A model for ultradian oscillations of insulin and glucose

A model is studied for the description of ultradian oscillations in human insulin secretion. According to this model, insulin and glucose concentrations obey a system of nonlinear ordinary differential equations with delay. This system is stable for the absence of delay and its reserve of stability decreases with the growth of delay. It is shown that oscillations of insulin and glucose arise due to the loss of stability in this system when the glucose delivery rate changes. The system is stable for small delivery rates which correspond to relatively small average concentrations of insulin and glucose (typical for healthy humans), as well as for relatively large concentrations of insulin and glucose (typical for developed diabetes in humans). Between these boundaries, the time-independent solution of the system becomes unstable and it transforms into a stable periodic solution describing the ultradian oscillations of insulin and glucose. In order to find the critical delivery rates, new sufficient stability conditions are derived for the linearized system. These conditions are developed in an explicit form. To verify them, direct numerical simulation of the nonlinear system is carried out. Its results show good agreement between the critical time of delay found by numerically simulating the initial nonlinear system and by employing the stability conditions derived for the linearized one. Based on the latter conditions, a sensitivity analysis is carried out, which reveals which parameters of the system are responsible for the insulin and glucose oscillations.