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arxiv: 2502.12984 · v4 · pith:ZDI7PNEInew · submitted 2025-02-18 · 🧮 math.DS · cs.NA· cs.SY· eess.SY· math.NA

On Erlang mixture approximations for differential equations with distributed time delays

Tobias K. S. Ritschel This is my paper

Pith reviewed 2026-05-23 02:48 UTC · model grok-4.3

classification 🧮 math.DS cs.NAcs.SYeess.SYmath.NA
keywords distributed delay differential equationsErlang mixture approximationlinear chain trickconvergencestability analysisordinary differential equationsnumerical approximation
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The pith

Erlang mixture approximations convert distributed-delay equations into ordinary differential equations whose solutions converge for continuous bounded kernels.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows how to approximate the kernel of a delay differential equation with a mixture of Erlang distributions. The linear chain trick then converts the approximated equation into a system of ordinary differential equations. The approximation is proven to converge when the number of mixture components increases fast enough, provided the kernel is continuous and bounded. This approach enables the use of ODE techniques for simulating and analyzing the original delay equations, including stability assessment. Numerical examples illustrate its application to logistic growth, chemotherapy models, and nuclear reactor kinetics.

Core claim

The paper claims that an Erlang mixture approximation to the delay kernel, when combined with the linear chain trick, produces a system of ordinary differential equations whose solutions converge to those of the original distributed-delay equation as the number of terms grows sufficiently rapidly, for any continuous and bounded kernel. The resulting ODE system can determine the stability of the DDE's steady states, and the convergence holds for the solutions themselves when the kernel is exponentially bounded as well. An auxiliary procedure using bisection and least-squares finds suitable coefficients for the mixture.

What carries the argument

Erlang mixture approximation of the delay kernel, transformed by the linear chain trick into an equivalent system of ordinary differential equations.

Load-bearing premise

The delay kernel is continuous and bounded.

What would settle it

A continuous and bounded kernel for which the Erlang mixture approximation fails to converge to the original DDE no matter how rapidly the number of terms increases.

Figures

Figures reproduced from arXiv: 2502.12984 by Tobias K. S. Ritschel.

Figure 1
Figure 1. Figure 1: Erlang kernels for a = 1.5 (left) and a = 1 (middle) and m = 0, . . . , 5, and an Erlang mixture kernel (right) for a = 1, M = 5, and the following values of the coefficients. Blue: cm = 1/(M + 1) for m = 0, . . . , M. Red: c0 = 0 and cm = 1/M for m = 1, . . . , M. Yellow: c0 = cM = 1/2 and cm = 0 for m = 1, . . . , M − 1 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The Erlang mixture delta family for a fixed rate parameter, a, and different values of t (left) and for fixed t and different values of a (right). For t = 0 in the left figure, the value of δa is 2 for s ∈ [0, 0.5) (see also Lemma 4.7). 4.1. Erlang mixture delta family. In this section, we introduce the Erlang mixture delta family and use it to derive an identity for an infinite-order Erlang mixture kernel… view at source ↗
Figure 3
Figure 3. Figure 3: Convergence analysis for the modified logistic equation. The state (top left) and kernel (bottom left) errors for Erlang mixture approximations of different orders, M, obtained with the proposed least-squares approach and the reference approach based on the theoretical expressions for the coefficients. The bottom right plot shows the state and kernel errors against each other, and the top right plot shows … view at source ↗
Figure 4
Figure 4. Figure 4: Convergence analysis for the myelosuppression model. The state (top left) and kernel (bottom left) errors for Erlang mixture approximations of different orders, M, obtained with the least-squares approach and the reference approach based on theoretical values of the coefficients. The kernel error is shown for Guglielmi and Hairer’s approach in the bottom right, and the true and approximate kernels are show… view at source ↗
Figure 5
Figure 5. Figure 5: Bifurcation analysis with respect to the model parameter σ (left column) and the kernel parameter µ2 (right column) for the modified logistic equation. First row: Eigenvalues. Second row: The largest real part of the eigenvalues. Third and bottom row: Simulations for selected parameter values (obtained with the numerical method described in Appendix B.1). of reaction rates: R(t) = S T (t)r(t), S(t) =   … view at source ↗
Figure 6
Figure 6. Figure 6: Top row: The true kernels and the corresponding Erlang mixture approximations for M = 100 (left) and M = 350 (right) obtained in connection with the bifurcation analysis shown in [PITH_FULL_IMAGE:figures/full_fig_p034_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Monte Carlo simulation of the molten salt reactor model. Top row: The neutron concentration (left) and the reactivity (right). For each point in time, the span shows the interval of the minimum and maximum state, and the 95% confidence interval spans the 2.5 and 97.5 percentiles. The mean is computed pointwise, and the sample is the simulation corresponding to the mean value of κ. Bottom row: The pointwise… view at source ↗
Figure 8
Figure 8. Figure 8: Left column: The Erlang mixture approximations of the kernels in (C.1) used in the Monte Carlo simulation shown in [PITH_FULL_IMAGE:figures/full_fig_p036_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Extended version of the results in the bottom left of [PITH_FULL_IMAGE:figures/full_fig_p046_9.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper proposes an Erlang mixture approximation to distributed delay kernels in DDEs, converted to ODEs via the linear chain trick. It claims to prove convergence of the approximation to the original DDE for continuous and bounded kernels when the number of mixture terms increases sufficiently fast under specific coefficient choices. It further claims that the resulting ODEs preserve the ability to assess stability of DDE equilibria and that solutions converge when the kernel is additionally exponentially bounded. A bisection-plus-least-squares method is given for selecting the mixture coefficients, and numerical examples on a modified logistic equation, a chemotherapy myelosuppression model, and a molten-salt reactor kinetics model illustrate accuracy, convergence rates, bifurcation analysis, and Monte Carlo simulation.

