{"id":"f22fa6f8-dd95-4f8a-91bd-4aeab3457de0","arxiv_id":"2505.00601","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In the large-population limit, an epidemic with last-infection memory converges to an age-and-trait PDE, and an endemic equilibrium exists when a susceptibility-weighted reproduction number crosses a threshold derived from the model curves.","lead":"This paper extends a known stochastic epidemic model so that each new infection can remember the previous one, through a trait that is redrawn by a memory kernel. In the infinite-population limit it derives a PDE and gives a threshold condition on the model's inputs that decides whether the disease stays endemic.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.6 is sound within its stated assumptions, but the no-overlap condition A2-(1)(c) is the load-bearing scope limit: outside it the threshold construction at Eq.","rationale":"The paper does what it sets out to do: it extends the FPPZ framework to Markovian trait memory and derives an endemicity threshold from the eigenfunction of the memory kernel. The FLLN (Theorem 3.1) follows from standard tightness/martingale arguments; the equilibrium proof is internally consistent under Assumptions A1-A2. A2-(3) and the support condition imply λ has bounded support, so the normalization ∫λS*=1 is not jeopardized by infinite integrals. The reader's weakest-assumption identification is accurate: A2-(1)(c) is exactly where Eq. (5.11) uses the no-overlap condition, and the theorem should not be read as covering overlapping infectivity/susceptibility regimes. However, because the paper states this assumption explicitly and repeatedly, and because the stability gap is honestly flagged as open in the memory case, the central claim is supported within its scope. No new internal inconsistency or unsupported assertion was found that would change the ACCEPT verdict.","tokens_in":32618,"tokens_out":8771,"duration_ms":107265,"concrete_test":"Specialize to a single trait Θ={θ0}, ν=δ, K=1, λ(a)=1_{[0,1]}(a), and γ(a)=1_{[0.5,∞)}(a), so infectivity and susceptibility overlap on [0.5,1] while all other smoothness assumptions hold. Seek a stationary density of the form used in Theorem 3.6, u*(a)=F S e^{-F∫γ}. With S=1, the normalization H(F)=F∫e^{-F∫γ}=1 has a positive solution, but the force-of-infection equation (5.4) becomes F = F∫λ e^{-F∫γ} = F e^{-0.5F} < F, so no such equilibrium exists. Recompute the construction with overlapping supports to confirm that the failure occurs exactly at Eq. (5.11), i.e. Assumption A2-(1)(c) is indispensable for the threshold.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central construction in Theorem 3.6 defines the equilibrium as u*(a,θ)=F*S*(θ)exp(-F*∫₀^a γ(s,θ)ds), with S* chosen so that ∫λS*=1. The proof of the force-of-infection equation (5.4) uses Assumption A2-(1)(c), λγ≡0 with supp λ before supp γ, to obtain Eq. (5.11): on the support of λ the exponential waning factor equals 1, so ∫λS*e^{-F*∫γ}=∫λS*=1. If λ and γ overlap, the exponential is strictly less than 1 on part of the λ support, so the same normalization gives ∫λS*e^{-F*∫γ}<1=∫λS*, and the stationary equation F=F∫λS*e^{-F*∫γ} cannot hold. The threshold (3.9) is therefore established only in the no-reinfection-while-infectious regime, not for general overlapping infectivity and susceptibility curves. This is an explicit modelling assumption, acknowledged in the text as preventing immediate reinfection; it is consistent with the SIS convention in [11,12], so it is a scoping limitation rather than an internal contradiction. The stability theorem is further restricted to the memory-free case with an unverified spectral condition, also acknowledged. Within its stated scope the FLLN and equilibrium arguments are coherent.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a stochastic individual-based epidemic model with age since last infection and a trait parameter that carries memory of the last infection. At each reinfection, the trait is redrawn according to a Markov kernel K, and the infectivity and susceptibility curves depend on age and trait. The authors prove a functional law of large numbers (Theorem 3.1) showing that the empirical measure of ages and traits converges to the unique solution of a nonlinear age-and-trait structured PDE. The main result is Theorem 3.6: under Assumptions A1 and A2, an endemic equilibrium exists exactly when the threshold condition ∫Θ (1/γ*(θ)) S*(θ) ν(dθ) < 1 holds, where S* is the positive eigenfunction