Functional Central Limit Theorem and SPDE for epidemic model with memory of the last infection and waning immunity

Arsene Brice Zotsa-Ngoufack

REVIEW 2 major objections 5 minor 28 references

Functional Central Limit Theorem and SPDE for epidemic model with memory of the last infection and waning immunity

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves that in a stochastic epidemic model with memory of the last infection and waning immunity, the rescaled fluctuations of the empirical infection measure converge to a unique Gaussian-driven stochastic limit, and derives…

desk verdict Useful FCLT/SPDE result for a structured epidemic model with memory, but Proposition 3.13 has a genuine topology gap in identifying the Gaussian limit. read the letter →

arxiv 2505.15617 v2 pith:TQY2CVAQ submitted 2025-05-21 math.PR

Arsene Brice Zotsa-Ngoufack This is my paper · ORCID

classification math.PR MSC 60F0560F1735Q9260K3560G55

keywords stochasticepidemicmodelwithmemoryage-structuredvaryinginfectivitywaningimmunityfunctionalcentrallimittheoremVolterraintegralequationSPDEmeasure-valuedprocess

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Epidemic models that remember an individual's last infection while allowing immunity to wane are hard to analyse because the past keeps acting on the present. This paper proves a functional central limit theorem for such a model: the random fluctuations of the empirical infection measure around its deterministic large-population limit, rescaled by the square root of the population size, converge to a unique continuous stochastic process. The limit solves a nonlinear stochastic integral equation driven by a centered Gaussian field whose covariance is computed explicitly from the deterministic solution and the memory kernel. From this equation the paper derives an equivalent stochastic partial differential equation when the limit has a density, and an alternative representation purely in terms of fluctuations of the average force of infection and average susceptibility. If correct, this gives a rigorous quantitative description of how a finite population's epidemic trajectory deviates from the mean-field prediction.

What carries the argument

The argument is carried by the fluctuation empirical measure $\hat\mu^N$, by a per-individual reinfection counter $A_k^N(t)$ used to compare the interacting system with independent copies of the deterministic mean-field trajectory, and by the operator $R$ that encodes the expected jump effect of a reinfection through the memory kernel $K$. The counter comparison gives the $O(1/\sqrt{N})$ coupling estimate (3.5) under Assumption 2.3; this feeds the moment inequalities that yield tightness in the weighted Sobolev dual space $W^{-(m_d+1),\alpha}_0$. The martingale term $W^N$ has quadratic variation converging to the covariance (2.14) of the limiting Gaussian process, and a standard tightness criterion for Hilbert-space-valued processes turns these bounds into convergence.

What would settle it

Simulate the age-and-trait epidemic for growing $N$ with a memory kernel that obeys the model's normalization $\int_\Theta K(\theta,\tilde\theta)\,\nu(d\tilde\theta)=1$ for every $\theta$ but violates $\int_\Theta \sup_{\theta\in\Theta} K(\theta,\tilde\theta)\,\nu(d\tilde\theta)<\infty$, for example a heavy-tailed trait distribution; the theorem predicts tightness and the Gaussian limit (2.15), so observing divergence or subsequence-dependent limits would refute it.

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Extended reading notes

Core claim

The central result is Theorem 2.6: under Assumptions 2.1–2.3, as $N\to\infty$, $\hat\mu^N=\sqrt{N}(\mu^N-\mu)$ converges in distribution in $D(\mathbb{R}_+, W^{-(m_d+1),\alpha}_0)$ to the unique continuous solution $\hat\mu$ of the stochastic integral equation (2.15), where $W$ is a centered Gaussian process with covariance (2.14). When $\hat\mu_t$ has a density, the density solves the SPDE (2.16); equivalently, the pair $(\hat F,\hat S)$—the fluctuations of the average force of infection and average susceptibility—solves the Volterra system (2.18)–(2.19). The paper thus claims that memory of previous infections changes the fluctuation limit only through the kernel $K$, and that the no-memory FCLT is recovered when $K(\theta,\tilde\theta)=K(\tilde\theta)$.

Load-bearing premise

The proof leans on the technical condition that the memory kernel $K$ satisfies $\int_\Theta \sup_{\theta\in\Theta} K(\theta,\tilde\theta)\,\nu(d\tilde\theta)<\infty$; the coupling estimate that drives the whole tightness argument collapses if this integrability bound fails.

