Stability Analysis of Divorce Dynamics Models

This article examines the mathematical model of divorce. This model consists of four population classes, namely the Married class (M), the population class who experiences separation of separated beds (S), the population class who is divorced by Divorce (D), and the population class who experiences depression or stress due to divorce Hardship (H). This study focuses on the stability analysis of divorce-free and endemic equilibrium points. Local stability was analyzed using linearization and eigenvalues methods. In addition, the basic reproduction number is provided via the next generation matrix method. The existence and stability of the equilibrium point are determined from . The results showed that the rate of interaction between population M and populations other than H is very influential on efforts to minimize divorce. Divorce can be minimized when the transmission rate is reduced to . Reducing the transmission rate and increasing the rate of transfer from split bed class to married class can turn divorce endemic cases into non-endemic cases. A numerical simulation is given to confirm the analysis results.


Introduction
In a marriage, everyone wants a happy, eternal, and prosperous home life, in accordance with the objectives of marriage as stated in Law No.1 of 1974 ( [1,9]). However, not everyone can form a family that they aspire to, this is due to divorce, both death divorce, divorce talaq, and divorce based on the judge's decision [1]. The problem of the dynamics of the divorce epidemic is one of the most familiar problems in society. Several previous studies have modeled problems related to the dynamics of this divorce [15]. For example, the mathematical model of divorce dynamics developed by Patience Pokuaa Gambrah, et al [6], which examines the impact of counseling in divorce cases. Then the research conducted by Patience Pokuaa Gambrah and Yvonne Adzadu [7] divided the mathematical model into 3 compartments, namely marriage (M), separated (S), Divorced (D). In this study, the researchers drew a conclusion that the divorce epidemic can not Syamsir, Syamsuddin Toaha, Kasbawati

Jurnal Matematika, Statistika & Komputasi
only be controlled by reducing the level of contact between marriage and divorce but also increasing the number of marriages that go into separation and educating themselves from divorce which can be useful in fighting the epidemic. In this research, researchers will make modifications by combining the two models mentioned above and adding some controls to minimize the occurrence of divorce. Researchers will examine the dynamics of divorce with optimal control.

Divorce Dynamics Model
The development of the model in this study is an amalgamation of the dynamics of divorce according to Patience Pokuaa Gambrah, et al [6] and Patience Pokuaa Gambrah and Yvonne Adzadu [7] then resolved with optimal control theory ( [5,13,14]   From the compartment diagram of the model, the following system of differential equations is obtained: (2.1) With the total population is assumed that the initial conditions of the system meet: From system (4.1) can be obtained: 2) The description of the variables and parameters used is described in the following table.

Person/year
The average interaction between M and individuals other than H.
The rate of people divorcing after separation 1/year The rate at which people experience depression or stress after divorce The rate of people getting married again after divorcing 1

1/year
The rate at which people return to married status after separation 2

1/year
The rate at which people return to marriage after experiencing depression or stress

The point of equilibrium and stability is divorce-free and endemic
The divorce dynamics model in model (2.1) can be found for its equilibrium point if it meets ( [2,3,5,13]). Based on these requirements, system (2.1) is obtained:

Syamsir, Syamsuddin Toaha, Kasbawati Jurnal Matematika, Statistika & Komputasi
(3.1) The divorce-free equilibrium point is a condition where there is no divorce problem in a population or the number of divorced individuals is zero [12]. The point of divorce-free equilibrium is expressed in ( ) by assuming Thus from system The equilibrium point of a divorce or endemic occurs when divorce is still occurring. Thus, endemic occurs when system (1) is stable but all compartments are always positive i.e. and or can be expressed in the form ( ) where and .

Basic Reproduction Numbers
In epidemiology, the rate of spread of an infectious disease is usually measured by a value called the basic reproduction number ( ) ( [4,8,13]).. In this case in order to be free from divorce, it must be set [10,11]. In this case each divorced individual can only affect on average less than one new individual, so that in the end the divorce will be zero. Meanwhile, if , then each individual who is divorced can cause or affect on average more than one new individual, so that in the end there will be an epidemic.
To find , because what you want to control is the population that causes divorce, only models S and D are needed in equation (2.1), namely: From this equation, which causes an increase in divorce cases, namely ( ) which can be considered as ( ). Whereas the rest is given as ( ) and negated, so that ( ). and ( ) are obtained as follows: Furthermore, we obtain the F and V matrices substituted by the divorce-free equilibrium point ( ) ( ) with the following results: ) ( Next will be determined the inverse of the matrix So that the basic reproduction value is obtained by multiplying the matrix by the inverse matrix as follows: Using the characteristic equation ( 1 ) , we get: which has a solution: Because the value of the basic reproduction number is obtained from the spectral radius or the largest value of the eigenvalues, the basic reproduction number for the system of equations above is  (   2 ) ( 2 ). Based on the results obtained for , to make , the denominator must be greater than the numerator. Mortality due to natural factors (μ) and the rate of depression or stress after divorce (γ) cannot be increased. Therefore what needs to be done is to increase the number of individuals who return to their married status after separating ( 2 ). In addition, the mean interactions of M with other individuals except H (β) and the proportion of M divorced (ρ) must also be derived. Whereas (P) is the chance of successful interaction between M and other populations except H. Thus the divorce rate will be reduced. So it can be said, from this analysis it will be known that the most influential parameters of all the parameters in the divorce dynamic model are the parameters β, ρ, and 2 . Suppose μ = 0.003, γ = 0.56, ρ = 0.03, P = 1, and Λ = 0.08, then the relationship between the average interaction of M with individuals other than H (β) to the rate of remarriage after parting ( 2 ) is illustrated in Figure 2 below.

Numerical Simulation
Simulations are given to provide a geometric picture related to the results of the analysis that has been carried out. Simulations are made using the Maple 18 program and carried out by providing values for each parameter according to the condition .

a. Simulation for
Based on the given model (2.2), and the values adopted from [6] and [7] and with a slight modification for the simulation are given = 0.03, β = 0.08, ρ = 0.052, γ = 0.00001, δ = 0.00001,    In Figure 5.9 it can be seen that the value of ( ) goes to as long as t increases. Then in Figure 5.10, Figure 5.11 and Figure 5.12, it can be seen that the values of ( ), ( ) and ( ) go to , and respectively as t increases. In other words, in Figures 5.9 to 5.12 it can be seen that ( ) . So the equilibrium point is stable for

Conclusion
The mathematical model of the dynamics of divorce has an equilibrium point of divorce-free and divorce. The analysis results from the model also obtained the basic reproduction number ( ). From the value of the basic reproduction number, it can be seen that whether all the eigenvalues associated with the divorce-free equilibrium point and the existence of divorce are all negative or not. There are several parameters that affect the value of , namely the proportion of married people who are divorced (ρ), the average interaction between people with married status and other individuals (β), the chances of successful interactions between people with married status and other populations (P ), the proportion of individuals who have avoided divorce due to the presence of external parties (Judge) ( 2 ) natural death rate (μ), rate of people returning to marriage status after separation ( 2 ) and the rate of people experiencing depression or stress after divorce ( γ). When , all eigenvalues associated with the divorce-free equilibrium point are negative. This means that the point of divorce-free equilibrium is stable. Whereas when , there is one eigenvalue associated with a positive divorce-free equilibrium point and all eigenvalues associated with the equilibrium point of the divorce are all negative.