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.
翻译：在许多建模应用中自然会出现 Gamma 分布延迟方程式( DDEs ) 。 然而, 目前尚不具备通用 Gamma 分布的 DDE 的适当数字方法 。 因此, 建模者往往使用Erlang 分布法和线性链技术来接近伽马分布法, 以得出等效的普通差异方程式系统。 在这项工作中, 我们开发了一种功能上连续的 Runge- Kutta 方法, 以数字方式将伽马分布的 DDE 进行数字整合, 并进行数字测试, 以确认数字方法的准确性。 由于大多数科学软件包中没有功能上连续的 Runge- Kutta 方法, 我们随后得出伽马分布的 DDE 的低度近似值。 我们用数字方法表明, 使用通用 Erlang 近似方法可以产生与基本伽马分布的 DDE 质量不同的解决方案, 我们的光度近比并不具有这一限制 。 最后, 我们使用我们的低度光度近比来对合成流行病学数据进行统计推断 。
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