of the memory kernel's integral operator normalized by ∫ λ S* = 1, and γ* is the long-run Cesàro limit of susceptibility. Uniqueness is obtained under a monotonicity condition (3.10). The paper also analyzes local stability in the memory-free case, conditional on a spectral condition (5.29), and verifies it for a step-susceptibility SIS-type model. Applications to vaccination policies are given, including a renewal-process vaccination model for which the threshold of [12] is recovered.","tokens_in":32797,"tokens_out":15369,"duration_ms":165838,"significance":"If correct, the paper provides a genuine extension of the Forien--Pang--Pardoux--Zotsa framework by allowing the new infectivity and susceptibility curves to depend on the previous infection via a trait Markov chain. The threshold is parameter-free and explicit: it is expressed in terms of the eigenfunction of the memory kernel, the susceptibility long-run limit, and the infectivity curves. The FLLN proof follows a standard tightness-and-martingale argument and is coherent. The equilibrium construction is algebraically consistent under A1 and A2, and the examples, including the vaccination models, are concrete and give interpretable thresholds. The paper also delivers, to my knowledge, a nontrivial local stability result for a memory-free SIS-type model, albeit only under an explicit no-eigenvalue condition.","major_comments":[{"comment":"The endemicity threshold is established under Assumption A2(1)(c), which imposes that the support of λ precedes the support of γ and that λγ ≡ 0. The proof uses Eq. (5.11), where the exponential waning factor is dropped on the support of λ. If the infectivity and susceptibility supports overlap, the exponential factor is strictly less than 1 on part of the λ-support, and the proposed stationary state does not satisfy the force-of-infection equation. The assumption is explicit and is consistent with [11,12], so this is a scoping limitation rather than an internal contradiction; however, the abstract and introduction should state prominently that the threshold is proved only in the no-reinfection-while-infectious regime.","section":"Theorem 3.6, Eq. (5.11)"},{"comment":"The local stability result is conditional on the non-existence of eigenvalues of the linearized operator with real part ≥ 0, as stated in Eq. (5.29). For the general memory-free model this condition is not verified; it is checked only for the special SIS-type model in Proposition 5.9. The text acknowledges this, but the statement of Theorem 3.8 should be more explicit that it is a conditional stability criterion for general curves, with a concrete verification in one model class, rather than a fully general stability theorem.","section":"Theorem 3.8, Eq. (5.29)"}],"minor_comments":[{"comment":"The notation Qk is used both for the original Poisson measure and for its compensated version; please distinguish the two, for example by writing Q̃k for the compensated measure.","section":"Section 2 and Eq. (4.1)"},{"comment":"The normalization condition ∫ λ S* = 1 requires that ∫ λ S* > 0; the case λ ≡ 0, which would make the threshold ill-defined, should be excluded or explicitly discussed.","section":"Proposition 5.3, Eq. (5.7)"},{"comment":"The class of infectivity curves λ used in the renewal vaccination model is not specified precisely; please state the assumptions on λ needed both for Assumption A2 and for the definition of R0.","section":"Section 6.2"},{"comment":"The manuscript contains numerous typographical and OCR-related artifacts, for example 'memor y', 'W ANING', 'd/greaterorequalslant1', and broken mathematical spacing. The final version should be carefully proofread.","section":"Throughout"},{"comment":"The probability measure E*ν is introduced only after the condition (3.9); it may be clearer to define E*ν before rewriting the threshold as E*ν[1/γ*] < R*0.","section":"Remark 3.7"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a solid contribution within the journal's scope. The two citations to the authors' own related work [11,12] are used as baselines and the new results are distinct. The main caveats are scope limitations that are explicitly assumed, so I do not see grounds for rejection. I recommend minor revision to make those limitations more prominent and to polish the presentation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nPunchline: this paper does something real but sharply scoped. It extends the Forien-Pang-Pardoux-Zotsa framework by letting the new trait at reinfection depend on the previous trait through a Markov kernel K. The main new result is a general endemicity threshold: under their assumptions there is an endemic equilibrium when ∫ (1/γ*) S* dν < 1, with S* the unique positive eigenfunction of the kernel operator and γ* the long-run Cesàro limit of susceptibility. That threshold is parameter-free—no constants fitted to the conclusion—and in the no-memory limit it reduces to the thresholds of [11] and [12], which is exactly what a proper generalization should do.