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arxiv: 2605.18571 · v1 · pith:4YWILCRYnew · submitted 2026-05-18 · 🧬 q-bio.PE · q-bio.QM

Incorporating vaccine effects into epidemiological models: common pitfalls and solutions

Casey E. Middleton , Oliver Eales , James M. McCaw , Freya M. Shearer This is my paper

Pith reviewed 2026-05-20 01:25 UTC · model grok-4.3

classification 🧬 q-bio.PE q-bio.QM
keywords vaccine efficacyleaky vaccinesepidemiological modelingherd immunityparameter estimationvaccine effectivenesspublic health
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The pith

Directly incorporating empirical vaccine effectiveness estimates into models underestimates the population impact of leaky vaccines.

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

Epidemiological models often incorporate vaccine effects using parameters derived from empirical estimates of vaccine efficacy or effectiveness. However, for vaccines that confer leaky protection by partially reducing the risk of infection rather than preventing it entirely, these estimates do not directly translate to the model's parameters. This mismatch means that using the estimates as-is leads to models that underpredict the vaccine's true effect at the population level. The paper introduces an adjustment to the parameterization that corrects for this and shows that it results in predictions of fewer total infections and lower herd immunity thresholds. These guidelines can help make models used for vaccine policy more accurate.

Core claim

To accurately parameterize epidemiological models with vaccine protection, both the vaccine's mechanism of action and the statistic used to infer vaccine efficacy from data must be considered. For leaky vaccines, common empirical estimation methods do not yield directly applicable parameter values. Naive incorporation of these estimates underestimates population-level vaccine impact. The proposed adjusted parameterization aligns the modeled effect with empirical data, leading to predictions of fewer total infections and lower herd immunity thresholds for leaky vaccines.

What carries the argument

An adjusted parameterization approach that accounts for the mismatch between empirical vaccine efficacy statistics and model protection parameters for leaky vaccines.

If this is right

  • Models with adjusted parameters predict fewer total infections compared to naive approaches.
  • Lower herd immunity thresholds are estimated for leaky vaccines under the adjusted method.
  • This leads to more accurate assessments of vaccine impact in public health planning.
  • Decision making for vaccination strategies benefits from reduced bias in impact estimates.

Where Pith is reading between the lines

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

  • Similar mismatches could exist in models of other partial interventions such as partial quarantine or treatment effects.
  • Empirical studies might benefit from estimating VE in ways that directly match common model structures.
  • Applying this adjustment could revise current estimates of required vaccination coverage for disease control.

Load-bearing premise

The vaccine mechanism is leaky protection, meaning it partially reduces individual infection risk, and the empirical VE statistic does not directly correspond to the model's protection parameter.

What would settle it

Comparing model predictions of total infections or the vaccination coverage needed for herd immunity using naive versus adjusted parameters against observed data from a real-world vaccination program for a leaky vaccine.

Figures

Figures reproduced from arXiv: 2605.18571 by Casey E. Middleton, Freya M. Shearer, James M. McCaw, Oliver Eales.

