We propose an optimal control strategy by conducting awareness campaigns for diabetics about the severity of complications of diabetes and the negative impact of an unbalanced lifestyle and the surrounding environment, as well as treatment and psychological follow-up. Pontryagin’s maximum principle is used to characterize the optimal controls, and the optimality system is solved by an iterative method. Finally, some numerical simulations are performed to verify the theoretical analysis using MATLAB.

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.

In this paper, we present a continuous mathematical model PMHTrTpQ of alcohol drinking with the influence of private and public addiction treatment centers. We study the dynamical behavior of this model and we discuss the basic properties of the system and determine its basic reproduction number R0. We also study the sensitivity analysis of model parameters to know the parameters that have a high impact on the reproduction number R0. The stability analysis of the model shows that the system is locally as well as globally asymptotically stable at drinking-free equilibrium E0 when R0≤1. When R0>1, drinking present equilibrium E∗ exists and the system is locally as well as globally asymptotically stable at alcohol present equilibrium E∗.

In this work, we propose a discrete mathematical model that describes the dynamics of citizens who have the right to vote and their electoral behavior with regard to a political party during an awareness program or an electoral campaign. Also, we propose an optimal strategy for an awareness program or an election campaign that helps politicians to distinguish between different categories of voters in order to increase the participation rate in the electoral process and obtain the greatest possible number of votes with a minimal effort. Pontryagin’s Maximum Principle, in discrete time, is used to characterize the optimal controls and the optimality system is solved by an iterative method. The numerical simulation is carried out using Matlab. Consequently, the obtained results confirm the performance of the optimization strategy.

Nowadays, Diabetes is one of the most common diseases, which has a huge and growing socio-economic burden affecting individuals, families, and the whole society. In this paper, we propose an optimal control approach modeling the evolution from pre-diabetes to diabetes with and without complications and the effect of living environment. We show the existence of an optimal control and then use a numerical implicit finite-difference method to monitor the size of population in each compartment.