Significance. If the stated convergence and stability results hold with the claimed rigor, the work supplies a systematic, theoretically supported route for replacing distributed-delay DDEs by finite-dimensional ODEs, which would be useful for simulation, bifurcation studies, and stochastic analysis in applications. The explicit proofs of convergence for continuous bounded kernels and of stability preservation constitute a clear strength, as does the provision of a concrete parameter-selection procedure and the demonstration on three distinct models.

major comments (2)
  1. [Convergence theorem (§3)] Convergence theorem (abstract and §3): the claim that convergence holds 'for specific choices of the coefficients if the number of terms increases sufficiently fast' is central, yet the manuscript supplies neither an explicit rate nor an a-priori error bound in terms of the number of terms and the kernel's modulus of continuity; without such a quantitative estimate the practical prescription for choosing the truncation level remains incomplete.
  2. [Stability assessment] Stability section: the assertion that the approximate ODEs 'can be used to assess the stability of the steady states of the original DDEs' requires showing that the characteristic roots (or Lyapunov exponents) of the ODE system converge to those of the DDE; the manuscript does not appear to supply a perturbation argument or spectral convergence result that would justify this transfer for finite truncations.
minor comments (3)
  1. [Method section] The description of the linear-chain-trick transformation would benefit from an explicit block-matrix form of the resulting ODE system to make the dimension and coupling transparent.
  2. [Numerical examples] Figure captions for the numerical examples should state the number of mixture terms used and the achieved L^∞ or L^2 error relative to a reference solution.
  3. [Parameter selection] The least-squares objective used in the bisection procedure is not written out; an explicit formula would clarify what quantity is being minimized.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The comments highlight important aspects that can strengthen the presentation of our results. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Convergence theorem (§3)] Convergence theorem (abstract and §3): the claim that convergence holds 'for specific choices of the coefficients if the number of terms increases sufficiently fast' is central, yet the manuscript supplies neither an explicit rate nor an a-priori error bound in terms of the number of terms and the kernel's modulus of continuity; without such a quantitative estimate the practical prescription for choosing the truncation level remains incomplete.

    Authors: The convergence theorem establishes that the Erlang mixture approximations converge to the original DDE for continuous bounded kernels under the specified conditions on the coefficients and sufficiently rapid increase in the number of terms. While the proof is based on approximation properties that guarantee convergence without an explicit rate, we recognize that providing an a-priori error bound would be beneficial for practical applications. We will revise Section 3 to include a quantitative error estimate derived from the modulus of continuity of the kernel, using standard results from approximation theory for mixtures of Erlang distributions. This will also clarify the choice of truncation level in the numerical procedure. revision: yes

  2. Referee: [Stability assessment] Stability section: the assertion that the approximate ODEs 'can be used to assess the stability of the steady states of the original DDEs' requires showing that the characteristic roots (or Lyapunov exponents) of the ODE system converge to those of the DDE; the manuscript does not appear to supply a perturbation argument or spectral convergence result that would justify this transfer for finite truncations.

    Authors: We show in the manuscript that the solutions converge when the kernel is exponentially bounded, which provides indirect support for stability assessment. However, to directly justify the use for stability analysis of finite truncations, a spectral convergence result is indeed desirable. We will add a new subsection or appendix providing a perturbation argument demonstrating that the characteristic roots of the approximate system converge to those of the DDE as the mixture approximation improves, leveraging the continuity of the spectrum with respect to the delay kernel in an appropriate topology. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper establishes convergence of an Erlang-mixture kernel approximation to DDEs (via the linear chain trick) for continuous bounded kernels when the number of terms grows sufficiently fast, plus stability and solution convergence under an added exponential bound. These results are stated as theorems with explicit domain assumptions and coefficient conditions; the bisection/least-squares parameter method is a separate computational procedure. No quoted step reduces a claimed prediction or uniqueness result to a fitted input, self-citation chain, or definitional equivalence by the paper's own equations. The central claims remain independent of the paper's fitted values or prior self-references.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on domain assumptions about the delay kernel properties and the introduction of the Erlang mixture as an approximation device; no new physical entities are postulated. The paper adds a modeling and proof technique rather than new axioms beyond standard analysis.

free parameters (1)
  • Erlang mixture coefficients
    Chosen via bisection and least-squares estimation to optimize the approximation for given kernels.
axioms (2)
  • domain assumption The delay kernel is continuous and bounded
    Invoked for the proof that the approximation converges when the number of terms increases sufficiently fast.
  • domain assumption The kernel is exponentially bounded
    Required for the statement that ODE solutions converge to the original DDE solutions.

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