Asymptotic properties of a delay differential equation model for the interaction of glucose with plasma and interstitial insulin

Abstract

Over the last 50 years, a variety of models for the interaction between glucose and insulin have been suggested in the literature. One of these, developed by Sturis et al. and consisting of six nonlinear ordinary differential equations, has been widely accepted as providing a possible mechanism for the origin of ultradian oscillations in pancreatic insulin secretion. However, the model contains non-observable auxiliary variables which are present to delay the action of insulin on glucose, but which have no clinical interpretation. In this paper we propose a model which incorporates this time delay explicitly. The resulting system consists of three differential delay equations. We prove theorems on positivity, boundedness, persistence and global asymptotic stability. For the latter, we employ a Lyapunov functional approach to obtain conditions which involve the time delay.