PROBABILISTIC DETERMINATION OF STABILITY FOR A DELAY-DIFFERENTIAL MODEL OF THE GLUCOSE-INSULIN DYNAMICAL SYSTEM

Abstract

Some of the questions involved in the formulation of a new model for a physiological phenomenon, when the model represents a dynamical system, concern its qualitative behavior. The determination of the stability of a particular dynamical system is usually made analytically, from a linearization of the system around an equilibrium point. This analytic proof may often be complicated or impossible, leading to the imposition of conditions on the relative magnitude of the model structural parameters or to other partial results. We discuss a general technique whereby a probabilistic judgement is made on the stability of a dynamical system, and we apply it to the study of a particular delay differential system modelling the relationship between insulin secretion and glucose uptake. This technique is applicable in case experimental material is available: a stability criterion is obtained via the usual linearization around an equilibrium point and its distribution is approximated from the estimated dispersion of the model parameters. The probability that the stability criterion is such as to make the model stable can then be computed directly. While the conclusion is probabilistic in nature, it is generally applicable to a vast class of models.