The regulatory system for diabetes mellitus: Modeling rates of glucose infusions and insulin injections

Novel mathematical models with open and closed-loop control for type 1 or type 2 diabetes mellitus were developed to improve understanding of the glucose-insulin regulatory system. A hybrid impulsive glucose-insulin model with different frequencies of glucose infusions and insulin injections was analyzed, and the existence and uniqueness of the positive periodic solution for type 1 diabetes, which is globally asymptotically stable, was studied analytically. Moreover, permanence of the system for type 2 diabetes was demonstrated which showed that the glucose concentration level is uniformly bounded above and below. To investigate how to prevent hyperinsulinemia and hyperglycemia being caused by this system, we developed a model involving periodic intakes of glucose with insulin injections applied only when the blood glucose level reached a given critical glucose threshold. In addition, our numerical analysis revealed that the period, the frequency and the dose of glucose infusions and insulin injections are crucial for insulin therapies, and the results provide clinical strategies for insulin-administration practices.