Huang, TomShu, SuWichitaksorn, NuttananSpencer, Kirsten2026-04-122026-04-122026-05-03Mathematics 2026, 14(9), 1550, ISSN: 2227-7390 (Print).2227-7390http://hdl.handle.net/10292/20904In this study, we propose a general mathematical modelling framework based on the characteristics of elite athletes’ movements in the touch rugby matches to investigate the dynamic relationship between physical workload and injury risk over time. Our framework extends the Cox-based model in the context of touch rugby by incorporating a time-scaling component and cluster-specific heterogeneity simultaneously. In addition, we allow for the inclusion of covariates (e.g., velocity variation) to capture their effects. We applied our model to high-frequency wearable sensor data collected from 27 elite athletes (15 men and 12 women). The empirical study results show that our model, time-scaled frailty model (TSFM), demonstrates better goodness-of-fit than traditional frailty and Andersen–Gill models. The results reveal that higher velocity variation, particularly during high-intensity phases, and longer time of continuous exposure to the workload spike state significantly increased overload risk, ultimately resulting in injury. It also highlights the importance of individual differences, even under the same exercise intensity. These insights provide coaches with an evidence-based framework for athlete monitoring, allowing for more personalized training loads, tactical deployment, and injury prevention strategies in elite touch rugby environments.Copyright: © 2026 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license.49 Mathematical sciencessurvival analysisfrailty modelrecurrent events modellingunobserved heterogeneityexternal workload analysispiecewise exponential additive mixed modelJournal ArticleOpenAccess10.3390/math14091550