Modeling Diabetes
摘要
This chapter presents a collection of classroom-ready modeling projects focused on diabetes onset and management, designed for undergraduate courses in Mathematical Modeling, Differential Equations, and Dynamical Systems. The projects progress from linear to more realistic nonlinear representations of the glucose–insulin regulatory system formulated as systems of ordinary differential equations. Each model includes a discussion on the assumptions and is analyzed using analytical and numerical techniques. The chapter begins with a 2 \(\times \) 2 linear ODE model of blood glucose regulation as a foundation for modeling diabetes. Students estimate parameters from patient-specific time-series data, practice curve fitting, and incorporate discontinuous inputs to represent food intake and treatment regimens. The system is solved using the Laplace transform, reinforcing mathematical techniques through a biomedical application. Students then investigate insulin treatment strategies using optimal control theory, combining analytical and numerical approaches to balance effective glucose regulation with minimal insulin use. Finally, diabetes onset is modeled as a bifurcation in a nonlinear system. Students analyze equilibria, stability, and parameter sensitivity to explore mechanisms driving disease onset. Each project includes exercises designed for classroom or independent study and is accompanied by MATLAB code hosted on GitHub. Instructors will find the materials adaptable for lectures, labs, or projects, providing students a hands-on experience in mathematical modeling.