{"id":"0654096b-80f4-48d0-b8f7-a4db30090fc7","arxiv_id":"2505.15617","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A functional central limit theorem and SPDE are derived for fluctuations of a structured epidemic model with memory of the last infection and waning immunity.","lead":"This paper proves a central limit theorem for large epidemics with fading immunity and memory of past infections. It gives the exact random fluctuation pattern around the average epidemic curve, which supports prediction intervals and yields a stochastic partial differential equation description.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.13 justifies the convergence of the quadratic variation by a total-variation continuity argument that cannot hold, since µ^N is atomic while µ typically has a density; the FLLN topology is strictly weaker.","rationale":"The reader's weakest assumption, Assumption 2.3(2), is genuinely restrictive and is used in Lemma 3.1 and Proposition 3.2, but it is an explicit hypothesis rather than an internal inconsistency. I see a more direct proof gap in Proposition 3.13: the paper proves H_φ is Lipschitz in total variation and applies the continuous mapping theorem via the FLLN, yet the FLLN cannot hold in total variation because the empirical measure is atomic while the limit typically has a density. Since λ and R^(2)φ are only bounded measurable, they are not test functions for the weak topology either, so the stated continuity argument does not establish the required convergence. The conclusion H_φ(µ^N) → H_φ(µ) is probably true and can likely be obtained from the paper's own coupling estimates, but that repair is not written down. This strengthens the case for a conditional verdict rather than full acceptance: the central claim is plausible, but the proof as printed has a concrete missing justification. I therefore keep the reader's CONDITIONAL verdict, while disagreeing that Assumption 2.3(2) is the single most load-bearing concern.","tokens_in":26209,"tokens_out":25094,"duration_ms":231883,"concrete_test":"Isolate the quadratic-variation step in Proposition 3.13. Set λ ≡ 1 and γ ≡ 1 so the particles are independent and the FLLN holds in the weak topology. Choose a smooth fixed φ and a kernel K such that f = R^(2)φ is a bounded, non-continuous function. Check whether the proof can be repaired by showing, for each fixed s, E|⟨µ^N_s, f⟩ − ⟨µ_s, f⟩| → 0 using the coupling bound of Lemma 3.1, and then applying dominated convergence to conclude H_φ(µ^N) → H_φ(µ). If this pointwise-plus-dominated-convergence route succeeds, replace the invalid TV-continuity argument and keep the result; if it fails, the identification of the Gaussian process in Theorem 2.6 is unproved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Proposition 3.13 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 the total variation distance and then invokes Theorem 2.1 via the continuous mapping theorem. This step is not justified: Theorem 2.1 cannot be convergence in total variation, because for any initial law with a density the FLLN solution µ_t has a density, whereas µ^N_t is atomic, so ‖µ^N_t − µ_t‖_TV = 2 for every N. Thus the Lipschitz estimate has no content in the actual mode of convergence. The functions λ and R^(2)φ are only assumed bounded measurable (Assumptions 2.1 and 2.2), so H_φ is not continuous in the weak Skorohod topology of D(R_+, P(R_+ × Θ)) either. The convergence of the Doob-Meyer process is essential for identifying the limiting Gaussian process W in (2.14); without a valid proof of H_φ(µ^N) → H_φ(µ), the martingale characterization in Proposition 3.13 is incomplete. The gap is likely repairable by proving pointwise convergence of ⟨µ^N_s, R^(2)φ⟩ with the coupling estimates in Lemma 3.1 and Proposition 3.2, followed by dominated convergence, but the manuscript does not supply this argument.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":26500,"tokens_out":16461,"duration_ms":145535,"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":[{"comment":"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.","section":"Proposition 3.13"},{"comment":"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.","section":"Proposition 3.14 / Theorem 2.6"}],"minor_comments":[{"comment":"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.","section":"Definition 2.5"},{"comment":"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.","section":"Definition 2.5"},{"comment":"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.","section":"Assumption 2.3"},{"comment":"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.","section":"Theorem 2.6"},{"comment":"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.","section":"Proposition 3.13"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nTwo 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.