Mathematical analysis of modified blood glucose insulin model through fractal fractional operators.

This study presents an advanced mathematical perspective of a generalized diabetes model, emphasizing the critical complications associated with this disease, such as cardiovascular disease, kidney failure, nerve damage, vision problems, and weakened immunity conditions, which can escalate into life-threatening conditions such as heart attacks, strokes, and blindness. Blood glucose, an essential energy source for the human body, is regulated by hormones such as insulin and glucagon. Diabetes emerges either due to the body's resistance to insulin or the autoimmune destruction of insulin-producing cells in the pancreas. Focusing on these physiological insights, we reformulate the blood glucose-insulin (MBGI) model by incorporating some novel parameters, introducing a dietary intake compartment, and employing a new fractional operator in the sense of a fractal-fractional derivative to better capture the complex dynamics of the disease. This study investigates the existence, uniqueness, and Hyers-Ulam stability of solutions via fixed-point approaches, particularly Leray-Schauder techniques. Furthermore, a numerical scheme based on Newton's polynomial interpolation is developed to visualize the behavior of the model. The attained results show that increasing both the fractal dimension and fractional order leads to a crucial reduction in glucose concentration, offering valuable insights for the effective management and control of diabetes.