Modeling and qualitative analysis of diabetes therapies with state feedback control

For the therapies of diabetes mellitus, a novel mathematical model with two state impulses: impulsive injection of insulin and impulsive injection of glucagon, is proposed. To avoid hypoglycemia and hyperglycemia, the injections of insulin and glucagon are determined by closely monitoring the plasma glucose level of the patients. By using differential equation geometry theory, the existence of periodic solution and the attraction region of the system have been obtained, which ensures that injections in such an automated way can keep the blood glucose concentration under control. The simulation results verify that the better insulin injection strategy in closed-loop control is a larger dose but longer interval rather than a smaller dose but shorter interval. Besides, our numerical analysis reveals that medicine studies and practice that slow down the insulin degradation are helpful for the plasma glucose control. Our findings can provide significant guidance in both design of artificial pancreas and clinical treatment.