On Erlang ODE approximations of differential equations with distributed time delays

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. We refer to this as the Erlang ODE approximation of the DDEs, and we prove that the Erlang mixture approximation converges for continuous and bounded kernels if the number of terms increases sufficiently fast. Furthermore, we show that if the kernel is also exponentially bounded, the Erlang ODE approximation can be used to assess the stability of the steady states of the original DDEs and that the solution to the ODE approximation converges. 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 rates of the approximations 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.