Distributed Delay Differential Equation Representations of Cyclic Differential Equations

TL;DR

This paper extends the equivalence between compartmental ODE models and scalar distributed delay differential equations to include non-linear transit rates and delays, providing new analytical tools for biological modeling.

Contribution

It demonstrates that complex compartmental models with non-linear and delayed transitions can be represented as scalar DDEs, simplifying analysis and revealing hidden physiological processes.

Findings

Derived scalar DDEs for biological models with non-linear transit rates

Calculated equilibria and characteristic functions without determinants

Identified physiological processes obscured in compartmental structures

Abstract

Compartmental ordinary differential equation (ODE) models are used extensively in mathematical biology. When transit between compartments occurs at a constant rate, the well-known linear chain trick can be used to show that the ODE model is equivalent to an Erlang distributed delay differential equation (DDE). Here, we demonstrate that compartmental models with non-linear transit rates and possibly delayed arguments are also equivalent to a scalar distributed delay differential equation. To illustrate the utility of these equivalences, we calculate the equilibria of the scalar DDE, and compute the characteristic function-- without calculating a determinant. We derive the equivalent scalar DDE for two examples of models in mathematical biology and use the DDE formulation to identify physiological processes that were otherwise hidden by the compartmental structure of the ODE model.