Global stability in a model of the glucose-insulin interaction with time delay

Abstract

A variety of models on the interaction between glucose and insulin have been suggested over the last 50 years. One, developed by Sturis et al. [19], and consisting of six nonlinear ordinary differential equations, has been widely accepted. However, the model has the disadvantage of containing auxiliary variables which have no clinical interpretation. In this paper we study an alternative model which incorporates a time delay explicitly, negating the need for the auxiliary equations. A simplifying assumption of having just one insulin compartment reduces the number of equations still further. We then study the resulting system of two differential delay equations, establishing results on positivity, boundedness, persistence and global asymptotic stability. For the latter, two quite different approaches are employed: comparison principles and Lyapunov functionals. The two approaches provide different sets of sufficient conditions for global stability, so that we investigate different regions of parameter space.