Modeling Impulsive Injections of Insulin: Towards Artificial Pancreas

We propose two novel mathematical models with impulsive injection of insulin or its analogues for type 1 and type 2 diabetes mellitus. One model incorporates the periodic impulsive injection of insulin. We analytically show the existence and uniqueness of a positive globally asymptotically stable periodic solution for type 1 diabetes, which implies that the perturbation caused by insulin injection will not disturb homeostasis but amend the glucose from a pathological level toward a healthier level. We also show that the system is uniformly permanent for type 2 diabetes, that is, the glucose concentration level is uniformly bounded above and below. The other model determines the insulin injection by closely monitoring the glucose level when it reaches or passes a predefined but adjustable threshold value. We analytically prove the existence and stability of the order one periodic solution, which ensures that the perturbation by the injection in such an automated way can keep the blood glucose concentration...