Viral population dynamics at the cellular level, considering the replication cycle

Seong Jun Park

arxiv: 2510.14481 · v3 · submitted 2025-10-16 · 🧬 q-bio.PE · q-bio.QM

Viral population dynamics at the cellular level, considering the replication cycle

Seong Jun Park This is my paper

Pith reviewed 2026-05-18 06:32 UTC · model grok-4.3

classification 🧬 q-bio.PE q-bio.QM

keywords viral population dynamicsreplication cyclerenewal processbirth-death processstochastic modelingnon-exponential growthstage duration variability

0 comments

The pith

Viral population growth follows analytical expressions that incorporate random durations across replication stages via a coupled renewal and birth-death model.

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

This paper builds a stochastic framework that tracks viral particles by coupling a birth-death process for their numbers to a renewal process that advances cells through replication stages with arbitrary random durations. It derives closed-form expressions for the expected particle count over time and classifies when growth remains exponential or becomes non-exponential due to timing variability. Classical deterministic or Markovian models omit these effects because they assume fixed or exponential stage times. Readers should care because the approach directly connects measurable intracellular timing statistics to whether an infection expands rapidly or stays limited at the cellular level. Stochastic simulations confirm the formulas hold across different stage-duration distributions.

Core claim

By representing the intracellular replication cycle as a renewal process with general stage-duration distributions and coupling it to a structured birth-death process, analytical expressions are derived for the expected number of viral particles over time. This formulation captures non-exponential waiting-time effects neglected in classical models and identifies conditions under which the population exhibits exponential expansion or non-exponential behavior.

What carries the argument

A renewal process for progression through replication stages with arbitrary duration distributions, coupled to a structured birth-death process for the viral particle population.

If this is right

Exact analytical formulas exist for the time-dependent expected viral count under general stage duration laws.
Viral populations exhibit non-exponential growth when stage durations have high variance, unlike Markovian models.
Growth regimes are classified according to properties of the stage duration distributions such as their Laplace transforms.
Stochastic simulations match the analytical predictions for deterministic, exponential, and heavy-tailed stage durations.

Where Pith is reading between the lines

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

The model could be applied to predict how drugs targeting specific replication stages change overall infection speed.
Cell-to-cell differences in stage timing may help explain why some infections persist rather than clear or explode.
Linking the framework to spatial models of tissue spread would test whether single-cell timing variability affects macroscopic infection patterns.

Load-bearing premise

The replication cycle inside each cell can be modeled as a renewal process with independent stage durations drawn from fixed general distributions.

What would settle it

Measurements of viral particle accumulation over time in infected cell cultures that cannot be matched by the derived expressions for any fitted stage-duration distribution would falsify the analytical results.

read the original abstract

We develop a stochastic framework for viral population dynamics at the cellular level that explicitly incorporates the replication cycle with random stage durations. The model is formulated as a structured birth-death process coupled with a renewal description of intracellular progression, allowing for general distributions of stage completion times. Within this framework, we derive analytical expressions for key population descriptors, including the expected number of viral particles over time. The formulation captures non-exponential waiting-time effects, which are typically neglected in classical deterministic or Markovian models, and reveals how variability in replication timing shapes population growth. We further analyze the model to characterize growth regimes and identify conditions under which the population exhibits exponential expansion or non-exponential behavior. Stochastic simulations are used to validate the analytical results and to illustrate the impact of different stage-duration distributions. Our results provide a mathematically tractable and generalizable approach to linking intracellular replication mechanisms with population level viral dynamics, offering new insight into how temporal heterogeneity influences infection outcomes.

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

The paper couples renewal theory to a structured birth-death process for viral replication stages but the analytical expressions for general distributions are likely just integral equations rather than closed forms.

