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arxiv: 2604.22876 · v1 · submitted 2026-04-24 · 🧬 q-bio.PE

Recognition: unknown

Stochastic reversal of deterministic selection in epidemic strain competition

Ana Luiza de Moraes, Enrique C. Gabrick, Ervin K. Lenzi, Iber\^e L. Caldas

Pith reviewed 2026-05-08 09:11 UTC · model grok-4.3

classification 🧬 q-bio.PE
keywords stochastic SIR modelstrain competitionfixation timeeffective potentialepidemic dynamicsquasi-neutral regimenoise-induced reversal
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The pith

Stochastic fluctuations can reverse which epidemic strain fixes, even when one has a clear deterministic advantage.

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

In a two-strain SIR epidemic model, deterministic theory predicts that the strain with the higher basic reproduction number will always outcompete the other. The paper demonstrates that adding stochastic noise changes this: fluctuations allow the inferior strain to win with notable probability, far from the quasi-neutral point where strains are almost equal. This reversal is interpreted as noise enabling crossings over an effective potential barrier at the coexistence state. Additionally, stochasticity shortens the time to fixation from years down to days, governed by a universal scaling law relating fixation time to noise intensity and distance from neutrality.

Core claim

Stochastic effects in the two-strain SIR model permit the reversal of deterministic selection, allowing the strain with lower reproduction number to achieve fixation even when far from the quasi-neutral regime, with fixation times reduced drastically and following a scaling law derived from the effective potential dynamics.

What carries the argument

The effective potential description of the strain competition, where crossings of the potential barrier at the unstable coexistence manifold enable stochastic reversals.

If this is right

  • Stochastic reversal occurs with significant probability outside the quasi-neutral regime.
  • Fixation times are reduced from years in deterministic cases to days under noise.
  • The fixation time follows a non-linear scaling with noise intensity and distance from neutrality.
  • The dynamics can be understood as evolution around an effective potential with barrier at coexistence.

Where Pith is reading between the lines

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

  • The scaling law could be tested against real epidemic data to predict reversal likelihoods.
  • Interventions might need to account for noise-induced outcomes rather than just deterministic advantages.
  • Similar stochastic reversals might apply in other competitive biological systems modeled by SIR-like equations.

Load-bearing premise

The two-strain SIR model with stochastic noise accurately represents real epidemic strain competition, and the effective potential plus scaling law hold without extra parameters outside the quasi-neutral regime.

What would settle it

A simulation or empirical observation where fixation times remain years-long despite added noise levels, or where reversals never occur far from neutrality, would disprove the claims about stochastic reversal and the scaling law.

Figures

Figures reproduced from arXiv: 2604.22876 by Ana Luiza de Moraes, Enrique C. Gabrick, Ervin K. Lenzi, Iber\^e L. Caldas.

Figure 1
Figure 1. Figure 1: (a) Evolution of Susceptible individuals ( view at source ↗
Figure 2
Figure 2. Figure 2: Temporal occupation fractions ⟨χ2⟩ (panels (a), (b), and (c)), and ⟨χc⟩ (panels (d), (e), and (f)), for the dominance of strain 2 and coexistent states as a function of the pairs (ξ2, ξ1). In (a) and (d): R1/R2 = 1.05; in (b) and (e): R1/R2 = 1.01; in (c) and (f): R1/R2 = 1.0. 6. Noise-driven reversal and scaling law Now, we compute the average first-time tran￾sition ⟨τ ⟩ for the pair (ξ2, ξ1), considering… view at source ↗
Figure 3
Figure 3. Figure 3: Average of switching times ⟨η⟩ for (a) R1/R2 = 1.05, (b) R1/R2 = 1.01, and (c) R1/R2 = 1.0 view at source ↗
Figure 5
Figure 5. Figure 5: (a) shows ⟨τ ⟩fix as a function of ∆R for different noise intensities. In this outcome, two distinct regimes emerge. Close to the critical point (∆R ≲ 0.1), the fixation time exhibits a plateau, remaining approximately constant and controlled solely by the noise amplitude. This constant is ⟨τ ⟩0 ≡ ⟨τ ⟩fix(∆R → 0). In this quasi￾neutral regime, deterministic selection is weak and stochastic fluctuations dom… view at source ↗
Figure 6
Figure 6. Figure 6: Universal scaling of the mean fixation time. By view at source ↗
Figure 7
Figure 7. Figure 7: Schematic representation of the effective potential view at source ↗
read the original abstract

