A Nonlinear State-Space Model for Fatigue Attention Dynamics in Online Learning Environments

Behavioural analytics remain a dominant approach for modelling learner engagement and predicting performance in digital learning environments. However, existing approaches are largely retrospective, relying on observable behavioural outcomes rather than modelling the underlying cognitive state dynamics that evolve during sustained learning. This study proposes a nonlinear state-space modelling framework that formalises the interaction between cognitive fatigue, attention, and learning as a continuous-time dynamical system. Fatigue is modelled as a latent state governed by load–recovery dynamics, attention is represented as a fatigue-coupled cognitive resource, and learning accumulation is expressed as an attention-mediated process under saturation constraints. The model is discretised and empirically estimated using time-indexed webcam-derived pilot data (N = 63) and further validated using a large-scale intervention dataset (N = 1245). Parameter estimation is performed using regression-based approximation of the discretised state equations, with cluster-robust inference applied to account for intra-session dependencies. The webcam-derived features were pre-processed using temporal windowing and normalisation to ensure consistency across sessions. The swarm-optimised intervention was implemented through adaptive control of instructional load and recovery scheduling, enabling real-time regulation of fatigue progression. Empirical results demonstrate statistically significant model validity, with fatigue dynamics showing moderate explanatory capability(R2 = 0.543, p<0.001) and attention dynamics also significant (R2 = 0.499, p = 0.004). At the system level, adaptive intervention significantly reduced fatigue and improved learning performance (t(1244) = 14.34, p < 0.001). The findings suggest a transition from retrospective behavioural modelling toward anticipatory cognitivestate regulation, contributing toward a computational foundation for fatigue-aware adaptive learning systems.