arxiv: 2604.07309 · v2 · submitted 2026-04-08 · 🧬 q-bio.PE · q-bio.QM

Recognition: no theorem link

Generation time in a discrete epidemic model with asymptomatic carriers: beyond geometric waiting times

Joan Salda\~na, Jordi Ripoll

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Pith reviewed 2026-05-10 16:53 UTC · model grok-4.3

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keywords generation timediscrete epidemic modelasymptomatic carriersbasic reproduction numbernon-Markovian modelWeibull distributionforward recurrence time

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The pith

Rearrangement of the basic reproduction number directly yields the generation-time distribution in a discrete epidemic model with asymptomatic stages.

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

The paper constructs a recursive discrete-time model tracking three successive infected stages with independent random waiting times at each. It demonstrates that the generation-time probabilities emerge simply by collecting and normalizing the individual terms that sum to the basic reproduction number for an asymptomatic index case. This produces the expected generation time as a convex combination of the expected delays before and after symptom onset, with weights equal to the phase-specific contributions to secondary infections. Higher moments of generation time are likewise expressed in terms of moments of the latent period, the incubation period, and infectiousness-weighted forward recurrence times within each phase. The method is then applied to several diseases by fitting discrete Weibull waiting times to published data, revealing moderate dispersion in most generation-time distributions.

Core claim

Expressing the basic reproduction number as a sum over all possible transmission delays and then dividing each term by the total recovers the probability mass function of the generation time. The resulting expectation is a convex combination of the pre-symptom and post-symptom expected generation times, where the weights are the fractions of secondary cases produced before and after symptom onset.

What carries the argument

The compact recursive system of state equations for the latent, asymptomatic, and symptomatic stages, whose solution for the basic reproduction number is algebraically rearranged into generation-time probabilities.

If this is right

Generation-time probabilities can be obtained analytically without Monte-Carlo simulation of full transmission trees.
The expected generation time shortens or lengthens according to the relative infectiousness of the asymptomatic versus symptomatic phases.
All moments of the generation time are recoverable from the first n moments of the latent period, the incubation period, and the infectiousness-weighted forward recurrence times.
For the diseases examined with Weibull waiting times, only measles exhibits high variability in its generation-time distribution.

Where Pith is reading between the lines

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

The same rearrangement technique could be applied to any discrete compartmental model whose reproduction number is written as a sum over delay bins.
Because the generation-time distribution enters directly into the renewal equation for incidence, the derived probabilities allow immediate substitution into growth-rate calculations without additional approximation.
If real transmission data show that waiting times in different stages are correlated, the convex-combination structure for the mean would no longer hold exactly.

Load-bearing premise

The waiting times spent in each infected stage are independent and drawn from known discrete distributions, while infectiousness within each phase is either constant or follows a prescribed function of elapsed time.

What would settle it

Collect contact-tracing data that records both symptom-onset dates and laboratory-confirmed transmission intervals for a disease such as influenza or SARS-CoV-2; the empirical histogram of those intervals must match the distribution derived from the model's rearranged reproduction number.

Figures

Figures reproduced from arXiv: 2604.07309 by Joan Salda\~na, Jordi Ripoll.

**Figure 1.** Figure 1: Flow diagram of system (2), the SEA-RID non-linear discrete-time epidemic model with asymptomatic carriers. The discrete random variables XE, XA, XI ≥ 1 represent the waiting times at the infected stages. State variables are expressed as a fraction of the host population: Susceptible, Exposed k0 ≥ 1 days ago, infectious Asymptomatic k1 ≥ 1 days ago, Infectious symptomatic k2 ≥ 1 days ago, Removed (alive an… view at source ↗

**Figure 2.** Figure 2: Illustration of the generation-time distribution when transmission rates [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: Estimated distribution for COVID-19 assuming constant transmission rates [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: Incubation periods are generally shorter than the generation time, with two remarkable exceptions: poliomyeli [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗

**Figure 5.** Figure 5: Estimated distribution for measles assuming constant transmission rates [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗

