Dynamics of a multi-delay glucose-insulin-glucagon system under dual-hormone impulsive control

This paper develops a glucose-insulin-glucagon model formulated as a nonlinear impulsive differential system that incorporates three key physiological delays alongside periodic dual-hormone injections. The well-posedness of the system is established by proving the positivity and boundedness of solutions and the system permanence. Furthermore, by employing a fixed point theorem and constructing a Lyapunov functional adapted to the hybrid structure, the existence and stability of a positive periodic solution are demonstrated. Numerical simulations validate the theoretical results and reveal how the delays modulate glucose oscillations through phase interactions with the dosing schedule. The analysis indicates that the insulin-glucagon delivery phase relationship significantly impacts long-term glucose profiles. Another key finding is that for both hormones, administering a fixed total dose via high-frequency, low-amplitude pulses yields superior therapeutic outcomes. These results offer novel dynamical insights into delayed dual-hormone regulation and provide a theoretical basis for optimizing pulsatile treatment strategies.