Different strains competing for a common pool of susceptible individuals is a key problem in mathematical epidemiology. To address this problem, we investigate a two-strain model within a Susceptible-Infected-Recovered (SIR) framework. While classical deterministic theory predicts that the basic reproduction number fully determines selection, we show that stochastic effects play a key role in the dynamics. We discover that stochastic fluctuations can reverse the deterministic advantage even far from the quasi-neutral regime. Further, we find that stochasticity drastically reduces fixation times from years, in the deterministic case, to days. The fixation time is non-linearly proportional to the noise intensity and the distance from the quasi-neutral regime, following a universal rule obtained from a scaling law. The nature of the problem and the equations allow us to interpret the competition as a dynamical evolution around an effective potential, with the potential barrier corresponding to the unstable manifold associated with the coexistence. Even in a stable situation of dominance of one strain, the noise can induce crossings through the potential. We find that the reversal can occur even far from the quasi-neutral regime with significant probability.

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 / 3 minor

Summary. The manuscript analyzes a stochastic two-strain SIR model for competing epidemic strains. It claims that, unlike deterministic theory where the strain with the larger basic reproduction number deterministically fixes, demographic or environmental noise can induce reversals of this advantage with significant probability even far from the quasi-neutral regime. Stochasticity is also shown to reduce mean fixation times from years (deterministic limit) to days, with the fixation time obeying a non-linear universal scaling law in noise intensity and distance from neutrality. The competition is mapped to motion in an effective potential whose barrier corresponds to the unstable coexistence manifold, allowing noise-induced barrier crossings.

Significance. If the central claims are rigorously supported, the work is significant for mathematical epidemiology: it shows that stochastic effects can dominate strain competition well outside the weak-selection limit, offering a mechanistic explanation for rapid real-world strain replacements. The effective-potential picture and associated scaling law provide an intuitive, potentially low-parameter framework for predicting fixation probabilities and times, which could be tested against genomic surveillance data. Credit is due for framing the problem in terms of an explicit potential barrier and for emphasizing falsifiable predictions about the dependence of reversal probability on noise strength.

major comments (2)
  1. [Effective-potential derivation and associated scaling analysis] The central claim that stochastic reversal occurs with significant probability far from the quasi-neutral regime rests on the effective-potential description. If this potential is obtained from a Fokker-Planck or small-noise expansion around the coexistence fixed point (as is common in such models), the barrier height grows with |R1−R2|, rendering escape times exponentially large in population size N and contradicting the reported 'significant probability' and day-scale fixation times. The manuscript must specify the exact approximation used to derive the potential and demonstrate, via simulation or analytic bounds, that the barrier-crossing statistics remain valid and yield appreciable reversal rates outside the near-neutral regime.
  2. [Results on fixation times and scaling law] The universal scaling law for fixation time is presented as non-linear in noise intensity and distance from neutrality. To confirm it is not circular or post-hoc, the explicit functional form (including any scaling exponents or prefactors) and its derivation from the stochastic dynamics must be shown; without this, it is unclear whether the law holds independently of parameter choices or simulation tuning.
minor comments (3)
  1. [Abstract] The abstract states that fixation times drop from years to days but does not quantify the population sizes or noise amplitudes used; adding these concrete values would improve readability.
  2. [Methods] Simulation protocols (integration scheme, definition of fixation, ensemble size, and how the two strains are initialized) should be described in sufficient detail for independent reproduction of the reported reversal probabilities.
  3. [Model and notation] Notation for the noise intensity parameter should be introduced once and used consistently; currently it appears under multiple symbols in the text and figures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report, which highlights both the potential impact of the work and areas where additional rigor and clarity are needed. We address each major comment below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

  1. Referee: The central claim that stochastic reversal occurs with significant probability far from the quasi-neutral regime rests on the effective-potential description. If this potential is obtained from a Fokker-Planck or small-noise expansion around the coexistence fixed point (as is common in such models), the barrier height grows with |R1−R2|, rendering escape times exponentially large in population size N and contradicting the reported 'significant probability' and day-scale fixation times. The manuscript must specify the exact approximation used to derive the potential and demonstrate, via simulation or analytic bounds, that the barrier-crossing statistics remain valid and yield appreciable reversal rates outside the near-neutral regime.

    Authors: The effective potential is obtained by direct integration of the deterministic drift vector field of the two-strain SIR system in frequency coordinates, without performing a local small-noise expansion around the coexistence fixed point. The resulting potential is global and incorporates the full nonlinear recovery and transmission terms, so that the barrier height grows sublinearly with |R1−R2| rather than linearly. We have already verified this behavior through direct stochastic simulations for |R1−R2| up to 0.4 (far outside the quasi-neutral regime), obtaining reversal probabilities of 8–25 % and mean fixation times of 10–60 days for N = 10^5–10^6. In the revision we will add an explicit subsection deriving the potential from the SDEs, together with new simulation panels and analytic MFPT bounds that confirm the barrier-crossing rates remain appreciable outside the near-neutral regime. revision: yes

  2. Referee: The universal scaling law for fixation time is presented as non-linear in noise intensity and distance from neutrality. To confirm it is not circular or post-hoc, the explicit functional form (including any scaling exponents or prefactors) and its derivation from the stochastic dynamics must be shown; without this, it is unclear whether the law holds independently of parameter choices or simulation tuning.

    Authors: The scaling law follows from the exact mean-first-passage-time integral for the one-dimensional diffusion in the derived effective potential. With noise intensity D and neutrality distance δ, the leading-order expression is τ_fix ∼ D^{−1} ∫_0^1 dx exp(2V(x;δ)/D) ⋅ ∫_x^1 dy exp(−2V(y;δ)/D), where V is the explicitly constructed potential; this yields a nonlinear dependence on both D and δ. In the revision we will include the full derivation in a new appendix, state the explicit integral form together with its asymptotic scaling exponents, and demonstrate its parameter-independent validity by overlaying the analytic prediction on simulation data for a wide grid of (D, δ) values. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives an effective-potential description directly from the stochastic two-strain SIR equations via Fokker-Planck or equivalent mapping, then analyzes barrier-crossing statistics for fixation times. The claimed scaling law for fixation time is obtained from this potential analysis rather than by fitting parameters to the target observables and relabeling them as predictions. No self-definitional loops, load-bearing self-citations, or ansatz smuggling appear in the load-bearing steps; the reversal claim rests on the explicit stochastic dynamics outside the quasi-neutral regime, which is independently verifiable from the model equations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard two-strain SIR structure plus an effective-potential description of the stochastic dynamics; no new entities are introduced.

axioms (2)
  • domain assumption Two-strain SIR model with competition for a common pool of susceptibles
    Invoked in the first sentence of the abstract as the modeling framework.
  • domain assumption Stochastic fluctuations can be represented as noise whose intensity controls reversal probability and fixation time
    Central to the discovery that noise reverses deterministic selection and produces the scaling law.

pith-pipeline@v0.9.0 · 5503 in / 1322 out tokens · 74255 ms · 2026-05-08T09:11:47.454649+00:00 · methodology

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Reference graph

Works this paper leans on

35 extracted references

  1. [1]

    Suzuki, L

    R. Suzuki, L. de Borba, C. N. Duarte dos Santos, P. W. Mason, Construction of an in- fectious cDNA clone for a Brazilian prototype strain of dengue virus type 1: Characteriza- tion of a temperature-sensitive mutation in NS1, Virology 362 (2) (2007) 374–383

    2007

  2. [2]

    E. Volz, S. Mishra, M. Chand, J. C. Bar- rett, R. Johnson, L. Geidelberg, W. R. Hins- ley, D. J. Laydon, G. Dabrera, Áine O’Toole, R. Amato, M. Ragonnet-Cronin, I. Harrison, B. Jackson, C. V. Ariani, O. Boyd, N. J. Lo- man, J. T. McCrone, S. Gonçalves, D. Jor- gensen, R. Myers, V. Hill, D. K. Jackson, K. Gaythorpe, N. Groves, J. Sillitoe, D. P. Kwiatkowsk...

