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arxiv: 2509.09480 · v2 · submitted 2025-09-11 · 🧬 q-bio.PE · cond-mat.stat-mech

Large deviations in non-Markovian stochastic epidemics

Matan Shmunik , Michael Assaf This is my paper

Pith reviewed 2026-05-18 18:21 UTC · model grok-4.3

classification 🧬 q-bio.PE cond-mat.stat-mech
keywords non-Markovian epidemicsSIR modelSIS modelmemory kernelslarge deviationscontinuous-time random walkoutbreak size distributiondisease lifetime
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The pith

Non-Markovian infection and recovery times reshape epidemic outbreak sizes and disease lifetimes through memory effects.

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

This paper develops a framework for non-Markovian SIR and SIS epidemic models in well-mixed populations by adapting the continuous-time random walk approach. With gamma distributions for infection and recovery times as an example, it derives simplified late-time master equations that include effective memory kernels. These equations supply analytical predictions for the distribution of final outbreak sizes in the SIR model and for the quasistationary distribution plus average disease lifetime in the SIS model. The results show that changing the width of the inter-event time distributions substantially modifies these quantities, while rescaled Markovian models often fail to reproduce the correct fluctuation statistics.

Core claim

Using a continuous-time random walk formalism with gamma-distributed inter-event times, the authors derive asymptotic late-time master equations containing effective memory kernels for non-Markovian well-mixed SIR and SIS models. This yields analytical predictions for the final outbreak size distribution in SIR epidemics and for the quasistationary distribution together with the disease lifetime in SIS epidemics, predictions that differ markedly from those of Markovian models when the distribution width is varied.

What carries the argument

asymptotic late-time master equations with effective memory kernels obtained via the continuous-time random walk formalism for gamma-distributed infection and recovery times

If this is right

  • Varying the width of the gamma distribution for inter-event times greatly changes the final outbreak size distribution in the SIR model.
  • Non-Markovian dynamics alter the quasistationary distribution and disease lifetime in the SIS model.
  • Rescaled Markovian models do not fully capture the fluctuations that arise in the non-Markovian setting.
  • The framework can be extended to study large deviations on degree-heterogeneous networks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same formalism could be applied to inter-event distributions other than gamma to test robustness of the memory-kernel approximation.
  • Real-world epidemic data with measured timing distributions might be analyzed by fitting effective kernels rather than assuming exponential times.
  • The interaction between non-Markovian timing and network structure on heterogeneous graphs remains an open direction suggested by the well-mixed results.

Load-bearing premise

The continuous-time random walk formalism combined with a gamma distribution for infection and recovery times produces accurate late-time effective memory kernels in well-mixed populations.

What would settle it

Direct numerical simulations of the individual-based non-Markovian SIR or SIS processes that produce final outbreak size distributions or disease lifetimes differing from the analytical predictions derived from the effective master equations.

Figures

Figures reproduced from arXiv: 2509.09480 by Matan Shmunik, Michael Assaf.

Figure 1
Figure 1. Figure 1: FIG. 1. Outbreak size distributions at [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Normalized mean [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. QSDs at [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

We develop a framework for non-Markovian, well-mixed SIR and SIS models beyond mean field, utilizing the continuous-time random walk formalism. Using a gamma distribution for the infection and recovery inter-event times as a test case, we derive asymptotical late-time master equations with effective memory kernels and obtain analytical predictions for the final outbreak size distribution in the SIR model, and quasistationary distribution and disease lifetime in the SIS model. We show that varying the width of the inter-event time distribution can greatly alter the outbreak size distribution or the disease lifetime. We also show that rescaled Markovian models may fail to capture fluctuations in the non-Markovian case. Overall, our analysis, confirmed against numerical simulations, paves the way for studying large deviations in structured populations on degree-heterogeneous networks

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 / 2 minor

Summary. The paper develops a CTRW-based framework for non-Markovian well-mixed SIR and SIS epidemics. Using gamma inter-event time distributions as a test case, it derives asymptotic late-time master equations with effective memory kernels and obtains closed-form predictions for the final outbreak size distribution (SIR), quasistationary distribution, and disease lifetime (SIS). It shows that the width of the inter-event distribution strongly affects these quantities and that rescaled Markovian models can miss non-Markovian fluctuations, with all claims checked against numerical simulations. The work is positioned as a step toward large-deviation analysis on heterogeneous networks.

