Dynamical analysis of impulsive insulin delivery in a glucose-insulin system with insulin-degrading enzyme effects

This paper investigated impulsive insulin therapy within an extended glucose-insulin regulatory system that included insulin-degrading enzyme dynamics. Two treatment modes were examined: periodic delivery and state-dependent feedback. For the periodic model, we established solution positivity and boundedness, proved the global asymptotic stability of a unique positive periodic solution for type I diabetes, and derived permanence bounds along with a safe dosing region for type II diabetes. For the state-dependent model, under mild conditions, we demonstrated the existence and global orbital asymptotic stability of a unique order-1 periodic orbit within a physiologically relevant glucose range. Numerical simulations supported the theoretical results and illustrated how the choice of dosing period, bolus size, and threshold level influenced long-term glucose control. The results provided a unified theoretical framework for comparing and optimizing impulsive insulin therapies.