Abstract
The aim of this paper is to study and investigate the optimal control strategy of a discrete mathematical model of drug consumption. The population that we are going to study is divided into six compartments: potential drug users, light drug users, heavy drug users, heavy drug users-dealers and providers, temporary quitters of drug consumption, and permanent quitters of drug consumption. Our objective is to find the best strategy to reduce the number of light drug users, heavy drug users, heavy drug users-dealers and providers, and temporary quitters of drug consumption. We use four control strategies which are awareness programs through media and education, preventing contact through security campaigns, treatment, and psychological support along with follow-up. Pontryagin’s maximum principle in discrete time is used to characterize the optimal controls. The numerical simulation is carried out using MATLAB. Consequently, the obtained results confirm the effectiveness of the optimization strategy.
Highlights
Drug addiction is a serious problem that affects a large population of the world causing some people diseases, psychological disorders, and in many cases it leads to death
When the number of drug users is so high, it is obligatory to resort to some strategies such as preventing contact through security campaigns in order to reduce the number of heavy drug users-dealers (Figure 3(c)); and to protect light drug users from contacting heavy drug users-dealers through security campaigns which has a positive effect on reducing the number of the heavy drug users (Figure 3(b)). erefore, we propose an optimal strategy by using the optimal control u2 in the beginning
Taking into consideration the importance and the effectiveness of this strategy on the individuals who temporarily quit drug, we propose an optimal strategy by using the optimal control u4 in the beginning which represents follow-up and psychological support to prevent the temporary quitters from reverting back to using the drugs occasionally (θ ≠ 0) (Figure 5(b)). e proposed strategy has an additional effect in decreasing clearly the number of temporary quitters of drug consumption
Summary
Drug addiction is a serious problem that affects a large population of the world causing some people diseases, psychological disorders, and in many cases it leads to death. Based on the aforementioned reasons, we will develop a discrete time model in order to study the dynamics of the population that uses drug and we will introduce two classes of drug users distributed into a compartment of heavy drug consumers, who do not influence other individuals, and the compartment of heavy drug users-dealers and providers of drug, who do influence other individuals. In order to find the best strategy to reduce the number of light drug users, heavy drug users, and temporary quitters of drugs, we will use four control strategies: awareness programs through media and education, preventing contact through security campaigns, treatment, and psychological support along with follow-up.
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