A Mathematical Overview of the Monogamous Marriage in a Multiregion Framework: Modelling and 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.
Mathematical models are useful tools for understanding the functioning of natural systems and for predicting their evolution. Among these models 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], and so forth.
Civil status is the situation of the person in the family and/or society. A person’s marital status is positioned in one of four categories: virgin, married, divorced, or widowed (VMDW). Several authors have discussed the social functioning, marital status, family stability, and social control of health behavior [12–18]. In , the authors have given a discrete mathematical model that describes the marital status of the monogamous marriage case and they also studied the optimal control that reduces the number of unmarried and divorced people and increases the number of married people. In this paper, we generalize the results established in  to the case of a multiregion system. We studied the effect of the population travel to different regions for the marriage, divorce, and widow.
Many studies on civil status have given statistics on the family situation of different regions (cities, rural regions, industrial regions, tourist regions, etc.), according to the age group of the population. For example in Morocco, the High Commissioner for Planning (HCP, public institution)  gives different statistics according to the age groups of the population; there are of single people for the age group of 20 to 24 and for the group of 25 to 29 years old. The percentage of divorce is for couples with less than 5 years of marriage and for married couples who have exceeded 20 years of marriages. 41.4% of married men and 41.8% of married women are identified by the HCP in the urban areas. The percentage of divorce is 0.8% for men and 3.1% for women. The widowed men are 0.6% of the population and women are 7.3% in urban areas. For the rural area, there are 40% of married men and 42.5% of married women; the percentage of divorce is 0.4% for married men and 1.4% for married women. Widowed men are represented by 0.6% of the population and women by 6.8% in rural areas.
In recent years, many attempts have been made to develop control strategies for different systems [5, 20–27]. There are a number of different methods for calculating the optimal control for a specific mathematical model. For example, Pontryagin et al.’s maximum principle  allows the characterization of the optimal control for an ordinary equation model system with a given constraint.
In , the authors have described a new modelling approach based on multiregion discrete-time SIR models aiming to describe the spatial-temporal evolution of epidemics that emerge in different geographical regions and to show the influence of one region on another region via infection travel.
In this paper and inspired by , we investigate an approach that determines an optimal control relative to a discrete VMDW model in a multiregion framework, which defines the evolution of the marital status of the marriage in a population, enabling decision-makers to develop very useful control strategies to reduce the virgin individuals and to increase the number of married individuals in a specific region.
The first control can be considered as public awareness campaigns showing to individuals the benefits of marriage on the psychic and social stability on persons and the society, or cultural entertained events to give people the chance to meet and to know and to allow themselves to get married. 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.
The optimal control problem was the 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: in Section 2, the model VMDW is described for a multiregion discrete model. In Section 3, we give some results concerning the existence of optimal control and we use Pontryagin’s maximum principle to investigate the analysis of control strategies and to determine the necessary condition for optimal control. Numerical simulations are given in Section 4. Finally, we conclude the paper in Section 5.
2. Mathematical Modelling
We consider a discrete-time model VMDW of the marital status of the family dynamic within a domain of interest which represents a country, a city, a town, or a small domain. We assume that there are geographical regions (domains) of the domain studied . Let , and let be the population of domain at time ; that is, the number of individuals who are residents in .
This model classifies the marital status of the family dynamics of a population into eight compartments in each region : virgin men , virgin women , married men , married women , divorced men , divorced women , widowed men , and widowed women .
The unit of time 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 units , can be considered as years. The following system describes a discrete model of the marital status of the monogamous marriage case of a region :where , , , , , , , and are the given initial state in the region . In equations (1)–(8), all parameters are nonnegative and defined in Table 1.
For equation (1) of the model, virgin men of a region can contact a virgin, divorced, or widowed woman of another region with , , and rates, respectively, and this contact can result in a marriage. The married couple can stay in the region or settle in the conjoin region . Thus, the number of virgin men decreases and the number of virgins at the instant is substituted for the number and a portion of this number that represents the part of married couples who decides to settle in is added at the time to the number of men marrying in region and the other portion is added to the number of men marrying in region . Then, we have , and .
Similarly in equation (2), the number of virgin women in region decreases at the instant by substituting the number of virgin women at the instant , the number which represents the number of married women after contact with a virgin, divorced, or widowed men in region with , , and levels, respectively. Also we have , and .
In equation (3), a divorced man in region can contact a virgin, divorced, or widowed woman of region with , , and rates, respectively, so this contact can result in marriage. Then the number is decreased by the number of divorced men at the time and added to the number of married men and we have and . In the model we propose here, it was considered that a divorced man remains in his region and the divorced woman returns to the region of her parents; this is the case for the majority of regions whether they are conservative or not. And so the number is added to the number of divorced men and the number of divorced women is added to the number of divorced women of the region with being the rate that a woman divorces a man from the region and returns to the region . Consequently, the total number of is added to the number of divorced women in the region with the relation . The same principle applies to equations (5) and (6) with .
For equation (7), the number of married men in the region increases at the instant by the number of virgin, divorced, and widowed men who are married by contacting virgin, divorced, or widowed women of the region and decreases the natural mortality with a and divorce rate with a rate. The same principle can be applied to equation (8).
In equations (7) and (8) from the model, the natural mortality of married men and women was considered. The mortality rate for married men is and the mortality rate for married women is . These rates appear in equations (5) and (6) which correspond to widowed men and women, respectively, so the number of mortalities which is married men and the number are added to the number of widowed women and similarly the number is added to the number of widowed men.
3. Methods and Results
3.1. An Optimal Control Approach
An optimal control approach has been applied to models (1)–(8) to reduce the virgin and divorced individuals and increase the number of married individuals along the control strategy period. For this, we introduce a control variable that characterizes the benefits of an awareness campaign to educate virgin men and women about the benefits of marriage for the individual and the society, especially the legal procedures, administrative complications, and the heavy financial and social consequences of divorces, respectively, in the abovementioned models (1)–(8). Then, the model is given by the following equations:
3.2. Characterization of the Optimal Control
For an initial state , , we consider an optimization criterion defined by the following objective function:subject to systems (9)–(16). Here, , , and are positive constants to keep a balance in the size of , , and , respectively. In the objective functional, and are the positive weight parameters which are associated with the controls and .
In other words, we seek the optimal controls such thatwhere is the set of admissible controls defined bywhere .
The sufficient condition for existence of an optimal control for problem (18) follows from standard results of . In order to find an optimal solution, first we find the Hamiltonian for the optimal control problem (18). In fact, the Hamiltonian of the optimal problem is given bywhere are the adjoint functions to be determined suitably.
At the same time by using Pontryagin et al.’s maximum principle , we derive necessary conditions for our optimal control. We obtain the following theorem.
Theorem 1. Let , , , , , , , and be optimal state solutions with associated optimal control for the optimal control problem (18). Then, there exist adjoint variables that satisfywith transversality conditionsFurthermore, the optimal control is given byfor .
Proof. Using Pontryagin et al.’s maximum principle  and setting , , , , , , , , and , we obtain the following adjoint equations: