Modeling Impulsive Intake of Glucose: The Significance of Diet to Glucose-Insulin System

It is important to get good dietary advice such as a daily value of exogenous glucose intake to keep blood glucose within reasonable range for everyone, especially for those with diabetes mellitus or with borderline diabetes mellitus. In this paper, two novel glucose-insulin models with impulsive glucose intake are considered to investigate the significance of diet to the glucose-insulin system. For the first model, we prove the positiveness, boundedness, and permanence of the solutions and then the existence and stability of the periodic solutions are demonstrated. For the second model, we consider the insulin secretion delay based on the first model. The sufficient delay-dependent conditions which determine the stability of the periodic solution are also obtained. In addition, the theoretical results are verified by numerical simulation. The results can give people some dietary advice how to prevent diabetes, and also, the intake amount and frequency of the exogenous glucose from diet are recommended for diabetic patients to assist in the treatment of diabetes.