Numerical methods and hypoexponential approximations for gamma distributed delay differential equations

TL;DR

This paper introduces a new numerical method for gamma distributed delay differential equations and proposes hypoexponential approximations that improve accuracy over traditional Erlang-based methods, with applications in epidemiology.

Contribution

The authors develop a functionally continuous Runge-Kutta method for gamma distributed DDEs and derive hypoexponential approximations that better capture the dynamics than Erlang approximations.

Findings

The new numerical method confirms high accuracy in integrating gamma distributed DDEs.

Hypoexponential approximations outperform Erlang approximations in preserving solution characteristics.

Application to epidemiological data demonstrates practical utility of the proposed methods.

Abstract

Gamma distributed delay differential equations (DDEs) arise naturally in many modelling applications. However, appropriate numerical methods for generic Gamma distributed DDEs are not currently available. Accordingly, modellers often resort to approximating the gamma distribution with an Erlang distribution and using the linear chain technique to derive an equivalent system of ordinary differential equations. In this work, we develop a functionally continuous Runge-Kutta method to numerically integrate the gamma distributed DDE and perform numerical tests to confirm the accuracy of the numerical method. As the functionally continuous Runge-Kutta method is not available in most scientific software packages, we then derive hypoexponential approximations of the gamma distributed DDE. Using our numerical method, we show that while using the common Erlang approximation can produce solutions that are qualitatively different from the underlying gamma distributed DDE, our hypoexponential approximations do not have this limitation. Finally, we implement our hypoexponential approximations to perform statistical inference on synthetic epidemiological data.