Figure 1
Figure 1. Figure 1: Infection dynamics vary based on vaccine mechanism of action. Partial model diagrams (A–C) and vaccine-derived protection distributions (D–H; c = 10−4 , 0.4, 5, 20, 104 ) for all-or-nothing, heteroge￾neous, and leaky vaccine mechanisms. Temporal infection dynamics for vaccinated (dashed) and unvac￾cinated (solid) populations (I) and time-varying VEmodeled,R (J) for each vaccine mechanism are depicted under… view at source ↗
Figure 2
Figure 2. Figure 2: The pipeline of empirical to modeled vaccine effectiveness (VE) estimates for a leaky vaccine. (A) Empirical VE estimates using risk-based (ARR-based) estimators are prone to waning VEmeasured,R over time, despite a constant true protection level (αindividual), due to differential depletion of susceptibles in vac￾cinated and unvaccinated populations. (B) When VEmeasured,R is used to inform a leaky model’s … view at source ↗
Figure 3
Figure 3. Figure 3: VEmodeled,R depends on the distribution of vaccine-derived protection. (A) Isoclines for VEmodeled,R under a given mean vaccine effect parameter (α¯) and varying distribution shapes (c). Three sample distributions are shown that produce VEmodeled,R = 0.3 when c → 0 (all-or-nothing, B), c = 10 (heterogeneous, C), and c → ∞ (leaky, D), with mean vaccine effect (α¯) depicted as a vertical dashed line. where P… view at source ↗
Figure 4
Figure 4. Figure 4: Model accuracy is high using the adjusted parameterization approach if model parameters align with study scenario. (A) Residual error using a VE estimate from a study setting with 56% of the population vaccinated (vertical line). Error is depicted for standard parameterization (α = VEmeasured,R, solid curve) and the adjusted parameterization (dashed curve) under a given model vaccine coverage. (B,C) Residu… view at source ↗
Figure 5
Figure 5. Figure 5: Adjusted parameterization accuracy is high for a heterogeneous vaccine if model correctly assumes true distribution shape. Residual error from the adjusted parameterization is shown, using a VE estimate from a study setting with a given shape parameter (ctrue). The study assumes R0 = 2.6 with a 35% vaccinated population and an average vaccine effect of α¯ = 0.7. The model assumes an identical R0 and vaccin… view at source ↗
Figure 6
Figure 6. Figure 6: The adjusted parameterization leads to lower predicted herd immunity thresholds for a leaky model. Predicted herd immunity threshold using the SIR model under standard parameterization (filled point or X) and adjusted parameterization range (lines and whiskers) for three R0 and VEmeasured,R scenarios. Adjusted parameterization ranges are shown for VE study vaccination coverage v ∈ [0.01, 0.99]. We first ca… view at source ↗
read the original abstract

Incorporating vaccination into mathematical models appears deceptively simple: models integrate vaccine-derived protections, such as reduced susceptibility to infection, using parameters informed by empirical estimates of vaccine efficacy or effectiveness (VE). In practice, however, empirical VE estimates often do not correspond directly to the parameters of epidemiological models. Here, we extend previous work to demonstrate that in order to accurately parameterize a model, one must consider both a vaccine's mechanism of action and the statistic used to infer VE from empirical data. When a vaccine confers leaky protection -- that is, vaccination partially rather than completely reduces individual infection risk -- we show that common empirical VE estimation methods do not provide directly applicable values for model parameters. Naive (i.e. direct) incorporation of these VE estimates into models results in an underestimate of population-level vaccine impact. To make progress when these estimates are the only available sources for VE, we introduce a parameterization approach which more accurately aligns the modeled effect of vaccination with empirical estimates. Under this adjusted parameterization approach, models predict fewer total infections and lower herd immunity thresholds for leaky vaccines than would be predicted under current parameterization practices. Our parameterization guidelines and adjustment approach can be used to improve accuracy in models that are used in vaccine decision making and public health planning.

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 manuscript argues that empirical estimates of vaccine efficacy (VE) for leaky vaccines, which confer partial rather than complete protection, do not map directly onto the susceptibility-reduction parameters used in standard compartmental epidemiological models. Direct (naive) insertion of these VE values into models produces an underestimate of population-level vaccine impact. The authors derive and present an adjusted parameterization that corrects for the mismatch between the empirical statistic and the model parameter, yielding predictions of fewer total infections and lower herd-immunity thresholds than current practice.

Significance. If the central adjustment holds, the work would improve the fidelity of models that inform vaccine policy and public-health planning. It supplies concrete parameterization guidelines and demonstrates the directional bias that arises from the mismatch, thereby offering a practical correction that could alter quantitative conclusions about vaccine impact for partially effective products.

