Age-Structured Absorbing Markov Modelling of Youth Fencing Dropout: Critical Windows and Mechanistic Drivers

Abstract

Adolescent dropout from organized sport is often a progressive process rather than an instantaneous event, with temporary disengagement frequently preceding permanent exit. This study presents an age-embedded absorbing Markov framework for analyzing youth fencing participation through three states: Active, Lapsed, and Dropped. The framework is designed for mechanistic analysis and intervention-timing comparisons rather than direct calibration to USA Fencing records. Age is incorporated directly into the state index, producing a sparse, time-homogeneous transition matrix while preserving age-specific transition behavior. Age-dependent transition probabilities are specified through logistic functions driven by three interpretable components: an age-localized vulnerability term, a bounded performance factor, and an encouragement term. Closed-form retention quantities are then obtained from the fundamental matrix of the absorbing chain, including expected seasons to dropout, expected time spent in Active and Lapsed states, and the full dropout-age distribution. In the illustrative baseline scenario, sensitivity analyses indicate that retention is most responsive to the timing and duration of age-localized vulnerability and to transitions involving temporary lapse. Approximately half of modeled dropout occurs through the Lapsed state, highlighting temporary disengagement as an important pathway rather than a minor deviation from continuous participation. Budget-constrained intervention experiments further show that, under equal total intervention magnitude, reducing lapse entry during early-to-mid adolescence produces larger retention gains than delayed or uniformly distributed strategies. The framework provides a tractable tool for examining dropout timing, pathway composition, and age-targeted intervention design in participation systems with reversible disengagement.