Model reduction for initial value ODEs

Abstract

Many physical phenomena in biology and physiology are described by mathematical models that comprise a system of initial value ordinary differential equations. Each differential equation may often be written as the sum of several terms, where each term represents a different physical entity. A wide range of techniques, ranging from heuristic observation to mathematically rigorous asymptotic analysis, may be used to simplify these equations allowing the identification of the key phenomena responsible for a given observed behaviour. In this study we extend an algorithm for automatically simplifying systems of initial value ordinary differential equations (Whiteley (2010)) that is based on a posteriori analysis of the full system of equations. Our extensions to the algorithm make the following contributions: (i) each equation in a system of differential equations may be written as a finite sum of contributions (including the derivative term), and any one of these terms may be neglected (if it is appropriate to do so) in the simplified model; and (ii) a simplified model is generated that allows accurate prediction of one or more components of the solution at all times. These extensions are illustrated using examples drawn from enzyme kinetics and cardiac electrophysiology.