On Erlang mixture approximations for differential equations with distributed time delays

TL;DR

This paper introduces an Erlang mixture approximation method for delay differential equations with distributed delays, transforming them into ODEs for easier analysis and simulation.

Contribution

The authors propose a novel Erlang mixture approximation and linear chain trick to analyze DDEs, with proven convergence and practical numerical applications.

Findings

Approximation converges for continuous, bounded kernels with increasing terms.

The method accurately assesses stability and bifurcations of DDEs.

Numerical examples demonstrate effectiveness in various models.

Abstract

In this paper, we propose a general approach for approximate simulation and analysis of delay differential equations (DDEs) with distributed time delays based on methods for ordinary differential equations (ODEs). The key innovation is that we 1) propose an Erlang mixture approximation of the kernel in the DDEs and 2) use the linear chain trick to transform the resulting approximate DDEs to ODEs. Furthermore, we prove that the approximation converges for continuous and bounded kernels and for specific choices of the coefficients if the number of terms increases sufficiently fast. We show that the approximate ODEs can be used to assess the stability of the steady states of the original DDEs and that the solution to the ODEs converges if the kernel is also exponentially bounded. Additionally, we propose an approach based on bisection and least-squares estimation for determining optimal parameter values in the approximation. Finally, we present numerical examples that demonstrate the accuracy and convergence rate obtained with the optimal parameters and the efficacy of the proposed approach for bifurcation analysis and Monte Carlo simulation. The numerical examples involve a modified logistic equation, chemotherapy-induced myelosuppression, and a point reactor kinetics model of a molten salt nuclear fission reactor.