A procedure for deriving new ODE models: Using the generalized linear chain trick to incorporate phase-type distributed delay and dwell time assumptions

Paul Hurtado,Cameron Richards

doi:10.5206/mase/10857

Paul Hurtado, Cameron Richards

Open Access

https://doi.org/10.5206/mase/10857

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Abstract

Ordinary differential equations models have a wide variety of applications in the fields of mathematics, statistics, and the sciences. Though they are widely used, these models are sometimes viewed as inflexible with respect to the incorporation of time delays. The Generalized Linear Chain Trick (GLCT) serves as a way for modelers to incorporate much more flexible delay or dwell time distribution assumptions than the usual exponential and Erlang distributions. In this paper we demonstrate how the GLCT can be used to generate new ODE models by generalizing or approximating existing models to yield much more general ODEs with phase-type distributed delays or dwell times.

Highlights

Ordinary differential equations (ODE) models are widely used in the sciences, in part because of the relative ease of formulating and analyzing ordinary differential equations (ODEs) models [35, 25, 3, 7, 1]
With the above preliminaries in hand, we may detail our generalized linear chain trick (GLCT)-based procedure which serves as a tool for implementing the GLCT to derive new ODE models with phase-type dwell time assumptions
We will illustrate the application of this GLCT-based procedure for deriving new ODE models by generalizing various biological models taken from the peer reviewed literature

Summary

Introduction

Ordinary differential equations (ODE) models are widely used in the sciences, in part because of the relative ease of formulating and analyzing ODE models [35, 25, 3, 7, 1] They are often criticized for their limited capacity to only incorporate a narrow range of delay and dwell time assumptions. In an ODE framework, the linear chain trick has long been used to incorporate exponential and Erlang distributed delays [23, 22, 26, 34, 6] Recently this technique has been generalized to a much broader family of delay or dwell time distributions [16]. These tools and techniques enable modelers to draw from a richer set of ODE model assumptions when constructing new models, and a framework for more clearly seeing

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