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

Arsene Brice Zotsa-Ngoufack; H\'el\`ene Gu\'erin

REVIEW 2 major objections 5 minor 1 cited by

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

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2505.00601 v1 pith:GAGQW3OS submitted 2025-05-01 math.PR q-bio.PE

Hélène Guérin , Arsene Brice Zotsa-Ngoufack This is my paper · ORCID

classification math.PRq-bio.PE MSC 60F1735Q9260K3535B4092D30

keywords stochasticepidemicmodelmemoryoflastinfectionwaningimmunitypiecewisedeterministicMarkovprocessage-structuredPDEendemicequilibriumendemicitythresholdvaccinationpolicy

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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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 $\lambda_* \le 2\rho e^{\rho a_*}$.

Reading between the lines

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

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.

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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

Dear colleague,

Punchline: 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.

What'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.

Soft 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.

My 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.

Recommendation. 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.

Referee Report

2 major / 5 minor

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.

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 (2)

[Theorem 3.6, Eq. (5.11)] 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.
[Theorem 3.8, Eq. (5.29)] 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.

minor comments (5)

[Section 2 and Eq. (4.1)] 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.
[Proposition 5.3, Eq. (5.7)] 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 6.2] 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.
[Throughout] 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.
[Remark 3.7] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the endemicity threshold is derived from the model's Perron eigenfunction and Cesàro susceptibility limit, and the authors' earlier papers are used only as baseline comparisons.

full rationale

The paper's central claim, Theorem 3.6, is not circular. The equilibrium is constructed from the stationary equations (5.1), which are reduced to the system (5.3)-(5.5). The function S* is defined as the normalized positive eigenfunction of the integral operator T from the memory kernel K, using the external spectral theorem of Schaefer [30], not from the desired threshold. The function H in (5.8)-(5.10) is computed from S* and the susceptibility curves, and its value at zero is obtained from the Cesàro limit in Assumption A2-(2), giving H(0)=∫(1/γ*)S*dν. The threshold (3.9) is therefore an output of the equilibrium equations, not an input. No parameter is fitted to the conclusion, and the change of measure in Remark 3.7 is merely a rewriting of (3.9). Equation (5.11), which drops the exponential waning factor on the support of λ, is a direct consequence of the explicitly stated disjoint-support assumption A2-(1)(c), not of a prior result by the same authors. The FLLN, Theorem 3.1, is proved in Section 4 by tightness, uniqueness, and martingale estimates, not by citation. The local stability result, Theorem 3.8, is stated conditionally on the spectral condition (5.29), and the paper explicitly acknowledges in Section 5.2 that a complete stability proof for the memory model remains open; this is an honest scope limitation rather than a circular reduction. The self-citations [11] and [12] are used as baselines: Remark 3.4 notes that the memory-free system coincides with [11], Example 5.2 recovers the [11] threshold after independent computation, and Section 6.2 recovers the [12] threshold by direct calculation. These prior-author results are not load-bearing assumptions in the proofs; they are confirmatory comparisons. Thus no circularity step can be exhibited from the paper's own equations.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The central claim rests on the stated structural assumptions A1 and A2 for existence, and A3 plus absolute continuity for stability. These are explicit, which is good, but they restrict the class of diseases: no reinfection while infectious, no demography, and a deterministic bound on infectiousness duration in the stability result. No new physical or biological entities are postulated; the trait θ is a mathematical indexing device, not an entity with independent empirical content.

assumptions (9)

domain assumption Assumption A1: K is a measurable kernel with ∫Θ K(θ, dθ~)ν(dθ~)=1 for each θ.
This is the memory mechanism: the new trait is drawn conditionally on the old one; used from Eq. (2.1) onward.
domain assumption Assumption A2-(1c): λγ≡0 and the support of λ lies before the support of γ.
Guarantees no reinfection during the infectious period and makes Eq. (5.11) hold, which is needed for the equilibrium identity; excludes overlapping infectivity and susceptibility.
domain assumption Assumption A2-(2): the Cesàro average of γ has a limit γ*(θ) ν-a.e.
Defines the threshold denominator 1/γ*; without it the endemicity condition has no meaning.
domain assumption Assumption A2-(3): ess supΘ ∫ exp(-x∫_0^a γ) da < ∞ for every x>0.
Technical integrability used to define H and to pass to limits in Lemma 5.4; excludes very slowly waning immunity.
domain assumption Assumption A2-(4): K>0 on Θ² and θ ↦ sup K ∈ L1(ν).
Ensures T² is compact so Perron-Frobenius arguments give the unique positive eigenfunction S* in Proposition 5.3.
domain assumption Assumption A3: γ(a,θ) ≥ σ 1_{a>a*} uniformly in θ.
Used only for local stability; the authors call it strong because it imposes a deterministic bound on infectivity duration.
domain assumption The population is homogeneous, demographic turnover is neglected, and at t=0 all individuals have been infected at least once.
Stated in Section 1; needed for the PDE normalization and for interpreting the equilibria as probability densities.
standard math Standard tools: tightness criteria for measure-valued processes, Gronwall, semigroup perturbation theory, spectral theory of positive integral operators.
Used in Sections 4 and 5 without formal machine-checking.
domain assumption For the stability section, Θ is open in Rd and ν is absolutely continuous with respect to Lebesgue measure.
Needed for the compactness arguments in Appendix B; excludes the discrete trait space used in Example 5.2-(4) for stability purposes.

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Pith. "Pith review of A stochastic epidemic model with memory of the last infection and waning immunity." pith.science (2026). https://pith.science/paper/GAGQW3OS

@misc{pith2026250500601,
  author       = {Pith},
  title        = {Pith review of: A stochastic epidemic model with memory of the last infection and waning immunity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GAGQW3OS}},
  note         = {Machine review of arXiv:2505.00601}
}

read the original abstract

We adapt the article of Forien, Pang, Pardoux and Zotsa: Arxiv preprint Arxiv2210.04667(2022), on epidemic models with varying infectivity and waning immunity, to incorporate the memory of the last infection. To this end, we introduce a parametric approach and consider a piecewise deterministic Markov process modeling both the evolution of the parameter, also called the trait, and the age of infection of individuals over time. At each new infection, a new trait is randomly chosen for the infected individual according to a Markov kernel, and their age is reset to zero. In the large population limit, we derive a partial differential equation (PDE) that describes the density of traits and ages. The main goal is to study the conditions under which endemic equilibria exist for the deterministic PDE model and to establish an endemicity threshold that depends on the model parameters. The local stability of these equilibria is also analyzed. The endemicity threshold is computed for several examples, including models that incorporate a vaccination policy, and a local stability result is obtained for a memory-free SIS-type model.

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Functional Central Limit Theorem and SPDE for epidemic model with memory of the last infection and waning immunity
math.PR 2025-05 conditional novelty 6.0 of 10

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.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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Forien, G

R. Forien, G. Pang, ´Etienne Pardoux, and A. B. Zotsa-Ngoufack. Stochastic epid emic models with varying infectivity and susceptibility. Annals of applied probability , 2025 (to appear)

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work page 2020