Editorial extensions

If this is right

Fluctuations of the empirical infection measure around the FLLN limit are of order $1/\sqrt{N}$ and converge to a Gaussian process, so forecast bands around mean-field epidemic curves become mathematically justified.
The limiting covariance (2.14) is explicit in terms of the deterministic solution, the infectivity and susceptibility functions, and the memory kernel, so it can be computed without simulating the full stochastic system.
When the limit measure has a density, the SPDE (2.16) gives a closed evolution equation for the fluctuation density, including the boundary condition at infection age zero.
The Volterra representation (2.18)–(2.19) shows that all fluctuations are captured by the pair $(\hat F,\hat S)$, and setting $K(\theta,\tilde\theta)=K(\tilde\theta)$ recovers the earlier no-memory FCLT.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The sup-integrability condition on $K$ is likely stronger than needed; one could test whether the same Gaussian limit emerges under weaker $L^1$-type kernels, which would extend the theorem to heavier-tailed trait distributions.
The explicit covariance (2.14) makes the limit directly usable for uncertainty quantification: compare the Gaussian fluctuations with Monte Carlo simulations of the $N$-individual process and use the mismatch to estimate the memory kernel.
The same Hilbert-space tightness route should transfer to other mean-field models with individual memory and unbounded initial ages, such as age-structured Hawkes or branching models, where bounded-age assumptions are unrealistic.
The SPDE (2.16) suggests a numerical discretization in age and trait; a particle method against the Gaussian limit would be a direct, implementable test of the theorem.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper gives the first FCLT and SPDE for the Guérin–Zotsa-Ngoufack model with memory of the last infection and waning immunity, extending the Pang–Pardoux/Forien program. The fluctuation equation is derived carefully, tightness is handled via Aldous in weighted Sobolev spaces, and uniqueness via Gronwall is clean. It also weakens assumptions used in earlier work, notably avoiding bounded initial age. That is a real step, and the alternative Volterra representation in Proposition 2.9 should be useful for applications.

The soft spot is Proposition 3.13. To identify the limiting Gaussian process, the paper needs H_φ(µ^N) → H_φ(µ), where H_φ(g)(t) = ∫_0^t ⟨g_s, λ⟩⟨g_s, R^(2)φ⟩ ds. The proof shows H_φ is Lipschitz in the sup norm of total variation, then invokes Theorem 2.1 via continuous mapping. That does not work. The FLLN converges in the weak (Prokhorov) topology, not in total variation; indeed µ^N_t is atomic while µ_t has a density, so the TV distance is 2 for every N. The Lipschitz estimate is therefore vacuous in the actual mode of convergence, and the continuous-mapping step is unjustified. The gap is load-bearing because the covariance in (2.14) is read off from this convergence. That said, the gap looks repairable: the coupling estimates in Lemma 3.1 and Proposition 3.2 should yield pointwise convergence of ⟨µ^N_s, R^(2)φ⟩ followed by dominated convergence. But the manuscript does not supply that argument, so the proof as written is incomplete.

Two smaller issues. Assumption 2.3(2) on the memory kernel is used critically to get the O(1/√N) coupling bound in Lemma 3.1, yet the paper gives no examples or discussion of kernels satisfying it. And the proof leans heavily on outsourced lemmas from [4], [9], [28], plus the companion FLLN in [13]; that is normal for the field, but it means a referee needs to check the dependencies. There are also numerous typos (“we mut consider”, duplicate “ˆµ1 and ˆµ1”) that should be cleaned up.

The central claim is likely correct, and the paper is a serious contribution to the FCLT/SPDE literature for non-Markovian epidemic models. It deserves a careful referee, with the explicit instruction that Proposition 3.13 must be fixed or the proof replaced. I would not desk-reject it, and I would read a corrected version closely.

Referee Report

2 major / 5 minor

Summary. The paper establishes a Functional Central Limit Theorem for a stochastic epidemic model with infection-age structure, individual traits, and memory of the last infection. The fluctuation process \hat\mu^N = \sqrt{N}(\mu^N - \mu) is shown to converge, in weighted Sobolev spaces, to a continuous Gaussian process solving a stochastic integral equation; the paper also derives an SPDE formulation and an alternative representation through fluctuations of the force of infection and average susceptibility. The proof follows the standard route of tightness via Aldous's criterion, identification of limits via martingale and quadratic-variation arguments, and uniqueness via Gronwall's inequality, with many technical estimates delegated to previous papers by the author and coauthors.

