Study On Three Delay Model Of Blood Glucose - Insulin Regulation System

The glucose-insulin regulated system is a part of vital movement. There is a risk of having diabetes as regulated system goes into functional disorder. The glucose content of normal body mainly regulated both by insulin secreted by cell β and glucagon secreted by cell a. For type-Ⅱ diabetes mellitus accounts for 90% of all diabetics, this paper formulates two models describing the dynamic behavior of glucose-insulin regulating system in the body of diabetics.In the part one, having taken into account the effect of physiological time-delay both for the glucose-insulin regulated system and its dynamical properties, three physiologically time-delays of the glucose-insulin regulated system are introduced and model of the system is formulated while existence, uniqueness, positive-ness, and boundness of solutions of ODE with time-delay are analyzed. In the circumstances of less insulin secreted by pancreas β-cell in the body of diabetics, the unique and positive equilibrium of two models are glob-ally asymptotical stable by the method of Lyapunov function. Correlated conditions for globally asymptotic stability are formulated and conditions for feasibility are analyzed either.In the part two, in order to simulate the pulsatile injection of insulin with regular daily ration inside the insulin pump, the modeling of the blood glucose-insulin regulated system with three time-delays and recurrent pulse is formulated. The existence of the periodic solution of impulsive differential equation is shown by Krasnoselskii’s Fixed Point Theorem. τs= 0 and τt,=τh= 0 as two conditions for stability are respectively analyzed in the circumstances where less insulin secreted by pancreas β-cell in the body of diabetics. By the method of Lyapunov function, the solutions of model under two conditions above are globally asymptotic stable and two different criteria are given.All conclusions above show that there is a globally asymptotic and positive equilibrium or periodical solution for type-Ⅱ diabetes mellitus glucose-insulin regulating system. Hence, we could rise up the level of glucose-insulin to normal just by choosing a proper injection period and a single dose in order to make a balance of the regulating system of glucose-insulin.