Authors : Davidon Jani; Senzenia Chakauya; Alice Chimhondoro; Ever Moyo; Faith Chiwungwe

Volume/Issue : Volume 10 - 2025, Issue 11 - November

Google Scholar : https://tinyurl.com/4u597rhb

Scribd : https://tinyurl.com/btrbjy7r

DOI : https://doi.org/10.38124/ijisrt/25nov870

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Abstract : In this paper, we present a rigorous mathematical analysis of a novel fractional-order dynamical model describing the evolution of a married couple’s relationship, with a specific focus on the pathway to divorce. We extend a classical three- state model (happy marriage, unhappy marriage, divorce) to the Caputo fractional-order framework to incorporate memory effects and hereditary traits, which are intrinsic to human interactions. We established the model’s well-posedness by proving the existence, uniqueness, non-negativity as well as boundedness of solutions in a biologically feasible region. We perform a detailed stability analysis of the equilibrium points, deriving a novel threshold parameter R0 and proving local stability using Matignon’s conditions. Our work is anchored in the formulation and solution of an optimal control problem, where we introduce two time-dependent control measures: one aimed at improving the relationship quality, for instance, counseling and another to prevent reconciliation from a state of divorce. We used Pontryagin’s Maximum Principle for fractional-order systems to derive the necessary conditions for optimality and characterize the optimal controls. We performed numerical simulations, utilizing the forward-backward sweep method with the Grünwald-Letnikov approximation to illustrate the dynamics of the uncontrolled and controlled systems. Our results quantitatively demonstrate that our proposed optimal control strategy can significantly reduce the number of divorces as well as increase the proportion of happy marriages. Thus, providing a mathematical basis for targeted interventional policies. The expanded analysis provides complete proof for all technical components, offering a robust foundation for further research in socio- mathematical modelling.

Keywords : Fractional-Order Calculus, Optimal Control, Divorce Dynamics, Caputo Derivative, Pontryagin’s Maximum Principle, Stability Analysis, Fixed Point Theorem.

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