Homogenization of Ordinary Differential Equations for the Fast Prediction of Diabetes Progression

Abstract: The impact of physical activity on a person's progression to type 2 diabetes is multifaceted. Systems of ordinary differential equations have been crucial in simulating this progression. However, such models often operate on multiple timescales, making them computationally expensive when simulating long-term effects. To overcome this, we propose a homogenized version of a two-timescale model that captures the short- and long-term effects of physical activity on blood glucose regulation. By invoking the homogenized contribution of a physical activity session into the long-term effects, we reduce the full model from 12 to 7 state variables, while preserving its key dynamics. The homogenized model offers a computational speedup of over 1000 times, since a numerical solver can take time steps at the scale of the long-term effects. We prove that the error introduced by the homogenization is bounded over time and validate the theoretical findings through a simulation study. The significant reduction in computational time opens the door to apply the homogenized model in medical decision support systems. It supports the development of personalized physical activity plans that can effectively reduce the risk of developing type 2 diabetes.