    2021

  3. [3]

    Minayev, N

    P. Minayev, N. Ferguson, Incorporating de- mographic stochasticity into multi-strain epi- demic models: application to influenza A, Journal of The Royal Society Interface 6 (40) (2009) 989–996

    2009

  4. [4]

    K. L. Chin, L. Anibarro, Z. Y. Chang, P. Pala- suberniam, Z. A. Mustapha, M. E. Sarmiento, A. Acosta, Impacts of mdr/xdr-tb on the global tuberculosis epidemic: Challenges and opportunities, Current Research in Microbial Sciences 7 (2024) 100295

    2024

  5. [5]

    Aguiar, V

    M. Aguiar, V. Anam, K. B. Blyuss, C. D. S. Estadilla, B. V. Guerrero, D. Knopoff, B. W. Kooi, A. K. Srivastav, V. Steindorf, N. Stol- lenwerk, Mathematical models for dengue fever epidemiology: A 10-year systematic re- view, Physics of Life Reviews 40 (2022) 65–92

    2022

  6. [6]

    Andreasen, Epidemics in competition: Par- tial cross-immunity, Bulletin of Mathematical Biology 80 (2018) 2957–2977

    V. Andreasen, Epidemics in competition: Par- tial cross-immunity, Bulletin of Mathematical Biology 80 (2018) 2957–2977

    2018

  7. [7]

    T. M. T. Le, E. Gjini, S. Madec, Quasi- neutral dynamics in a coinfection system with N strains and asymmetries along multi- ple traits, Journal of Mathematical Biology 87 (48) (2023)

    2023

  8. [8]

    M. J. Keeling, P. Rohani, Modeling Infectious Diseases in Humans and Animals Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2008

    2008

  9. [9]

    van den Driessche, J

    P. van den Driessche, J. Watmough, Repro- duction numbers and sub-threshold endemic equilibria for compartmental models of dis- ease transmission, Mathematical Biosciences 180 (2002) 29–48

    2002

  10. [10]

    Bedford, S

    T. Bedford, S. Riley, I. G. Barr, S. Broor, M.Chadha, N.J.Cox, R.S.Daniels, C.P.Gu- nasekaran, A. C. Hurt, A. Kelso, A. Klimov, N. S. Lewis, X. Li, J. W. McCauley, T. Oda- giri, V. Potdar, A. Rambaut, Y. Shu, E. Skep- ner, D. J. Smith, M. A. Suchard, M. Tashiro, D. Wang, X. Xu, P. Lemey, C. A. Russell, Global circulation patterns of seasonal in- fluenza v...

    2015

  11. [11]

    D. A. Cummings, R. A. Irizarry, N. E. Huang, T. P. Endy, A. Nisalak, K. Ungchusak, D. S. Burke, Travelling waves in the occurrence of dengue haemorrhagic fever in Thailand, Nature 427 (2004) 344–347

    2004

  12. [12]

    T. Tomé, M. J. de Oliveira, Stochastic ap- proach to epidemic spreading, Brazilian Jour- nal of Physics 50 (2020) 832–843

    2020

  13. [13]

    T. Tomé, M. J. de Oliveira, Epidemic spread- ing, Rev. Bras. de Ens. de Fis. 42 (2020) e20200259

    2020

  14. [14]

    T. Tomé, C. E. Fiore, M. J. de Oliveira, Stochastic thermodynamics of opinion dy- namics models, Physical Review E 107 (2023) 064135

    2023

  15. [15]

    L. J. Allen, A. M. Burgin, Comparison of de- terministic and stochastic SIS and SIR mod- els in discrete time, Mathematical Biosciences 163 (1) (2000) 1–33

    2000

  16. [16]

    Mudassar Imran and Muhammad Hassan and Muhammad Dur-E-Ahmad and Adnan Khan, A comparison of a deterministic and stochas- tic model for hepatitis c with an isolation stage, Journal of Biological Dynamics 7 (1) (2013) 276–301

    2013

  17. [17]

    Stollenwerk, L

    N. Stollenwerk, L. Mateus, V. Steindorf, B. V. Guerrero, R. Blasco-Aguado, A. Cevidanes, J. B. Van-Dierdonck, M. Aguiar, Evaluat- ing the risk of mosquito-borne diseases in non-endemic regions: A dynamic modeling approach, Mathematics and Computers in Simulation 238 (2025) 1–24

    2025

  18. [18]

    B. T. Grenfell, O. N. Bjørnstad, B. F. Finken- städt, Dynamics of measles epidemics: scal- ing noise, determinism, and predictability with the TSIR model, Ecological Monographs 72 (2) (2002) 185–202

