Investigation of fractional order model for glucose-insulin monitoring with PID and controllability.

The global prevalence of diabetes, a chronic condition that disrupts glucose homeostasis, is rapidly increasing. Patients with diabetes face heightened challenges due to the COVID-19 pandemic, which exacerbates symptoms associated with the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection. In this study, we developed a mathematical model utilizing the Mittag-Leffler kernel in conjunction with a generalized fractal fractional operator to explore the complex dynamics of diabetes progression and control. This model effectively captures the disease's inherent memory effects and delayed responses, demonstrating improved accuracy over traditional integer-order models. We identified a single equilibrium point that represents the stable glucose level in healthy individuals. To establish the existence and uniqueness of the model, we employed fixed point theory alongside the Lipschitz condition. The Ulam-Hyers stability of the proposed model was also examined. Subsequently, we analyzed the chaotic behavior of the diabetic model using feedback control approaches, focusing on controllability and PID techniques. The application of chaos theory revealed that glucose-insulin dynamics are highly sensitive to initial conditions, leading to complex oscillatory behavior that can result in unstable glucose levels. By implementing fractional-order PID controllers, we effectively stabilized chaotic glucose dynamics, achieving more reliable blood sugar regulation compared to conventional methods, with a notable reduction in oscillation amplitude. We conducted numerical simulations to validate our findings, employing the Newton polynomial method across various fractal and fractional order values to assess the robustness of the results. A discussion of the graphical outcomes from the numerical simulations, conducted using MATLAB version 18, is provided, illustrating the dynamics of glucose regulation under different fractal-fractional orders. This comprehensive approach enhances our understanding of the underlying mechanics driving chaotic behavior in glucose-insulin dynamics.