\n\nThe 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.","headline":"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.","tokens_in":27061,"tokens_out":3232,"would_cite":true,"duration_ms":29089,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F05","60F17","35Q92","60K35","60G55"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["stochastic epidemic model with memory","age-structured model","varying infectivity","waning immunity","functional central limit theorem","stochastic Volterra integral equation","SPDE","measure-valued process"],"falsifier":"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.","tokens_in":25986,"feed_emoji":"🦠","tokens_out":9919,"duration_ms":78556,"temperature":0.7,"pith_summary":"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.","feed_headline":"Fluctuations of memory epidemics converge to a Gaussian limit","feed_subtitle":"Random deviations around the mean-field epidemic curve now have an explicit Gaussian limit","key_machinery":"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.","core_discovery":"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 $K(\\theta,\\tilde\\theta)=K(\\tilde\\theta)$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"defines the age-and-trait epidemic model with memory kernel $K$ and supplies the FLLN deterministic limit $\\mu$ around which fluctuations are taken.","marker":"[13]"},{"why":"provides the no-memory FCLT that this paper extends and to which the Volterra representation reduces when $K$ does not depend on the previous trait.","marker":"[28]"},{"why":"supplies the varying-infectivity and susceptibility model whose independent trajectories are used as the coupling process $(a_k,\\theta_k)$.","marker":"[9]"},{"why":"contributes the Sobolev-space tightness method and moment inequalities adapted to the unbounded-initial-age setting.","marker":"[4]"},{"why":"provides the Hilbertian fluctuation framework and the Sobolev embedding results used for the tightness space.","marker":"[25]"},{"why":"gives the tightness criterion for Hilbert-space-valued semimartingales used to prove $C$-tightness of the martingale and fluctuation processes.","marker":"[14]"}],"fun_headline_variants":["Memory epidemics: fluctuations converge to Gaussian","FCLT solves epidemic fluctuations with memory","Epidemic SPDE: memory kernel drives Gaussian noise","Fluctuations of memory epidemics hit Gaussian limit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Memory epidemics: fluctuations converge to Gaussian","FCLT solves epidemic fluctuations with memory","Epidemic SPDE: memory kernel drives Gaussian noise","Fluctuations of memory epidemics hit Gaussian limit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000527,"raw_usage":{"total_tokens":2527,"prompt_tokens":914,"completion_tokens":1613,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":530,"completion_tokens_details":{"reasoning_tokens":1555}},"tokens_in":530,"tokens_out":1613,"duration_ms":10936,"temperature":1.0,"reasoning_tokens":1555,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T15:14:07.766021+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"A stochastic epidemic model with memory of the last infection and waning immunity","cited_arxiv_id":"2505.00601","evidence_quote":"defines the age-and-trait epidemic model with memory kernel $K$ and supplies the FLLN deterministic limit $\\mu$ around which fluctuations are taken."},{"cited_title":"Functional central limit theorems for epidemic models with varying infectivity and waning immunity.ESAIM: Probability and Statistics, 29:45–112, 2025","cited_arxiv_id":null,"evidence_quote":"provides the no-memory FCLT that this paper extends and to which the Volterra representation reduces when $K$ does not depend on the previous trait."},{"cited_title":"Stochastic epidemic models with varying infectivity and waning immunity","cited_arxiv_id":"2210.04667","evidence_quote":"supplies the varying-infectivity and susceptibility model whose independent trajectories are used as the coupling process $(a_k,\\theta_k)$."},{"cited_title":"Fluctuations for mean-field interacting age-dependent Hawkes processes.Electronic Journal of Probability, 22, 2017","cited_arxiv_id":null,"evidence_quote":"contributes the Sobolev-space tightness method and moment inequalities adapted to the unbounded-initial-age setting."},{"cited_title":"PhD thesis, Universit´ e de Nanterre-Paris X, 2006","cited_arxiv_id":null,"evidence_quote":"provides the Hilbertian fluctuation framework and the Sobolev embedding results used for the tightness space."},{"cited_title":"Weak convergence of sequences of semimartingales with applications to multitype branching processes.Advances in Applied Probability, 18(1):20–65, 1986","cited_arxiv_id":null,"evidence_quote":"gives the tightness criterion for Hilbert-space-valued semimartingales used to prove $C$-tightness of the martingale and fluctuation processes."}],"review_version":1}