In this paper, we consider a discrete model of the marital status of the family dynamics. It is assumed that individuals in the society can be classed in one of the eight compartments: virgin men, virgin women, married men, married women, divorced men, divorced women, widowed men and widowed women. The objective of this work is to treat the modeling and control the system that describes the case of monogamous marriage. We determine two controls which allow to reduce the number of virgin, divorced individuals and increase the number of married individuals. The first control is the benefits of an awareness campaign to educate virgin men and women about the benefits of marriage for the individual and for the society, the second control characterizes the legal procedures, administrative complications and the heavy financial and social consequences of divorces. The optimal control problems are procured based on a discrete version of Pontryagin's maximum principle and determined numerically using a progressive-regressive discrete schema that converges following a convenient test related to the Forward-Backward Sweep Method (FBSM) on the optimal control.

Since the beginning of the COVID-19 pandemic, vaccination has been the main strategy to contain the spread of the coronavirus. However, with the administration of many types of vaccines and the constant mutation of viruses, the issue of how effective these vaccines are in protecting the population is raised. This work aimed to present a mathematical model that investigates the imperfect vaccine and finds the additional measures needed to help reduce the burden of disease. We determine the R0 threshold of disease spread and use stability analysis to determine the condition that will result in disease eradication. We also fitted our model to COVID-19 data from Morocco to estimate the parameters of the model. The sensitivity analysis of the basic reproduction number, with respect to the parameters of the model, is simulated for the four possible scenarios of the disease progress. Finally, we investigate the optimal containment measures that could be implemented with vaccination. To illustrate our results, we perform the numerical simulations of optimal control.

The main objective of this paper is to develop a new mathematical model to study, analyze, and control the family status in several regions and to discuss the impact of the connectivity of regions and the mobility of residents on the marital status of the family, by adopting a multiregion discrete-time model. The modelling and the control process of the system that describes the case of monogamous marriages in a multiregion framework are considered. Two combined control strategies are proposed, which allow reducing the virgin and divorced individuals and increasing the number of married individuals in a specific region. The first control is considered as the impact of public awareness campaigns to educate virgin men and women about the benefits of marriage for the individual and the society; the second control characterizes the legal procedures, administrative complications, and the heavy financial and social consequences of divorces. The optimal control theory is applied to characterize such optimal strategies and determined numerically using a progressive-regressive discrete scheme to discuss the obtained results.

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In the present paper, we consider a mathematical model of a SEIR with immigration of infectives. The optimal control theory is applied to reduce the latent and infectious groups, increase the number of recovered individuals and this with an optimal cost. We use two controls representing the effort that reduces the contact between the infectious and susceptible individuals and a therapeutic treatment. We presents an approach that investigates a free terminal optimal time control witch give a minimum duration of a vaccination campaign. The Pontryagin's maximum principle is used to characterize the optimal controls and the optimal final time. We obtained an optimality system that we sought to solve numerically by an iterative discrete scheme that converges following an appropriate test similar the one related to the forward-backward sweep method.