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Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level

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Numerical methods and hypoexponential approximations for gamma distributed delay differential equations.

Tyler Cassidy¹, Peter Gillich², Antony R Humphries³

¹Theoretical Biology and Biophysics, Los Alamos National Laboratory, Los Alamos, NM 87545, USA.

IMA Journal of Applied Mathematics

|January 24, 2023

Summary

This summary is machine-generated.

New numerical methods for gamma distributed delay differential equations (DDEs) offer improved accuracy. A functional continuous Runge-Kutta method and a hypoexponential approximation overcome limitations of existing Erlang approximations for DDE modeling.

Keywords:

delay differential equations functional continuous Runge–Kutta methods infinite delay equation linear chain trick

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Area of Science:

Mathematical modeling
Numerical analysis
Computational mathematics

Background:

Gamma distributed delay differential equations (DDEs) are crucial in various modeling applications.
Existing numerical methods often rely on approximations like the Erlang distribution, which can lead to inaccuracies.
There is a need for more accurate and robust numerical techniques for gamma distributed DDEs.

Purpose of the Study:

To develop and validate novel numerical methods for gamma distributed DDEs.
To address the limitations of the common Erlang approximation in DDE modeling.
To introduce a more accurate hypoexponential approximation for gamma distributed DDEs.

Main Methods:

Development of a fourth-order accurate functional continuous Runge-Kutta (FCRK) method for direct integration of gamma distributed DDEs.
Derivation of a hypoexponential approximation for gamma distributed DDEs, enabling solution via standard ordinary differential equation (ODE) solvers.
Comparison of solutions obtained from FCRK, hypoexponential approximation, and the traditional Erlang approximation using reference solutions.

Main Results:

The FCRK method demonstrates fourth-order convergence and high accuracy for gamma distributed DDEs.
The common Erlang approximation can yield qualitatively different solutions compared to the underlying gamma distributed DDE.
The proposed hypoexponential approximation provides a more accurate representation of gamma distributed DDEs without the limitations of the Erlang method.
The hypoexponential approximation was successfully applied to statistical inference on synthetic epidemiological data.

Conclusions:

The developed FCRK method and hypoexponential approximation offer significant improvements for modeling with gamma distributed DDEs.
The hypoexponential approximation is a superior alternative to the Erlang approximation for accurately solving gamma distributed DDEs.
These new methods enhance the reliability of DDE-based modeling in fields like epidemiology.