Significance. If the main theorem is valid, the paper makes a useful contribution to the non-Markovian epidemic FCLT literature: it incorporates memory of previous infections through a trait kernel, avoids a bounded-initial-age assumption, and uses weighted Sobolev spaces to handle the fluctuations. The derived SPDE and the comparison with the memoryless case in [28] are valuable. However, the proof as written contains load-bearing gaps in the identification of the limiting covariance and in the well-posedness/uniqueness step, so the central claim is not yet fully established.

major comments (2)

[Proposition 3.13] The convergence of the Doob-Meyer process H_φ(µ^N) to H_φ(µ) is not justified. The proof shows only that H_φ is Lipschitz with respect to the sup-norm total variation distance and then invokes Theorem 2.1 via the continuous mapping theorem. But Theorem 2.1 gives convergence in the Skorohod weak topology, not in total variation; when µ_0 has a density, µ_t has a density while µ^N_t is atomic, so ‖µ^N_t − µ_t‖_TV = 2 for every N. Moreover, λ and R^(2)φ are only bounded measurable under Assumptions 2.1–2.2, so H_φ is not continuous in the weak topology used in Theorem 2.1. Since this convergence identifies the covariance of the limiting Gaussian process in (2.14), the argument is load-bearing. A direct proof using the coupling estimates of Lemma 3.1 and Proposition 3.2 to obtain pointwise convergence of ⟨µ^N_s, f⟩ for bounded measurable f, followed by dominated convergence in s, would likely repair the gap, but the manuscript does not supply it.
[Proposition 3.14 / Theorem 2.6] The uniqueness argument requires that Rφ_{t−s} belongs to the Sobolev space W^{md+2,α}_0 so that ⟨\hatµ_s, Rφ_{t−s}⟩ is a valid dual pairing for \hatµ_s ∈ W^{−(md+2),α}_0. The proof of (3.29) only gives the pointwise bound |Rφ(a,θ)| ≤ 2‖φ‖_{C_b^{md+2}}; it does not control derivatives in θ or decay in a. Indeed, for φ with compact support in a and γ not decaying, Rφ(a,θ) → γ(a,θ)∫ φ(0,θ̃)K(θ,θ̃)ν(dθ̃) as a→∞, so Rφ need not lie in the weighted Sobolev space at all. Thus equation (2.15) may not even be well defined for the stated class of test functions, and the Gronwall argument does not establish uniqueness. The authors need either stronger regularity/decay assumptions on K and γ, a different solution space, or a different uniqueness argument.

minor comments (5)

[Definition 2.5] Definition 2.5 says W takes values in W^{md+1,α}, but the convergence result is in W^{−(md+1),α}; W should be a W^{−(md+1),α}-valued process, or the text should clarify the intended embedding.
[Definition 2.5] In the covariance formula, the right-hand side uses φ_s and ψ_s while the left-hand side uses φ_t and ψ_{t′}; the notation should be reconciled, and there is a typo where ϕ_s appears in the definition of eR.
[Assumption 2.3] The condition ∫_Θ sup_{θ∈Θ} K(θ,θ̃)ν(dθ̃) < ∞ is used crucially in Lemma 3.1 and Proposition 3.2, but the paper gives no examples or discussion of natural memory kernels satisfying it, nor does it discuss whether it can be relaxed. A short remark would improve applicability.
[Theorem 2.6] The theorem says the test function φ is bounded measurable and C^1 in the first two variables, and also φ_t ∈ W^{md+2,α}_0; such a φ need not belong to W^{md+2,α}_0 from the C^1 assumption alone. The regularity assumptions on φ should be stated precisely.
[Proposition 3.13] The statement says the convergence takes place in C(R_+, W^{−(md+1),α}_0), while W^N was defined and shown tight in W^{−md,α}; the embedding W^{−md,α} ↪ W^{−(md+1),α} should be explicitly invoked.

