Designing the Sports-Longevity Relationship Utilizing a 3D-ABC Fractional Framework

Abstract

This study proposes a novel fractional dynamic modeling framework to investigate the long-term impact of sports participation on human health and lifespan. Using publicly available data from international athletes, we construct a five-dimensional dynamic system that integrates both oscillatory adaptation variables, representing short-term training responses, and cumulative physiological indicators, reflecting long-term health effects. The framework is governed by the three-dimensional Atangana-Baleanu-Caputo (3D-ABC) fractional derivative, which enables a realistic representation of memory effects through its non-singular kernel and adjustable parameters controlling decay and adaptation profiles. Cosine-type memory dynamics are shown to effectively capture the periodic behavior of endurance and training load variables, while stress levels and longevity indices exhibit smooth, long-term accumulation patterns. Model parameters are estimated through data fitting, and sensitivity analysis demonstrates how varying the memory depth and decay rates significantly influence physiological outcomes. Furthermore, a decision-tree-based parameter tuning strategy is developed to guide practical model application. The results highlight the advantages of employing fractional calculus, particularly the 3D-ABC approach, in capturing both rapid and delayed biological adaptation processes. The proposed framework provides a robust theoretical and computational tool for understanding the link between structured physical activity and lifespan extension, while also offering insights for optimizing personalized sports training regimens.