and can be implemented by nonmathematicians Besides these works, we will study the dynamics of a mathematical alcohol model PMHTrTpQ which contains the following additions:. In order to achieve this purpose, we use optimal control strategies associated with three types of controls: the first represents awareness programs for potential drinkers, the second is the effort to encourage the rich to go to private treatment centers and the poor heavy drinkers to go to public treatment centers, and the third represents follow-up and the psychological support for temporary quitters of drinking.
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