Circularity Check

0 steps flagged · score 1.0 of 10

The FCLT derivation is conditional on the companion FLLN [13] and on the author's prior moment estimates [28], but these are independent inputs; no step reduces the target conclusion to its own assumptions.

full rationale

The claimed FCLT is not circular. The covariance (2.14) of the limiting Gaussian process is not postulated; it is obtained as the limit of the quadratic variation (3.4) of the compensated-noise martingale W^N. The proof of convergence of the fluctuation process is via Aldous tightness (Prop. 3.9) and a Gronwall uniqueness argument (Prop. 3.14), and the identification of the limit's Doob-Meyer process (Prop. 3.13) uses the FLLN of the companion paper [13] as an external input. Heavy self-citation occurs — Theorem 2.1 is from the authors' companion paper, and Prop. 3.2 adapts [28, Prop. 5.4] — but these are independent prior theorems, not restatements of the target result; the equivalence with [28] in Prop. 2.9 is a consistency check. One genuine technical gap, not a circularity: in Prop. 3.13 the paper proves H_phi is Lipschitz in total variation and invokes Theorem 2.1, although Theorem 2.1 supplies only weak convergence in D(R_+, P(R_+ × Theta)); since mu^N_t is atomic while mu_t typically has a density, ||mu^N_t - mu_t||_TV = 2 for all N. This leaves the convergence of the quadratic variation unproved as written, but the missing argument (pointwise convergence of <mu^N_s, R^(2)phi> via the coupling estimates of Lemma 3.1/Prop. 3.2 plus dominated convergence) would still not make the result equivalent to its input. Thus the circularity score is low, reflecting only the heavy but non-circular reliance on the author's own prior work.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters or invented entities appear. The Gaussian process W and the white noise zeta are limits derived from the model's Poisson noise. The axioms are standard Sobolev-space facts plus the companion FLLN result and the absolute-continuity condition on the trait distribution.

assumptions (4)

domain assumption The FLLN from Guerin and Zotsa-Ngoufack [13, Theorem 2.1]: mu^N converges to the unique solution mu of (2.5).
The FCLT is centered on this deterministic limit. If the FLLN failed, the fluctuation process sqrt(N)(mu^N - mu) would be meaningless. The theorem is quoted from a companion preprint by the same group.
domain assumption nu is absolutely continuous with respect to Lebesgue measure.
Stated at the start of Section 2.3.1; needed to define the weighted Sobolev spaces W^{j,alpha}_0 on R+ x Theta with norm involving da nu(d theta).
standard math Sobolev embedding theorems (2.10)-(2.13), including Hilbert-Schmidt embeddings.
Used to pass from W^{m_d+k,alpha} to C^{k,alpha} and to obtain compact embeddings for tightness; cited from [25] and [7].
standard math Aldous tightness criterion for Hilbert space-valued processes (Definition A.1) and the martingale characterization of Gaussian limits.
Used in Propositions 3.9 and 3.13 to prove tightness and identify the limit Gaussian process via its deterministic bracket.

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Pith. "Pith review of Functional Central Limit Theorem and SPDE for epidemic model with memory of the last infection and waning immunity." pith.science (2026). https://pith.science/paper/TQY2CVAQ

@misc{pith2026250515617,
  author       = {Pith},
  title        = {Pith review of: Functional Central Limit Theorem and SPDE for epidemic model with memory of the last infection and waning immunity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TQY2CVAQ}},
  note         = {Machine review of arXiv:2505.15617}
}

read the original abstract

We study the fluctuations of a stochastic epidemic model with memory of the last infections, varying infectivity, and waning immunity, as introduced in Guerin and Zotsa-Ngoufack:arXiv preprint arXiv:2505.00601. The dynamics of the epidemic model are described by a measure-valued process with respect to infection age and individual traits. The Functional Law of Large Numbers (FLLN) is formulated as an integral equation, which is solved by a deterministic measure. In this article, we establish the Functional Central Limit Theorem (FCLT), capturing the fluctuations of the stochastic model around its deterministic limit. The limit of the FCLT is given by a nonlinear stochastic integral equation which is solved by a random signed-measure. We further derive the weak solution in the form of a stochastic partial differential equation (SPDE) and propose an alternative representation of the FCLT, as fluctuations in the average total force of infection and average susceptibility.

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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Fluctuations for mean-field interacting age-dependent Hawkes processes.Electronic Journal of Probability, 22, 2017

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work page Pith review arXiv 2022

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A stochastic epidemic model with memory of the last infection and waning immunity

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work page Pith review arXiv 2025

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