\n\nWhat's good. The FLLN (Theorem 3.1) is a standard tightness and martingale argument, cleanly written. The equilibrium construction is algebraically coherent given A1 and A2. The examples are worthwhile: they compute S* for several kernels, and Section 6 recovers the vaccination-policy threshold of [12] without their independence assumption. The paper also states its own gaps plainly—stability with memory is left open, and the stability result that is proved is conditional on a spectral condition that is not checked.\n\nSoft spots. The load-bearing assumption is A2-(1)(c): the infectivity and susceptibility supports are disjoint, with λγ=0 and the support of λ entirely before the support of γ. This is what makes equation (5.11) work: on the support of λ, the waning exponential in the stationary equation is identically 1, so the normalization ∫λS*=1 closes the fixed point. If λ and γ overlap, that identity fails and the threshold argument in Theorem 3.6 is not established. This is not a hidden flaw—the text explicitly says the assumption prevents immediate reinfection and is standard in the prior literature—but it is a genuine scope limit. The reader's stress-test note has it right. Less central: local stability is memory-free only, and condition (5.29) is an unverified no-eigenvalue condition, so the paper cannot claim to settle stability in the setting that motivated it.\n\nMy take. The central claim is sound within its stated assumptions, and the limits are honest. The paper is an extension of an established program rather than a paradigm shift, but for the subfield of trait-and-age epidemic models with waning immunity it is a useful step. It is not self-contained—[11] and [12] are doing heavy lifting—but citation dependence is normal in this line of work and not a defect here.\n\nRecommendation. Send it out. A serious referee will have concrete work to do on the overlap question, but the main theorem and the FLLN deserve scrutiny in the literature. I would cite it for the threshold and would bring it to a reading group focused on structured epidemic models.","headline":"A careful extension of the FPPZ framework that earns a serious referee; the endemicity threshold is real, and the main scope limit (no reinfection while infectious) is explicit and acknowledged.","tokens_in":33420,"tokens_out":3183,"would_cite":true,"duration_ms":28303,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F17","35Q92","60K35","35B40","92D30"],"pacs":[],"model":"deepseek-v4-flash","headline":"An epidemic with waning immunity stays endemic precisely when a single computable threshold built from infectivity, susceptibility, and memory of the last infection falls below 1.","keywords":["stochastic epidemic model","memory of last infection","waning immunity","piecewise deterministic Markov process","age-structured PDE","endemic equilibrium","endemicity threshold","vaccination policy"],"falsifier":"Drop the separation assumption $\\lambda\\gamma\\equiv 0$ while keeping everything else, and numerically solve the stationary equations (5.1) for a two-trait example: if an endemic equilibrium appears while the computed threshold (3.9) is above 1, or fails to appear while the threshold is below 1, then the separation assumption is load-bearing and the threshold claim does not extend beyond it.","tokens_in":32336,"feed_emoji":"🦠","tokens_out":6249,"duration_ms":65393,"temperature":0.7,"pith_summary":"The paper asks when a disease that does not confer permanent immunity can remain endemic in a large population. It extends an established stochastic epidemic model by letting each new infection carry a trait—think of severity or immune imprint—chosen randomly with a distribution that depends on the trait of the previous infection, so the model remembers the last infection. In the infinite-population limit the age-and-trait structure is governed by a partial differential equation, and the paper derives a threshold condition for the existence of a non-zero steady state. The threshold is a single integral comparing the long-run average susceptibility with the stationary distribution of traits produced by the memory kernel. This matters because it turns the question \"does the disease die out or persist?