Numerical recipes in C: the art of scientific computing
Numerical recipes in C: the art of scientific computing
Applied Numerical Mathematics - The third international conference on the numerical solutions of volterra and delay equations, May 2004, Tempe, AZ
Applied Numerical Mathematics
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One of the questions involved in the formulation of a new model for a physiological phenomenon, when the model represents a dynamical system, is that concerning 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 very complex or impossible, leading to the imposition of conditions on the relative magnitude of the structural model parameters or to other partial results. We discuss a general technique whereby a probabilistic judgment 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 from which to estimate the dispersion of the model parameters. A stability criterion is obtained via the usual linearization around an equilibrium point, it is approximated as a Taylor series in the parameters truncated after the first term, and its variance is then computed from the dispersion of the parameters. While the conclusion is probabilistic in nature, it can be obtained for a wide class of models and from either sample or individual experimental subject's parameter estimates.