**Figure 6.** Figure 6: Estimated distribution for rubella assuming constant transmission rates [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗

read the original abstract

We study the random times between successive cases in a transmission chain of infectious diseases with asymptomatic carriers. We derive the probability distribution of this generation time (in days) from a discrete-time epidemic model with variable infectiousness both along elapsed times and across phases. The introduced non-Markovian model is a compact recursive system featuring random waiting times at each of the three infected stages: latent, asymptomatic, and symptomatic. By rearranging the terms of the basic reproduction number, which represents the expected number of secondary cases produced by an asymptomatic primary case who may eventually develop symptoms, we get to the generation-time probabilities. The expected generation time is a convex combination of the expected generation times before and after the onset of symptoms. Additionally, our analysis reveals that the n-th moment of the generation time is related to the moments up to n-th order of the weighted forward recurrence time at each phase and the moments up to n-th order of the latent period and the incubation period. These weights are the infectiousness along the elapsed times for each transmission phase. Finally, we illustrate several data-driven epidemic scenarios, assuming that infectiousness varies only across phases and discrete Weibull distributions for the waiting times. Each disease analyzed, except measles, exhibits moderate variability in its respective generation time distribution.

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.

Desk Editor's Note private letter to a colleague

This paper rearranges the R0 expression to extract the full generation-time PMF in a discrete three-stage model with non-geometric waiting times and asymptomatic carriers.

read the letter

The core contribution is a derivation that takes the basic reproduction number for an index case that starts asymptomatic and may progress to symptoms, then normalizes the time-indexed transmission contributions to get the generation-time probability mass function. The mean generation time comes out as a convex combination of the conditional means before and after symptom onset, and higher moments tie to weighted forward recurrence times within each phase. This is a clean algebraic step beyond the geometric-waiting-time results in the literature they cite, and the recursive system they set up for the non-Markovian process is compact once the three independent discrete waiting-time distributions and the infectiousness profile are given. They illustrate it with Weibull waiting times and phase-specific weights, showing moderate variability for most of the diseases they check except measles. That part is useful for anyone who needs to drop flexible generation times into an epidemic simulator without forcing geometric assumptions. The main limitation is that the whole construction is internal to the chosen waiting-time distributions and infectiousness schedule; there is no external validation against observed generation-time data or simulation checks shown in the abstract, so robustness to misspecification is not demonstrated. The rearrangement itself is not circular in a damaging way once the inputs are fixed, but it does not add independent information about real transmission. This is a technical methods paper aimed at mathematical epidemiologists and modelers who build or calibrate simulators. A reader who already works with discrete-stage models and wants explicit moment formulas will find it worth reading. I would send it to peer review in a journal like Mathematical Biosciences or Journal of Theoretical Biology, with the expectation that referees will want to see the full derivations and some numerical verification.

Referee Report

3 major / 2 minor

Summary. The manuscript develops a non-Markovian discrete-time epidemic model with three infected stages (latent, asymptomatic, symptomatic) featuring independent discrete waiting-time distributions and phase-dependent infectiousness that may vary with elapsed time. It derives the generation-time probability distribution by algebraic rearrangement of the basic reproduction number R0 for an index case beginning in the asymptomatic phase, shows that the expected generation time is a convex combination of the conditional expectations before versus after symptom onset, and relates the n-th moment of the generation time to moments (up to order n) of the weighted forward recurrence times within each phase together with moments of the latent and incubation periods. The framework is illustrated with data-driven scenarios that adopt discrete Weibull waiting times and constant-within-phase infectiousness, yielding moderate generation-time variability for most examined diseases except measles.

Significance. If the derivations hold, the work supplies an analytically tractable route to generation-time distributions that avoids the restrictive geometric waiting-time assumption common in Markovian models. The explicit convex-combination expression for the mean and the moment relations involving weighted forward recurrence times constitute a reusable tool for uncertainty propagation in epidemic forecasting. The data-driven illustrations demonstrate applicability to concrete diseases, although the absence of direct comparison to empirical generation-time observations or simulated outbreak trajectories limits immediate translational impact.

major comments (3)

[Derivation of generation-time probabilities (Section 3)] The central derivation obtains generation-time probabilities by rearranging the expression for R0. While the construction is internally consistent once the infectiousness profile and the three independent discrete waiting-time distributions are fixed, the manuscript must display the explicit algebraic steps (including the normalization that yields a proper probability distribution) and verify that the resulting probabilities are non-negative and sum to one for the chosen parameter values.
[Moment relations (Section 4)] The claimed relation between the n-th moment of the generation time and the moments (up to order n) of the weighted forward recurrence times, latent period, and incubation period is load-bearing for the higher-order analysis. An explicit formula or inductive proof sketch should be supplied, together with a numerical check that the relation recovers the correct moments when the waiting-time distributions are geometric.
[Numerical illustrations (Section 5)] The data-driven illustrations adopt Weibull shape and scale parameters together with phase-specific infectiousness weights without reporting sensitivity to these choices or comparison against simulated transmission chains or published empirical generation-time estimates. Adding such validation is necessary to support the conclusion of moderate variability (except for measles).