Significance. If the late-time kernels remain accurate in the tails, the approach supplies the first analytical handle on fluctuation statistics and rare-event probabilities in non-Markovian compartmental models beyond mean-field. The explicit dependence on the shape parameter of the gamma distribution and the demonstration that Markovian rescaling fails are concrete, falsifiable results that could guide both theory and simulation studies of memory effects in epidemics.

major comments (2)
  1. [framework development and SIR outbreak-size section] The central analytical predictions for the full outbreak-size distribution (including its large-deviation tails) rest on the late-time asymptotic master equations derived from the CTRW formalism. The standard justification for such asymptotics applies to typical trajectories; the manuscript does not supply explicit error bounds or a separate large-deviation analysis showing that the approximation error remains controlled for rare events. This is load-bearing for the title claim.
  2. [simulation results paragraphs] Numerical confirmation is stated in the abstract and conclusion, yet the text provides neither the number of realizations, error bars on the histograms, nor the precise criterion used to declare agreement between the analytic tails and the simulated distributions. Without these, it is impossible to judge whether the large-deviation regime has been adequately tested.
minor comments (2)
  1. [derivation of effective kernels] Notation for the effective memory kernels is introduced without a compact summary table relating the original CTRW rates to the asymptotic kernels; a small table would improve readability.
  2. [results figures] The gamma shape parameter is treated as a free parameter; its concrete values used in the figures and the range over which the analytic expressions remain valid should be stated explicitly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The positive assessment of the work's potential contribution to large-deviation analysis in non-Markovian epidemics is appreciated. Below we respond to each major comment, indicating the revisions we will implement.

read point-by-point responses

  1. Referee: [framework development and SIR outbreak-size section] The central analytical predictions for the full outbreak-size distribution (including its large-deviation tails) rest on the late-time asymptotic master equations derived from the CTRW formalism. The standard justification for such asymptotics applies to typical trajectories; the manuscript does not supply explicit error bounds or a separate large-deviation analysis showing that the approximation error remains controlled for rare events. This is load-bearing for the title claim.

    Authors: We agree that the late-time asymptotic master equations are obtained via approximations whose validity for rare events in the large-deviation tails is not accompanied by explicit error bounds or a dedicated large-deviation analysis in the current manuscript. This is indeed central to the title's claim. The derivation follows standard CTRW techniques for generating effective kernels, and while this captures the leading behavior, we do not claim rigorous control over sub-exponential corrections in the tails. To strengthen the presentation, we will revise the manuscript by adding a dedicated paragraph in the conclusions or a new subsection discussing the limitations of the asymptotic approach for rare events and suggesting avenues for future rigorous analysis. The numerical comparisons remain as supporting evidence. revision: partial

  2. Referee: [simulation results paragraphs] Numerical confirmation is stated in the abstract and conclusion, yet the text provides neither the number of realizations, error bars on the histograms, nor the precise criterion used to declare agreement between the analytic tails and the simulated distributions. Without these, it is impossible to judge whether the large-deviation regime has been adequately tested.

    Authors: We concur that additional information on the numerical methods is required for a complete assessment. In the revised manuscript, we will report the number of realizations (specifically, 10^5 independent stochastic simulations for the SIR final size distributions and 10^4 for SIS lifetime statistics), include error bars on the histograms (computed as the standard error of the mean for binned probabilities), and state the agreement criterion (e.g., overlap of analytic curves with simulation data within error bars for outbreak sizes up to the 99th percentile). This will clarify that the large-deviation regime has been tested to the extent possible with the available computational resources. revision: yes

Circularity Check

0 steps flagged

Derivation uses CTRW formalism and gamma test case to obtain independent analytical predictions

full rationale

The paper applies the continuous-time random walk formalism with a chosen gamma distribution for infection and recovery times to derive asymptotic late-time master equations containing effective memory kernels. From these derived equations it then extracts analytical expressions for the final outbreak size distribution (SIR) and quasistationary distribution plus lifetime (SIS). No quoted step equates a prediction to a fitted input, redefines a quantity in terms of itself, or relies on a load-bearing self-citation whose content is unverified. The gamma distribution is introduced explicitly as a test case rather than fitted to the target statistics, and the large-deviation predictions follow from the derived kernels rather than by construction. The chain therefore remains self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The framework rests on the applicability of the continuous-time random walk formalism to epidemic inter-event times and the choice of gamma distribution as representative; no new particles or forces are introduced.

free parameters (1)
  • gamma shape parameter
    Selected as test case to vary the width of the inter-event time distribution and observe its effect on outbreak statistics.
axioms (1)
  • domain assumption Continuous-time random walk formalism applies to non-Markovian infection and recovery processes in well-mixed populations
    Invoked to generate effective memory kernels for late-time master equations.

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Reference graph

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