Numerical methods and hypoexponential approximations for gamma distributed delay differential equations
Published in IMA Journal of Applied Mathematics, 2022
A Gamma distributed delay differential equation can be written as
\[\frac{dX}{dt} = F\left(X(t), \int_{0}^{\infty} X(t-s) g_a^j(s)ds\right)\]where \(g_a^j\) is the PDF of the Gamma distribution with rate parameter \(a\) and shape parameter \(j\). When \(j \in \mathbb{N}\), this single ODE is equivalent with a finite-dimensional system of ODEs, which is often referred to as the linear chain trick. This “trick” is often used in modeling to get more realistic residence time distributions in compartmental models. By default, the residence time in a compartment is exponentially distributed. Adding auxiliary or “pseudo” compartments turns this into an Erlang distribution (recall that Erlang is Gamma with integer shape). However, what if \(j\) is not an integer? This can be useful if we want to estimate \(j\) from observations, and don’t want to deal with the complications of integer-valued model parameters. In this paper, we develop numerical integration schemes to solve delay differential equation. A complication is that the delay \(s\) has an unbounded domain. We also develop some approximation schemes that resemble the linear chain trick, but allow \(j\) to be real-valued.
Recommended citation: Cassidy T. et al (2022). "Numerical methods and hypoexponential approximations for gamma distributed delay differential equations." IMA Journal of Applied Mathematics. 87(6): 1043–1089.
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