A MATHEMATICAL MODEL ANALYSIS OF MARRIAGE DIVORCE

HAILEYESUS TESSEMA1,∗, ISSAKA HARUNA2, SHAIBU OSMAN3, ENDESHAW KASSA4 1Department of Mathematics, College of Natural and Computational Sciences, University of Gondar, Gondar, Ethiopia 2Pan African University Institute of Basic Sciences, Technology and Innovation, Nairobi, Kenya 3Department of Basic Sciences, School of Basic and Biomedical Sciences, University of Health and Allied Sciences, Ghana 4Department of Mathematics, College of Natural and Computational Sciences, Debre Markose University, Debre Markose, Ethiopia


INTRODUCTION
Marriage is the sole means to enter family life in any community, and it is the foundational element that plays the primary goal of providing the needs and requirements of its members and society [1]. Marriage is a socially recognized and approved union of couples who dedicate to one another with the aspirations of a permanent and long intimate relationship [2]. There may be a disagreement inside a family that leads to separation and divorce for a variety of reasons. Divorces have become a common part of American life, affecting children of all ethnic backgrounds, religions, and social position [4]. Divorce rates have risen year after year in England and Wales [3] in Spain, a disparity in marital satisfaction, as well as the economy, has a significant impact on divorce [5].
Divorce is a common phenomenon in Africa, with immediate and long-term consequences [5,6]. Divorce rates among young people have risen in South Africa, which has one of the highest rates in the world [7]. Divorce has wider societal, cultural, economic, psychological, and political consequences [10,11] and divorce affects all the children [12,14]. Stable families produce a wealthy nation and a stable world, whereas unstable families produce a country and a world in disorder. This must first be decided out of the family for a country and region or the world at large to be at peace [6].
Mathematical modeling is an important tool used in analyzing the dynamics of infectious diseases [15]. The mathematical modeling of marriage divorce as a social epidemiology has received relatively little attention. For example, the study by [16] proposed and analyzed a nonlinear MSD ( Married, Separated and Divorced ) mathematical model to study the dynamics of divorce epidemic in Ghana. The existence and stability of the divorce free and endemic equilibria was proved using the computed basic reproduction number. They concluded that, reducing the contact rate between the marriaged and divorce, increasing the number of marriage that go into separation and educating separators to refrain from divorce can be useful in combating the divorce epidemic. Duato & L. J.odar [5] proposed and analyzed a mathematical modeling of divorce propagation allowing the estimation of the future divorced population. A sensitivity analysis of the growth of the divorced population with respect to the contagion rate is included.
A discrete model of the marital status of the family dynamics also studied by [18].
In this paper, we proposed a mathematical model of marriage divorce considering divorce as a transmitted disease that transmit in between human propagated by divorced women/ man over married ones.

MODEL DESCRIPTION AND FORMULATION
The model divides the entire population into four subpopulations: those who reach the age of getting married are single individuals, S(t); those who got married individuals are denoted by, M(t); those who separate bud not divorced are broken marriage individuals, B(t) and those who are divorced marriage are denoted by D(t).
π is the recruitment rate of individuals being single when he/she reaches the age of getting married. This individuals got married at rate of β . The married individuals got broken and move to the broken compartment due to the contact with the divorced individuals at a rate of α. The broken marriage recovered from their conflicts and renew their marriage at a rate of ε and live as the previous style. Some of these broken marriage got a permanent divorce at a rate of δ . this divorced people join the single subpopulation at a rate of ρ and some of the will die due to divorce at a rate of σ . The whole population has µ as an average death rate.
In addition we assume that sex, race and social status do not affect the probability of being divorced and members mix homogeneously (have the same interaction to the same degree).
The state variables of the model are represented and described in Table 1 and Table 2 shows the description of model parameters. The compartmental flow diagram for the model is shown in Figure 1.
With regards to the above assumptions, the model is governed by the following system of differential equation: If there is no death due to the divorce, we get After solving equation (3) and evaluating it as t −→ ∞, we got which is the feasible solution set for the model (1)  Then the solution set (S(t), M(t), B(t), D(t)) of system (1) is positive for all t ≥ 0.
Therefore, all the solution sets are positive for t ≥ 0.

Divorce free equilibrium point(DFEP).
When there is no divorce in marriage, I.e D = B = 0, the divorce free equilibrium occur and is obtained by taking the right side of Eq. (1) equal to zero. Therefore the divorce free equilibrium point is given by: , 0, 0 .

Basic reproduction number.
We calculate the basic reproduction number R 0 of the system by applying the next generation matrix approach as laid out by [19] and so it is the spectral radius of the next-generation matrix. Hence, Then by the principle, we obtained: The Jacobian matrices at DFEP is given by Therefore, the basic reproduction number is given us

Local Stability of DFEP.
Theorem 3.2. The Divorce free equilibrium point is locally asymptotically stable if R 0 < 1 and Proof. The Jacobian matrix of system (1) evaluated at the divorce-free equilibrium, we get The characteristic polynomial is given as Where From the Equation (9), we see that We applied Routh-Hurwitz criteria and by the principle equation (10) has strictly negative real root iff ψ 1 > 0 , ψ 2 > 0 and ψ 1 ψ 2 > 0. Clearly we see that ψ 1 > 0 because it is the sum of positive parameters and Hence the DFE is locally asymptotically stable if R 0 < 1.

