  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.
