Married and discrete

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Advances in Difference Equations volumeArticle : Cite this article. Metrics details. In Married and discrete 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 of virgin, divorced individuals and increase the 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.

There are different of marriage. The most recognized one is the monogamous marriage, when a man marries a single woman. The second type of marriage is polygamy, it is the fact that a man marries several women. Lately another type of marriage has emerged, it is the marriage of homosexuals, it is the union of two people of the same sex.

In Married and discrete work we will treat the modeling and control of a system that describes the case of monogamous marriage. Mathematical models are tools for understanding the functioning of natural systems and for predicting their evolution. Among these models, there are those that study the dynamics of populations and ecosystems. Many researchers have studied models of population dynamics: prey and predator dynamics [ 1 — 4 ], epidemic dynamics in a population [ 5 — 8 ], molecular systems [ 9 — 11 ], etc. During longitudinal studies [ 12 — 19 ] and [ 20 ], changes in mental, physical, moral and social functioning were assessed of married, unmarried and widowed women.

These studies have determined that health effects differ between never-married women, divorced and separated women. There is support for social behavior and health behavior among married couples. Following the modeling of the dynamics of the population and inspired by population census studies in different countries, and in particular by the official of the population census in Morocco prepared by the High Commissioner for Planning HCP, public institution [ 21 ], which provided statistical data on the evolution of the demographic and socio-economic situation of women, we are interested in this work in defining a model that presents the evolution of the civil state of a population.

Civil status is the situation of a person in the family and society. The of The phenomenon of celibacy at Married and discrete age of 55 appears more in urban areas than cities. However, the news of spinsterhood or celibacy did not all rise. The showed that the proportion of single and married women who reached or exceeded the age of 15 the official marriage age in Morocco is 18 years old declined from In contrast, the proportion of married couples rose to The indicated the age at the first marriage, which was In the case of persons with special needs, there was a clear rise up to According to the official data of the HCP, in Morocco, 6.

Women with at least one boy divorce half as much as the others 8. Using these statistical studies and the evolution of marital status, we define a discrete model VMDW which defines the evolution of the marital status of the marriage in a population. In this article we are interested in the control of this system. The controls have been established to reduce the of unmarried and divorced individuals and to increase the of married individuals. The first control was considered as an accompaniment to the sensitization of the individuals to the benefit of marriage for the psychic and social stability of persons and also for the society.

The second control is determined for the persons who have initiated the divorce proceedings. This control is considered as a long and costly legal procedure. As we know, mathematics is an abstract tool that is used to solve real problems. In our work we assume but we are not obliged to do so that the control theory is more appropriate to solve our problem. In recent years, many attempts have been made to develop the control strategy for different systems [ 522 — 27 ] and [ 28 ].

There are a of different methods for calculating the optimal control for a specific mathematical model. In this paper, we investigate an approach that determines an optimal control relative to a discrete marital status model which allows to inoculate a control awarenesswhich allows to reduce the virgin individuals and to increase the of married individuals with a minimal cost.

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The optimal control problem was subject of an optimization criterion represented by the minimization of an objective function. The optimality system is solved based on an iterative discrete schema that converges following an appropriate test similar to the one related to the Forward-Backward Sweep Method FBSM. The paper is organized as follows. Numerical simulations are given in Section 4.

Finally, we conclude the paper in Section 5. The unit of i can correspond to days, months or years, it depends on the frequency of the survey and demographic studies as needed. However, demographic statistics are generally done annually, so the unit of i can be considered as years.

The following system describes a discrete model of the marital status of the monogamous marriage case:. In equations 1 - 8 Married and discrete, all parameters are nonnegative and defined in Table 1. The same principle applies to equations 67 and 8. The same principle can be applied to equations 4. In equations 3 and 4 from the model, the natural mortality of married men and women was considered.

Optimal control approach has been applied to models 1 - 8 to reduce the of virgin, divorced individuals and increase the of married individuals along the control strategy period.

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Then the model is given by the following equations:. The model 9 - 16 fulfils the constant population through time constraint, i. Note that this assertion proves that the constant population through time is independent of the control strategy. In order to find an optimal solution, first we find the Hamiltonian for the optimal control problem In fact, the Hamiltonian H of the optimal problem is given by.

We obtain the following theorem. We present now the obtained numerical simulations in the case when the studied domain represents cities or towns. We use data from Table 2. The HCP gives different statistics according to the age groups of the population. There are Then we update the optimal controls values 1819 using the values of state and costate variables obtained in the steps. Finally, the steps are executed until we reach a tolerance criterion.

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Considering the critical level of control, we give an optimal control sufficient to reduce the of virgin and divorced individuals and to increase the of married individuals. In the following, we can see that the optimal control function has a very desirable effect upon the population of virgin and divorced which decreases, while the married population increases for almost the entire length of process.

The graphs below allow us to compare changes in the of virgin, divorced and married individuals before and after the introduction of control. The time evolution of the respective populations with and without control is displayed in Figures 2 and 3respectively.

In Married and discrete population of 1, individuals, the of uncontrolled virgin men decreases to virgin men. On the other hand, by applying the control law, this decreases rapidly to reach out the end of the companion awareness of 20 virgin men. This result is similar to virgin women as shown in Figures 4 and 5. Figures 6 and 7 show that the of married men and women grows after 5 years and reaches a maximum value of about for married men and for married women, then it declines to for men and for women. This decline is due to the fact that the of divorced men and women develops after five years, but it is still quite high as far as the uncontrolled of married men and women, with a maximum value of for men and for women.

Figures 8 and 9 also show the effect of applying the control law by indicating that the of divorced individuals decreases faster at the beginning of treatment. Then we notice that the increases and we can justify it by the fact that the of divorced individuals is proportional to Married and discrete of married individuals. Since the of married individuals increases along the treatment, it is normal that the of divorced individuals also increases.

In this paper, we consider a discrete-time marital status model, we treat the modeling and control of a system that describes the case of monogamous marriage. An optimal control is investigated to reduce the of virgin, divorced population and to increase the married one. The first control introduced the benefits of an awareness campaign to educate virgin men and women about the benefits of marriage for individuals and for the society, and the second control characterized the legal procedures, administrative complications and the heavy financial and social consequences of divorces.

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Canale, RP: An analysis of models describing predator-prey interaction. XII Article Google Scholar. SIAM J. Galbraith, JK: The predator state. Discrete Dyn. Agusto, FB: Optimal isolation control strategies and cost effectiveness analysis of a two-strain avian influenza model. Biosystems 3 Wiley, New York Texts in Applied Mathematics, vol. Springer, New York Nicholson, JK: Global systems biology, personalized medicine and molecular epidemiology.

Married and discrete

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