major comments (2)
  1. [§3.2] §3.2 (or equivalent section on empirical estimators): The mismatch and resulting underestimate are demonstrated for direct risk-ratio estimators, but the manuscript does not examine whether the same systematic discrepancy arises when VE is obtained from Cox proportional-hazards or Poisson regression models with time-varying hazards—the estimators most frequently reported in the VE literature. Because the direction and magnitude of the proposed correction rest on this mismatch, the generality of the adjustment requires explicit verification across these standard procedures.
  2. [§4] §4, derivation of the adjusted susceptibility parameter: The adjustment formula is presented under the assumption of a constant leaky protection mechanism; it is not shown whether the same closed-form correction remains valid when protection wanes, when there is heterogeneity in individual response, or when the model includes age structure or contact networks. These extensions are load-bearing for the claim that the adjusted parameterization should be adopted in routine modeling practice.
minor comments (2)
  1. Notation for the leaky protection parameter (e.g., the multiplier on susceptibility) is introduced without a clear side-by-side comparison to the empirical VE statistic; a small table or equation block making the distinction explicit would improve readability.
  2. [Results] The numerical examples in the results section would benefit from reporting both the naive and adjusted predictions on the same axes (or in a single table) so that the magnitude of the difference is immediately visible to readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments on our manuscript. We address each of the major comments below and outline the revisions we will make to improve the clarity and generality of our findings.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (or equivalent section on empirical estimators): The mismatch and resulting underestimate are demonstrated for direct risk-ratio estimators, but the manuscript does not examine whether the same systematic discrepancy arises when VE is obtained from Cox proportional-hazards or Poisson regression models with time-varying hazards—the estimators most frequently reported in the VE literature. Because the direction and magnitude of the proposed correction rest on this mismatch, the generality of the adjustment requires explicit verification across these standard procedures.

    Authors: We acknowledge that our primary demonstrations in §3.2 use direct risk-ratio estimators for simplicity and to illustrate the core mismatch. However, the adjustment arises from the fundamental difference between the empirical VE (which measures relative reduction in infection probability or rate) and the model parameter (which scales the force of infection for vaccinated individuals in a leaky manner). For Cox proportional-hazards models, the estimated hazard ratio under proportional hazards directly informs the susceptibility reduction, but when embedded in a dynamic model, the population-level impact still requires the same correction to avoid underestimating vaccine effects. Similarly for Poisson regression. We will add an appendix with analytical arguments and numerical simulations confirming that the adjustment applies equivalently to these estimators, thereby strengthening the generality of our recommendations. revision: yes

  2. Referee: [§4] §4, derivation of the adjusted susceptibility parameter: The adjustment formula is presented under the assumption of a constant leaky protection mechanism; it is not shown whether the same closed-form correction remains valid when protection wanes, when there is heterogeneity in individual response, or when the model includes age structure or contact networks. These extensions are load-bearing for the claim that the adjusted parameterization should be adopted in routine modeling practice.

    Authors: The derivation in §4 is indeed for a basic model with constant leaky protection. We agree that demonstrating validity under waning immunity, response heterogeneity, age structure, or network contacts would further support broad adoption. The core adjustment corrects for the fact that empirical VE reflects an average effect that interacts with the nonlinear dynamics of transmission. In extensions, the same logic applies by adjusting the instantaneous susceptibility parameter at each time or for each subgroup. We will revise §4 and add a discussion section addressing these cases, providing guidance on how to apply the adjustment in more complex models without claiming a single closed-form for all scenarios. This addresses the load-bearing concern by clarifying the scope and providing a pathway for extensions. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper identifies an external mismatch between common empirical VE estimation procedures (e.g., risk-ratio based) and the protection parameters required by leaky-vaccine models, then proposes an adjusted parameterization to align them. This mismatch is demonstrated by comparing the statistic used in data inference against the model's susceptibility multiplier, without any reduction of the central claim to quantities defined inside the model by construction. No fitted inputs are relabeled as predictions, no uniqueness theorem is imported from self-citation, and the adjustment is presented as a guideline rather than a renaming of known results. The derivation relies on external benchmarks of VE methods and remains independent of any self-referential loop.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the vaccine acts via leaky protection and on the standard epidemiological modeling framework; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Vaccine confers leaky protection rather than all-or-nothing protection
    Invoked when stating that common VE estimation methods do not provide directly applicable values for model parameters.

pith-pipeline@v0.9.0 · 5764 in / 1139 out tokens · 37415 ms · 2026-05-20T01:25:47.647272+00:00 · methodology

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Reference graph

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    The vaccinated population is thus divided into3Kcompartments {VS,k, V I,k , V R,k}, and the unvaccinated compartments{S, I, R}follow the same equations as before

    and has a class-specificαofα k. The vaccinated population is thus divided into3Kcompartments {VS,k, V I,k , V R,k}, and the unvaccinated compartments{S, I, R}follow the same equations as before. The force of infection is λ(t) = β I(t) +PK k=1 VI,k(t) N ,(S20) 22 and the dynamics within each vaccinated stratum are dVS,k dt =−(1−α k)λ V S,k,(S21) dVI,k dt =...

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