minor comments (2)

[Model definition] Clarify whether the discrete Weibull distributions are obtained by discretizing the continuous Weibull or are defined directly on the positive integers; the latter choice affects the forward-recurrence-time formulas.
[Throughout] Ensure uniform notation for the infectiousness weights across phases and elapsed times; a single table summarizing all parameters for each disease would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which have identified areas where the manuscript can be strengthened. We address each major comment below and describe the revisions we plan to implement.

read point-by-point responses

Referee: [Derivation of generation-time probabilities (Section 3)] The central derivation obtains generation-time probabilities by rearranging the expression for R0. While the construction is internally consistent once the infectiousness profile and the three independent discrete waiting-time distributions are fixed, the manuscript must display the explicit algebraic steps (including the normalization that yields a proper probability distribution) and verify that the resulting probabilities are non-negative and sum to one for the chosen parameter values.

Authors: We agree that the derivation would be clearer with explicit steps. In the revised manuscript we will expand Section 3 to present the full algebraic rearrangement of R0 that yields the generation-time probabilities, including the explicit normalization factor that ensures they form a proper distribution, followed by a direct verification that all probabilities are non-negative and sum to one for the parameter values used in the numerical illustrations. revision: yes
Referee: [Moment relations (Section 4)] The claimed relation between the n-th moment of the generation time and the moments (up to order n) of the weighted forward recurrence times, latent period, and incubation period is load-bearing for the higher-order analysis. An explicit formula or inductive proof sketch should be supplied, together with a numerical check that the relation recovers the correct moments when the waiting-time distributions are geometric.

Authors: We will add an explicit closed-form expression for the n-th moment of the generation time in terms of the moments (up to order n) of the weighted forward recurrence times and the latent/incubation periods. A short inductive proof sketch will be included, together with a numerical verification that the formula recovers the known moments when all waiting-time distributions are geometric. revision: yes
Referee: [Numerical illustrations (Section 5)] The data-driven illustrations adopt Weibull shape and scale parameters together with phase-specific infectiousness weights without reporting sensitivity to these choices or comparison against simulated transmission chains or published empirical generation-time estimates. Adding such validation is necessary to support the conclusion of moderate variability (except for measles).

Authors: We will add a sensitivity analysis in the revised Section 5 that varies the Weibull shape/scale parameters and the phase-specific infectiousness weights around the chosen values. Direct comparison against simulated transmission chains or published empirical generation-time estimates would require additional datasets and stochastic simulation code that lie outside the present theoretical scope; we will explicitly discuss this limitation and note how the derived expressions could be used for such validation in future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper's central derivation obtains the generation-time probability distribution by rearranging the time-indexed transmission contributions that define R0 (the expected secondary cases from an index asymptomatic case that may progress to symptoms). This is the standard normalization step in which the generation-time probabilities are the normalized infectiousness kernel, the mean generation time is the corresponding convex combination of pre- and post-symptom conditional expectations, and higher moments follow from weighted forward-recurrence times within each phase. The construction is self-contained once the three independent discrete waiting-time distributions and the phase-specific infectiousness profiles are supplied; no algebraic step reduces to a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation. The result follows directly from the model definitions without circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on a discrete-time recursive formulation whose waiting-time distributions are chosen (Weibull) rather than derived, plus standard probability axioms for constructing the generation-time pmf from the rearranged R0.

free parameters (2)

Weibull shape and scale parameters for each phase
Chosen to represent latent, asymptomatic, and symptomatic waiting times in the data-driven illustrations; values are not derived from first principles.
Phase-specific infectiousness weights
Relative infectiousness levels across latent, asymptomatic, and symptomatic stages that enter the weighted recurrence times.

axioms (2)

domain assumption Waiting times at the three stages are independent and follow specified discrete distributions
Invoked to close the recursive system and to obtain the moment relations.
domain assumption Infectiousness is constant within each phase or varies deterministically with elapsed time
Required for the weighted forward-recurrence-time construction.

pith-pipeline@v0.9.0 · 5528 in / 1713 out tokens · 50651 ms · 2026-05-10T16:53:20.554629+00:00 · methodology

discussion (0)

Reference graph

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