\" into computable conditions on infectivity, susceptibility, and the way infections shape future immunity.","feed_headline":"One threshold decides if fading-immunity diseases persist","feed_subtitle":"Persistence depends on one integral combining susceptibility, infectivity, and the memory kernel.","key_machinery":"The machinery is a piecewise deterministic Markov process on (age, trait), with infection rate $F(t)\\gamma(a,\\theta)$, age reset to zero at infection, and trait jumping according to the kernel $K$; its large-population limit is a nonlinear transport PDE whose boundary condition feeds reinfections back in. The threshold is carried by the memory operator $T(B)(\\theta)=\\int_{\\Theta} K(\\tilde{\\theta},\\theta)B(\\tilde{\\theta})\\nu(d\\tilde{\\theta})$: its Perron eigenfunction $S^*$ is the stationary trait distribution generated by reinfections, and the function $H(x)=x\\int_{\\mathbb{R}_+\\times\\Theta} \\exp(-x\\int_0^a \\gamma(s,\\theta)ds)\\, S^*(\\theta)\\, da\\, \\nu(d\\theta)$ satisfies $H(0)=\\int_\\Theta S^*/\\gamma^* \\, d\\nu$, so solving $H(x)=1$ yields the equilibrium force of infection.","core_discovery":"The central claim is Theorem 3.6: under assumptions that force susceptibility to vanish while an individual is infectious and to have a well-defined long-run Cesàro limit $\\gamma^*(\\theta)$, the PDE limit has an endemic equilibrium when $\\int_{\\Theta} \\frac{1}{\\gamma^*(\\theta)} S^*(\\theta) \\nu(d\\theta) < 1$, where $S^*$ is the unique positive eigenfunction of the memory operator $T$ with eigenvalue 1, normalized so that $\\int_{\\mathbb{R}_+ \\times \\Theta} \\lambda(a,\\theta) S^*(\\theta) \\, da\\, \\nu(d\\theta) = 1$. Equivalently, after reweighting the trait distribution by $S^*$, the average number of infections produced in a fully susceptible population exceeds $E^*_\\nu[1/\\gamma^*]$. Under a monotonicity condition on susceptibility, the same inequality is necessary and sufficient; when it fails, only the disease-free equilibrium exists. The paper also proves local stability of the endemic equilibrium for a memory-free SIS-type model with bounded infectivity duration, a step that earlier approaches could not reach.","pith_inferences":["Editorial inference: the threshold formula suggests that memory affects endemicity only through the stationary trait distribution $S^*$, so transient correlations between consecutive infections should not move the endemicity boundary, only the shape of $S^*$.","Editorial inference: if infectivity and susceptibility overlap in real pathogens, the separation assumption fails; a natural numerical test is to solve the stationary equations without $\\lambda\\gamma\\equiv 0$ and see whether a modified threshold emerges.","Editorial inference: when the auxiliary function $H$ is non-monotone, the model can have two endemic equilibria, which hints at hysteresis—temporarily lowering the force of infection could push a population from the higher persistent state to the disease-free state.","Editorial inference: the stability proof is limited to bounded infectivity duration, so a plausible extension is to analyze the eigenvalue equation (5.29) for unbounded durations using Laplace-transform sign arguments."],"forward_implications":["Under the theorem's assumptions, the classical $R_0>1$ condition is replaced by $E^*_\\nu[1/\\gamma^*] < R^*_0$, with the memory kernel entering only through the reweighted trait distribution $S^*\\nu$.","With susceptibility non-decreasing in age, the threshold is necessary and sufficient: the disease-free state is the only equilibrium above it.","Vaccination enters through susceptibility curves; for a renewal vaccination scheme the paper recovers the previously known threshold without assuming infectivity and susceptibility are independent.","For the one-shot-vaccine model there are parameter regions with two endemic equilibria, so the same model can admit both a mild and a severe persistent state.","Local stability is proved for a memory-free SIS-type model with bounded infectivity duration, under the explicit parameter condition $\\lambda_* \\le 2\\rho e^{\\rho a_*}$."],"supporting_citations":[{"why":"Supplies the baseline stochastic epidemic model with varying infectivity and waning immunity that this paper adapts to