Global Stability of DFEP.
Theorem 3.3. The divorce free equilibrium point E 0 of the model (1) is globally asymptotically Proof. Consider the following Lyapunov function Differentiating equation (11) with respect to t gives Substituting dB dt and dD dt from the model (1), we get: Here take c 1 = δ δ +ε+µ c 2 , then we have Taking c 2 = 1, and substituting R 0 we get for M ≤ M 0 = β π µ(β +µ) and dV dt ≤ 0 for R 0 < 1 and dV dt = 0 if and only if D = 0. This implies that the only trajectory of the system (1) on which dV dt ≤ 0 is E 0 . Therefore by Lasalle's invariance principle, E 0 is globally asymptotically stable in Ω. Socially, this implies that divorce can be eliminated irrespective of the initial population of divorced humans provided R 0 < 1.
Theorem 3.4. The endemic equilibrium E * of system (1) is locally asymptotically stable in Ω if R 0 > 1.

Bifurcation analysis.
A bifurcation is a qualitative change in the nature of the solution trajectories due to a parameter change. We investigate the nature of the bifurcation by using the method introduced in [17,21], which is based on the use of the central manifold theory. In short, the method is summarized by Theorem 4.1 in [25]. In the theorem, there are two basic parameters a and b that decides the bifurcation type of the model.
Where x = 0 is an equilibrium point for the system in eq (15). That is f (0, φ ) ≡ 0 for all φ .
Assume the following 0)) is the linearization matrix of the system given by (15) around the equilibrium 0 with φ evaluated at 0. Zero is a simple eigenvalue of A and other eigenvalues The local dynamics of (15) around 0 are totally determined by a and b.
In particular, if a < 0 and b > 0,then the bifurcation is forward; if a > 0 and b > 0,then the bifurcation is backward. Using this approach, the following result may be obtained: Theorem 3.6. The model in system (1) exhibits forward bifurcation at R 0 = 1.
Proof. : We proved the theorem using the concept center manifold theorem [25] the possibility of bifurcation at R 0 = 1. Then the following change of variables was made S = z 1 , M = z 2 , B = z 3 and D = z 4 . In addition, using vector notation z = (z 1 , z 2 , z 3 , z 4 ) T , and dz dt = G(x), with G = (g 1 , g 2 , g 3 , g 4 ) T , then model in system (1) re-written in the form: We consider the contact and transmission rate α as a bifurcation parameters so that R 0 = 1 iff The divorce free equilibrium is given by (z 1 = π β +µ , z 2 = πβ µ(β +µ) , z 3 = 0, z 4 = 0). Then the Jacobian matrix of the system (16) at a divorce free equilibrium is given by: The right eigenvector, w = (w 1 , w 2 , w 3 , w 4 ) T , associated with this simple zero eigenvalue can be obtained from Jw = 0. The system becomes From Eq. (18) we obtain Here we have taken into account the expression for α * . Next we compute the left eigenvector, , associated with this simple zero eigenvalue can be obtained from vJ = 0 and the system becomes Since the first and second component of v are zero, we don't need the partial derivatives of f 1 and f 2 . From the partial derivatives of f 3 and f 4 , the only ones that are nonzero are : and all the other partial derivatives are zero. The signs of the bifurcation coefficients a and b, obtained from the above partial derivatives, given respectively by Since the coefficient b is always positive and a is negative. Therefore, system (1) exhibits forward bifurcation at R 0 = 1.

Sensitivity
Analysis. Sensitivity analysis notifies us how significant each parameter to divorce transmission. To go through sensitivity analysis, we used the normalized sensitivity index definition as defined in [20] as it has done in [13,27].
Definition. The Normalized forward sensitivity index of a variable, R 0 , that depends differentiably on a parameter, p, is defined as: for p represents all the basic parameters and R 0 = αβ πδ µ(µ+β )(µ+ρ+σ )(µ+ε+δ ) For the sensitivity index of R 0 to the parameters: And it is similar with respect to the remaining parameters.
The sensitivity indices of the basic reproductive number with respect to main parameters are found in Table 3.

NUMERICAL SIMULATION
Numerical simulations of the model (1) are carried out, in order to illustrate some of the analytical results of the study. A set of reasonable parameter values given in Table 3 Table 4.

CONCLUSION
In this paper, we proposed and analysed a deterministic mathematical model for marriage divorce in a population. From the model analysis we obtained a region where the model is well-posed mathematically and epidemiologically meaningful. We determined the divorce free and endemic equilibra points and their local and global stability analysis in relation to R 0 .
The model bifurcation analysis is done and exhibits forward bifurcation at R 0 = 1. Sensitivity analysis of the model was performed and identified the positive and negative indices parameters.
Numerical simulation was performed and displayed graphically to justify the analytical results.
Therefore, from the above results we recommend stakeholders and policy makers to give a positive feedback for parameters that has negative indices and put negative feedback on positive indices in order to control marriage divorce in a population. Doing on the parameters α and ε is an effective control to combat divorce. This mean that, reducing the contact rate between the marriage and divorce and educating separators to refrain from divorce and renew their marriage are important ways of control divorce in a population.

ACKNOWLEDGMENTS
We would like to express our appreciation to the anonymous reviewers of the paper.