read the letter

Hey, the main thing to know is that this paper sets up a stochastic model for viral particles at the cell level by combining a structured birth-death process with a renewal description of the replication cycle, allowing arbitrary stage-duration distributions instead of sticking to exponentials. That coupling is the core new piece. They derive expressions for expected particle counts over time, characterize growth regimes, and run simulations to show how timing variability affects expansion. The simulations look like a reasonable check on the framework and help illustrate the non-exponential effects that classical models ignore. That part is done cleanly enough. The soft spot sits with the tractability claim. For arbitrary distributions the expectation equation is typically a linear Volterra integral equation of the second kind, which does not yield simple closed-form time-domain solutions and usually needs numerical methods or Laplace inversion. If the paper presents the integral form or its transform as the analytical result, then the practical gain over Markovian models shrinks to modeling flexibility rather than easier computation. The abstract emphasizes usable analytical expressions, so this distinction matters. The work is aimed at modelers in mathematical virology who care about intracellular timing details feeding into population dynamics. A reader who builds or extends stochastic infection models would find the setup worth looking at. It has enough formal structure and validation effort to deserve a serious referee who can check the derivations and clarify exactly what counts as analytical here. I would send it out for peer review.

Referee Report

1 major / 1 minor

Summary. The manuscript develops a stochastic framework for viral population dynamics at the cellular level by coupling a structured birth-death process with a renewal process to model intracellular replication stages with arbitrary duration distributions. It claims to derive analytical expressions for the expected number of viral particles over time, characterize exponential versus non-exponential growth regimes, and validate results via stochastic simulations, emphasizing the role of non-exponential waiting times neglected in classical Markovian models.

Significance. If the central derivations yield tractable analytical forms (or readily solvable expressions) for general stage-duration distributions, the work would meaningfully extend standard viral dynamics models by linking mechanistic intracellular timing variability to population-level outcomes. The explicit use of renewal theory to capture temporal heterogeneity is a potential strength, offering a generalizable approach that could improve predictions of infection dynamics when stage durations are empirically non-exponential.

major comments (1)

[Abstract and derivation of expected population size] Abstract and the section deriving expected particle counts: the claim that 'analytical expressions' are derived for the expected number of viral particles over time for arbitrary (non-exponential) stage-duration distributions is load-bearing for the central advance. In a renewal-structured birth-death model the expectation typically satisfies a linear Volterra integral equation of the second kind whose kernel is the stage-completion density; closed-form time-domain solutions exist only for exponential or phase-type distributions. For general distributions the equation remains unsolved analytically and requires numerical solution or Laplace inversion. The manuscript must clarify whether the 'analytical expressions' consist of the integral equation (or its transform) itself or explicit invertible forms, and demonstrate tractability with at least one non-exponential example. This directly (

minor comments (1)

[Introduction] The abstract and introduction would benefit from a brief comparison to prior applications of renewal theory in viral or cell-cycle models to better situate the novelty.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful and insightful review. We address the major comment below, agree on the technical characterization of the integral equation, and have revised the manuscript to clarify the form of our analytical results while adding an explicit non-exponential demonstration.

read point-by-point responses

Referee: Abstract and the section deriving expected particle counts: the claim that 'analytical expressions' are derived for the expected number of viral particles over time for arbitrary (non-exponential) stage-duration distributions is load-bearing for the central advance. In a renewal-structured birth-death model the expectation typically satisfies a linear Volterra integral equation of the second kind whose kernel is the stage-completion density; closed-form time-domain solutions exist only for exponential or phase-type distributions. For general distributions the equation remains unsolved analytically and requires numerical solution or Laplace inversion. The manuscript must clarify whether the 'analytical expressions' consist of the integral equation (or its transform) itself or explicit invertible forms, and demonstrate tractability with at least one non-exponential example. This directly (

Authors: We agree that closed-form time-domain solutions are unavailable for arbitrary distributions and that the expectation satisfies a Volterra integral equation. In the manuscript the derived analytical expressions consist precisely of this integral equation (obtained via the renewal-theory coupling) together with its Laplace transform, which is the standard tractable representation in renewal theory and permits both analytical asymptotics and numerical inversion. We have revised the abstract and the derivation section to state this explicitly and to avoid any implication of closed-form time-domain solutions. We have also added a new subsection that solves the integral equation numerically for a gamma-distributed stage duration (shape parameter 2, non-exponential) and validates the result against stochastic simulations, thereby demonstrating tractability for a concrete non-exponential case. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation applies standard renewal theory to structured birth-death without self-referential reduction

full rationale

The paper sets up a renewal process for intracellular stages coupled to a structured birth-death process for viral particles, then derives the expectation via the standard integral renewal equation whose kernel is the stage-completion density. This is a direct application of classical renewal theory and branching process methods to the viral context; the resulting Volterra equation (or its Laplace transform) is the explicit analytical form claimed, not a fitted parameter or self-defined quantity. No self-citations are load-bearing for the central result, no parameters are fitted to subsets and relabeled as predictions, and the framework does not rename known empirical patterns or smuggle ansatzes via prior work. The derivation is therefore self-contained against external mathematical benchmarks and receives score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The model rests on standard assumptions from stochastic processes and renewal theory without introducing new free parameters or invented entities in the described framework.