incorporate memory of the last infection.","marker":"[11]"},{"why":"Supplies the method used to study stationary states of selection-mutation age-structured equations, adapted here to identify endemic equilibria.","marker":"[4]"},{"why":"Provides the vaccination-policy model whose endemicity threshold is recovered as a special case in the memory-free renewal vaccination example.","marker":"[12]"},{"why":"Gives the Hille-Tamarkin compactness result used to show the memory operator $T$ has the spectral properties needed for its Perron eigenfunction $S^*$.","marker":"[19]"},{"why":"Provides the Perron-type spectral theorem guaranteeing a unique normalized positive eigenfunction for the integral operator $T$.","marker":"[30]"},{"why":"Supplies the semilinear Cauchy problem framework used to linearize the PDE around an endemic equilibrium and conclude local stability.","marker":"[31]"},{"why":"Provides the spectral theory, essential growth bounds, and stability criteria for linearized age-structured semigroups used in the stability proof.","marker":"[35]"}],"fun_headline_variants":["Memory of last infection sets one threshold for disease persistence","Single integral predicts if waning-immunity epidemics persist","Endemicity threshold depends on infection memory and immune waning","When immunity wanes, one condition decides disease persistence","Persistence threshold for epidemics with infection memory and waning immunity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that no one can be reinfected while still infectious, so an infected person's susceptibility is zero during the entire infectious period; if infectivity and susceptibility overlap, the stationary state constructed in the proof no longer satisfies the equilibrium equations and the threshold is not proved.","fun_headline_variants_meta":{"raw":{"variants":["Memory of last infection sets one threshold for disease persistence","Single integral predicts if waning-immunity epidemics persist","Endemicity threshold depends on infection memory and immune waning","When immunity wanes, one condition decides disease persistence","Persistence threshold for epidemics with infection memory and waning immunity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000761,"raw_usage":{"total_tokens":3392,"prompt_tokens":969,"completion_tokens":2423,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":585,"completion_tokens_details":{"reasoning_tokens":2342}},"tokens_in":585,"tokens_out":2423,"duration_ms":16770,"temperature":1.0,"reasoning_tokens":2342,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:39:03.203223+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Drop the separation assumption $\\lambda\\gamma\\equiv 0$ while keeping everything else, and numerically solve the stationary equations (5.1) for a two-trait example: if an endemic equilibrium appears while the computed threshold (3.9) is above 1, or fails to appear while the threshold is below 1, then the separation assumption is load-bearing and the threshold claim does not extend beyond it.","supporting_citations":[{"cited_title":"Forien, G","cited_arxiv_id":null,"evidence_quote":"Supplies the baseline stochastic epidemic model with varying infectivity and waning immunity that this paper adapts to incorporate memory of the last infection."},{"cited_title":"Calsina and J","cited_arxiv_id":null,"evidence_quote":"Supplies the method used to study stationary states of selection-mutation age-structured equations, adapted here to identify endemic equilibria."},{"cited_title":"Foutel-Rodier, A","cited_arxiv_id":null,"evidence_quote":"Provides the vaccination-policy model whose endemicity threshold is recovered as a special case in the memory-free renewal vaccination example."},{"cited_title":"J¨ orgens","cited_arxiv_id":null,"evidence_quote":"Gives the Hille-Tamarkin compactness result used to show the memory operator $T$ has the spectral properties needed for its Perron eigenfunction $S^*$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Perron-type spectral theorem guaranteeing a unique normalized positive eigenfunction for the integral operator $T$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the semilinear Cauchy problem framework used to linearize the PDE around an endemic equilibrium and conclude local stability."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the spectral theory, essential growth bounds, and stability criteria for linearized age-structured semigroups used in the stability proof."}],"review_version":1}