We discuss mathematical models for endurance exercise. Such models are needed to accurately assess athletes' fitness (e.g. for guiding training or identifying talent); and to predict performances (e.g. for optimising race strategies and pacing in track-cycling events). So-called W`-balance models, based around the "critical power" paradigm, are the current state-of-the-art. Unfortunately, W`-balance models cannot account for many essential qualitative features of endurance exercise. For instance, they fail to adequately capture that: (a) low ("heavy" or "moderate") exercise intensities cannot be sustained indefinitely; (b) pacing impacts exercise tolerance; (c) exercise modality affects subsequent recovery; (d) fatigue from prolonged exercise changes the power-duration relationship (a.k.a. the "durability" concept). We introduce the exercise, fatigue and recovery tracking (EFRT) model - a novel and rigorous framework for endurance exercise; and demonstrate that the EFRT model is more realistic than W`-balance models in the sense that it can capture all the above-mentioned essential qualitative features of endurance exercise (and many more). The EFRT model is also highly parsimonious: it requires only a small number of parameters and all of these have meaningful interpretations: they represent speed, endurance, durability, and recovery.
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