Fuzzy Smbd Mathematical Model for Marriage Divorce

Abstract

In this study, a deterministic SMBD model for marriage and divorce in a population is put out and qualitatively analyzed using the stability theory of differential equations. Using a next-generation matrix technique, the basic reproduction number in relation to the divorce-free equilibrium was determined. The parameters for divorce-free equilibrium is local asymptotic stability have been determined. Fuzzy analysis has been done by taking into account the heartbroken couple overcame their differences and decided to get back together and some will pass away as a result of divorcing as membership functions of fuzzy numbers. The sensitivity indices of the parameters with regard to preventing or increasing divorce in marriage were found. To support the analytical findings, numerical simulation was carried out and visually presented using Euler method

Introduction: Marriage plays a fundamental role in shaping families, fulfilling emotional, cultural, and societal needs. However, the rising prevalence of conflicts within families often leads to separation and divorce, which have profound implications for children and society as a whole. Globally, divorce rates have been increasing, with South Africa ranking among the countries with the highest rates. These trends have sparked interest in understanding the dynamics of marriage, separation, and divorce, as they significantly influence societal well-being. While mathematical models are frequently applied to study biological phenomena such as infeoctious diseases, their use in analyzing social issues like marriage and divorce remains limited. Existing models, such as the Married-Separated-Divorced (MSD) framework, provide foundational insights into these dynamics. This study seeks to extend these approaches by introducing a fuzzy Single-Married-Brken-Divorced (SMBD) model, incorporating fuzzy logic to address the inherent uncertainties in human behaviour and societal influences on divorce.

Objectives: Develop a deterministic SMBD model to examine the dynamics   of marriage, separation, and divorce. Identify the factors influencing the spread of divorce using fuzzy logic to account for uncertainties in reconciliation probabilities and divorce-related risks. Perform sensitivity analyses to determine which parameters most significantly impact divorce rates. Simulate real-world scenarios and suggest effective strategies to mitigate rising divorce rates. Rating fuzzy logic to address uncertainties in social factors.

Methods: The study develops a mathematical model dividing the population into distinct groups representing single, married, separated, and divorced individuals. It considers transitions between these groups influenced by factors such as marriage rates, separation rates, reconciliation efforts, and societal factors that contribute to divorce. To address uncertainties in human behaviour, fuzzy logic is applied to represent variables like reconciliation probabilities and divorce-related risks. The analysis includes calculating a measure similar to the reproduction number in epidemiology, which determines the potential for divorce to spread within a population. Stability analysis is conducted to evaluate whether divorce rates stabilize or escalate under different conditions. Sensitivity analysis is also performed to identify the most influential factors driving divorce rates. Finally, numerical simulations are used

Results: The study reveals that divorce dynamics are influenced by several key factors, including marriage rates, reconciliation probabilities, and divorce-related risks. Fuzzy logic analysis shows that uncertainties in social behaviour play a critical role, with higher levels of uncertainty leading to increased divorce rates. Sensitivity analysis identifies reconciliation efforts and separation rates as the most significant parameters affecting divorce trends. Simulations demonstrate that promoting reconciliation and reducing separation rates can stabilize divorce rates within a population. Scenarios with high reconciliation rates show a decrease in the spread of divorce, while increased separation rates lead to higher divorce prevalence. These results emphasize the importance of targeting social factors to manage and control the rising trends in divorce.

Conclusions: The findings of this study underline the importance of mathematical modeling in understanding complex social phenomena such as marriage and divorce. By incorporating fuzzy logic, the model effectively addresses uncertainties inherent in social behaviours and provides a more comprehensive understanding of the dynamics at play. The results suggest that targeted interventions, such as reconciliation programs and counselling initiatives, can significantly reduce divorce rates and stabilize family structures. This study provides a foundation for further exploration into the dynamics of social systems. Future research could expand the model to include additional factors such as economic and cultural influences. Additionally, analyzing the impact of intervention strategies, such as awareness campaigns and legal reforms, could further inform policies aimed at reducing divorce rates and promoting family stability.