A reduced model for the long-term effects of physical activity on type 2 diabetes

Type 2 diabetes progresses slowly and may be reversed through lifestyle changes, but quantifying the long-term impact of regular physical activity remains challenging due to sparse longitudinal data. Mechanistic models offer a powerful tool by simulating metabolic processes over extended timescales. However, multi-scale formulations that capture both the short-term effects of exercise sessions and the slow evolution of disease tend to be computationally demanding, limiting their practical use in personalized decision support. To address this limitation, we derived a reduced version of a two-scale model that captures the short- and long-term effects of physical activity on blood glucose regulation. By analytically averaging the short-term effects induced by exercise, we developed a homogenized formulation that transmits the average contribution of physical activity to the slower glucose-insulin dynamics. This reduction preserves the key model dynamics while decreasing computational complexity by almost a factor 2000. We prove that the approximation error remains bounded and confirm the model's accuracy through a parameter-based simulation study. The resulting model provides a mathematically grounded reduction that retains key physiological mechanisms while enabling fast long-term simulations. This substantial computational gain makes it suitable for integration into medical decision support systems, where it can be used to design and evaluate personalized physical activity plans aimed at reducing the risk of type 2 diabetes.