axioms (1)

domain assumption Intracellular viral replication proceeds through distinct stages whose completion times follow general probability distributions.
Explicitly invoked to capture non-exponential waiting-time effects neglected in Markovian models.

pith-pipeline@v0.9.0 · 5687 in / 1218 out tokens · 31771 ms · 2026-05-18T06:32:31.516373+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

derive an analytical formula for viral populations at the cellular level, based on viewing viral replication as a birth-death process. The mean viral count is expressed via probability density functions... (Eq. 1)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

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Molecular Cell Biology

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[1] [1]

Fields Virology, 6th Edition

Liu L. Fields Virology, 6th Edition. Clinical Infectious Diseases 2014; 59:613

work page 2014

[2] [2]

Disease-causing human viruses: Novelty and legacy

Forni D, Cagliani R, Clerici M, et al. Disease-causing human viruses: Novelty and legacy. Trends Microbiol 2022; 30:1232–1242

work page 2022

[3] [3]

Principles of Virology, Volume 1: Molecular Biology

Flint J, Racaniello VR, Rall GF , et al. Principles of Virology, Volume 1: Molecular Biology. 2020

work page 2020

[4] [4]

Viral Infections of Humans: Epidemiology and Control

Kaslow RA, Stanberry LR, Duc JWL. Viral Infections of Humans: Epidemiology and Control. 2014

work page 2014

[5] [5]

The Global Virome Project

Carroll D, Daszak P , Wolfe ND, et al. The Global Virome Project. Science (1979) 2018; 359:872–874

work page 1979

[6] [6]

Using artificial intelligence to document the hidden RNA virosphere

Hou X, He Y , Fang P , et al. Using artificial intelligence to document the hidden RNA virosphere. Cell 2024; 187:6929-6942.e16

work page 2024

[7] [7]

Human viruses: Discovery and emergence

Woolhouse M, Scott F , Hudson Z, et al. Human viruses: Discovery and emergence. Philosophical Transactions of the Royal Society B: Biological Sciences 2012; 367:2864– 2871

work page 2012

[8] [8]

Emerging zoonotic diseases originating in mammals: A systematic review of effects of anthropogenic land-use change

White RJ, Razgour O. Emerging zoonotic diseases originating in mammals: A systematic review of effects of anthropogenic land-use change. Mamm Rev 2020; 50:336–352

work page 2020

[9] [9]

The Baltimore Classification of Viruses 50 years later: How does it stand in the light of virus evolution? Microbiology and Molecular Biology Reviews 2021; 85:10.1128/mmbr.00053-21

Koonin EV , Krupovic M, Agol VI. The Baltimore Classification of Viruses 50 years later: How does it stand in the light of virus evolution? Microbiology and Molecular Biology Reviews 2021; 85:10.1128/mmbr.00053-21

work page doi:10.1128/mmbr.00053-21 2021

[10] [10]

Encyclopedia of Cell Biology

Bradshaw RA, Stahl PD. Encyclopedia of Cell Biology. 2015

work page 2015

[11] [11]

Textbook of General Virology

Kumar S. Textbook of General Virology. 2025

work page 2025

[12] [12]

Viral kinetic modeling: State of the art

Canini L, Perelson AS. Viral kinetic modeling: State of the art. J Pharmacokinet Pharmacodyn 2014; 41:431–443

work page 2014

[13] [13]

Kinetics of virus production from single cells

Timm A, Yin J. Kinetics of virus production from single cells. Virology 2012; 424:11–17

work page 2012

[14] [14]

Kinetic modeling of virus growth in cells

Yin J, Redovich J. Kinetic modeling of virus growth in cells. Microbiology and Molecular Biology Reviews 2018; 82:10.1128/mmbr.00066-17

work page doi:10.1128/mmbr.00066-17 2018

[15] [15]