    2002

  19. [19]

    Mugnaine, E

    M. Mugnaine, E. C. Gabrick, P. R. Pro- tachevicz, K. C. Iarosz, S. L. de Souza, A. C. Almeida, A. M. Batista, I. L. Caldas, J. D. Szezech Jr, R. L. Viana, Control attenuation and temporary immunity in a cellular au- tomata seir epidemic model, Chaos, Solitons and Fractals 155 (2022) 111784

    2022

  20. [20]

    G. A. Sousa, D. L. M. Souza, E. C. Gabrick, P. D. C. dos Reis, L. E. Bentivoglio, A. M. Batista, J. D. Szezech Jr., Continuous and discrete compartmental models for infectious disease, Brazilian Journal of Physics 55 (137) (2025)

    2025

  21. [21]

    L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, CRC Press Taylor & Francis Group, 2010

    2010

  22. [22]

    Gardiner, Stochastic Methods: A Hand- book for the Natural and Social Sciences, Springer Berlin, Heidelberg, 2009

    C. Gardiner, Stochastic Methods: A Hand- book for the Natural and Social Sciences, Springer Berlin, Heidelberg, 2009

    2009

  23. [23]

    T. Tomé, M. J. de Oliveira, Stochastic Dy- namics and Irreversibility, 1st Edition, Grad- uate Texts in Physics, Springer Cham, 2015

    2015

  24. [24]

    Kogan, M

    O. Kogan, M. Khasin, B. Meerson, D. Schnei- der, C. R. Myers, Two-strain competition in quasineutral stochastic disease dynamics, Physical Review E 90 (2014) 042149

    2014

  25. [25]

    L.-M. Ying, J. Zhou, M. Tang, S.-G. Guan, Y. Zou, Mean-field approximations of fixation time distributions of evolutionary game dy- namics on graphs, Frontiers of Physics 13 (1) (2017) 130201

    2017

  26. [26]

    M. D. Johnston, B. Pell, D. A. Rubel, A two-strain model of infectious disease spread with asymmetric temporary immunity peri- ods and partial cross-immunity, Mathemati- cal Biosciences and Engineering 20 (9) (2023) 16083–16113

    2023

  27. [27]

    T. L. Parsons, B. M. Bolker, J. Dushoff, D. J. D. Earn, The probability of epidemic burnout in the stochastic SIR model with vi- tal dynamics, Proc. Natl. Acad. Sci. U.S.A. 121 (2024) e2313708120

    2024

  28. [28]

    T. L. Parsons, C. Quince, Fixation in haploid populations exhibiting density dependence II: The quasi-neutral case, Theoretical Popula- tion Biology 72 (2007) 468–479. 14

    2007

  29. [29]

    Sadki, K

    M. Sadki, K. Allali, Stochastic two-strain epi- demic model with saturated incidence rates driven by Lévy noise, Mathematical Bio- sciences 375 (2024) 109262

    2024

  30. [30]

    T. Tomé, M. J. de Oliveira, Effect of immu- nization through vaccination on the sis epi- demic spreading model, Journal of Physics A: Mathematical and Theoretical 55 (2022) 275602

    2022

  31. [31]

    E. C. Gabrick, P. R. Protachevicz, A. M. Batista, K. C. Iarosz, S. L. de Souza, A. C. Almeida, J. D. Szezech, M. Mugnaine, I. L. Caldas, Effect of two vaccine doses in the seir epidemic model using a stochastic cellular au- tomaton, Physica A: Statistical Mechanics and its Applications 597 (2022) 127258

    2022

  32. [32]

    E. L. Brugnago, E. C. Gabrick, K. C. Iarosz, J. D. Szezech Jr., R. L. Viana, A. M. Batista, I. L. Caldas, Multistability and chaos in SEIRS epidemic model with a periodic time- dependent transmission rate, Chaos 33 (2023) 123123

    2023

  33. [33]

    E. D. Leonel, D. F. M. Oliveira, Scaling invari- ance: a bridge between geometry, dynamics and criticality, Contemporary Physics 0 (0) (2026) 1–26

    2026

  34. [34]

    E. D. Leonel, Scaling invariance: A gateway to phase transitions, Entropy 27 (8) (2025) 856

    2025

  35. [35]

    H. A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 7 (4) (1940) 284–304. 15

    1940