Modelling viral and immune system dynamics

Perelson AS. Modelling viral and immune system dynamics. Nat Rev Immunol 2002; 2:28–36

work page 2002

[16] [16]

Modeling suggests that virion production cycles within individual cells is key to understanding acute hepatitis B virus infection kinetics

Hailegiorgis A, Ishida Y , Collier N, et al. Modeling suggests that virion production cycles within individual cells is key to understanding acute hepatitis B virus infection kinetics. PLoS Comput Biol 2023; 19:e1011309

work page 2023

[17] [17]

Measurements of the self-assembly kinetics of individual viral capsids around their RNA genome

Garmann RF , Goldfain AM, Manoharan VN. Measurements of the self-assembly kinetics of individual viral capsids around their RNA genome. Proceedings of the National Academy of Sciences 2019; 116:22485–22490

work page 2019

[18] [18]

Stochastic kinetics of reproduction of virions inside a cell

Zhdanov VP . Stochastic kinetics of reproduction of virions inside a cell. Biosystems 2004; 77:143–150

work page 2004

[19] [19]

Effects of ribosomes on the kinetics of Qβ replication

Usui K, Ichihashi N, Kazuta Y , et al. Effects of ribosomes on the kinetics of Qβ replication. FEBS Lett 2014; 588:117–123

work page 2014

[20] [20]

A detailed kinetic model of Eastern equine encephalitis virus replication in a susceptible host cell

Larkin CI, Dunn MD, Shoemaker JE, et al. A detailed kinetic model of Eastern equine encephalitis virus replication in a susceptible host cell. PLoS Comput Biol 2025; 21:e1013082

work page 2025

[21] [21]

Decoding the role of temperature in RNA virus infections

Bisht K, te Velthuis AJW. Decoding the role of temperature in RNA virus infections. mBio 2022; 13:e02021-22

work page 2022

[22] [22]

Effects of high temperature on pandemic and seasonal human influenza viral replication and infection-induced damage in primary human tracheal epithelial cell cultures

Yamaya M, Nishimura H, Lusamba Kalonji N, et al. Effects of high temperature on pandemic and seasonal human influenza viral replication and infection-induced damage in primary human tracheal epithelial cell cultures. Heliyon 2019; 5:e01149

work page 2019

[23] [23]

Influenza virus with increased pH of hemagglutinin activation has improved replication in cell culture but at the cost of infectivity in human airway epithelium

Singanayagam, A, Zambon M, Barclay, WS. Influenza virus with increased pH of hemagglutinin activation has improved replication in cell culture but at the cost of infectivity in human airway epithelium. J Virol 2019; 93:10.1128/jvi.00058-19

work page doi:10.1128/jvi.00058-19 2019

[24] [24]

Principles of Molecular Virology

Cann A. Principles of Molecular Virology. 2001

work page 2001

[25] [25]

Molecular and Cellular Biology of Viruses

Lostroh P . Molecular and Cellular Biology of Viruses. 2024

work page 2024

[26] [26]

Computational modeling of protracted HCMV replication using genome substrates and protein temporal profiles

Monti CE, Mokry RL, Schumacher ML, et al. Computational modeling of protracted HCMV replication using genome substrates and protein temporal profiles. Proceedings of the National Academy of Sciences 2022; 119:e2201787119

work page 2022

[27] [27]

Single-virus fusion measurements reveal multiple mechanistically equivalent pathways for SARS-CoV-2 entry

Sengar, A, Cervantes, M, Bandalapati, ST, et al. Single-virus fusion measurements reveal multiple mechanistically equivalent pathways for SARS-CoV-2 entry. J Virol 2023; 97:e01992-22

work page 2023

[28] [28]

Campbell NA, Reece JB. Biology. 2005

work page 2005

[29] [29]

Molecular Cell Biology

Lodish HF . Molecular Cell Biology. 2007. Supplementary Material Viral population dynamics at the cellular level, considering the replication cycle Derivation of Eq. (1) We examine the number of virions at the inter- and intracellular levels. Figure 1 describes the virion dynamics. This work assumes that an initial virion multiplies by infecting a host ce...

work page 2007