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arxiv: 2604.27312 · v1 · submitted 2026-04-30 · 🧬 q-bio.PE · physics.soc-ph

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Epidemic Extinction in a Continuous SIRS Model with Vaccination

Germano Hartmann Brill , Pablo Enrique Jurado Silvestrin , Sebastian Gon\c{c}alves
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Pith reviewed 2026-05-07 10:06 UTC · model grok-4.3

classification 🧬 q-bio.PE physics.soc-ph
keywords epidemic extinctionSIRS modelvaccinationcontinuous ODEstochastic effectsepidemic fade-outwaning immunitydiscrete population
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The pith

Continuous SIRS models with vaccination let epidemics persist unrealistically after the infected count falls below one person.

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

The paper studies a deterministic SIRS system of ordinary differential equations that includes time-dependent vaccination and waning immunity from recovered to susceptible. It tracks how changes in infection, recovery, and immunity-loss rates shape whether the epidemic reaches an endemic state or appears to decay toward zero. The analysis reveals that the continuous equations routinely produce nonphysical behavior once the infected fraction drops below the equivalent of a single individual, including extended persistence and secondary peaks that cannot occur in a real finite population. By contrasting these trajectories with the expectation that real epidemics end when no infected individuals remain, the work shows that continuous models cannot reliably describe extinction and that stochastic or discrete descriptions are required instead.

Core claim

In the continuous SIRS ODE model with vaccination, the infected compartment can fall below one individual while remaining positive, allowing the epidemic to persist or generate artificial secondary waves. Parameter sweeps across infection, recovery, and waning rates identify regimes of apparent extinction versus endemicity, yet all such outcomes remain artifacts once population discreteness is considered. The authors therefore conclude that accurate prediction of epidemic fade-out demands incorporation of stochasticity or discrete effects rather than reliance on deterministic continuous dynamics.

What carries the argument

The deterministic system of three ordinary differential equations for the susceptible, infected, and recovered fractions, augmented by a vaccination rate term and a constant waning-immunity rate.

If this is right

  • In regimes of rapid recovery the continuous model produces quick apparent decay followed by long nonphysical tails.
  • Vaccination lowers the endemic infected level but leaves the fractional-individual artifact unchanged near extinction.
  • Secondary peaks appear in the ODE trajectories after near-zero infected values, which would be impossible once infection has truly reached zero.
  • Real-world fade-out thresholds are systematically mismatched by the continuous dynamics whenever infected numbers become small.

Where Pith is reading between the lines

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

  • Forecasting tools should switch from deterministic ODEs to stochastic rules once prevalence falls below a few dozen cases to avoid overestimating outbreak length.
  • Policy decisions based solely on continuous-model projections may mis-time interventions at the tail of an epidemic.
  • Hybrid modeling frameworks that remain deterministic at high prevalence and become discrete or stochastic at low prevalence would better match observed extinction timing.

Load-bearing premise

The continuous ODE equations remain a valid description of epidemic dynamics even when the infected population is smaller than one person.

What would settle it

A fully stochastic individual-based simulation of the identical SIRS process with the same rates, in which the epidemic reaches exact zero infected individuals and stays extinct without secondary peaks once the last case recovers.

Figures

Figures reproduced from arXiv: 2604.27312 by Germano Hartmann Brill, Pablo Enrique Jurado Silvestrin, Sebastian Gon\c{c}alves.

Figure 1
Figure 1. Figure 1: Time series of the SIRS Model with vaccination with view at source ↗
Figure 2
Figure 2. Figure 2: Extinction probability heatmap for different values of R0 and vaccine counts. The gray region delimits the R0 and vaccine values where the extinction probability is defined as 1 due to susceptible depletion. The population in each subplot is (a) 104 , (b) 105 , (c) 106 , (d) 107 . Fixed parameter values: γ = 0.1, α = 0.01, t0 = 80, δ = 0.005. As observed in Fig. (2) there is a specific value where the numb… view at source ↗
Figure 4
Figure 4. Figure 4: Extinction probability heatmap with a Gaussian filter for di view at source ↗
Figure 3
Figure 3. Figure 3: Extinction probability heatmap for different values of R0 and number of vaccines in the agent-based Model. Fixed parameters values: γ = 0.1, α = 0.01, t0 = 80, δ = 0.005 and a population of N = 104 . As shown in Fig. (3), the same behavior observed in the con￾tinuous model regarding a region of epidemic persistence at low values of R0 is also exhibited in the agent-based model. This suggests that such beha… view at source ↗
Figure 5
Figure 5. Figure 5: Extinction probability heatmap for different values of R0 and number of vaccines for a vaccination campaign beginning after the stationary state. The gray region delimitates the R0 and vaccine number values where the extinction probability is defined as 1 because of the lack of susceptible. The population in each subplot is (a) 104 , (b) 105 , (c) 106 , (d) 107 . Fixed parameters values: γ = 0.1, α = 0.01,… view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of the oscillatory behavior of the normalized infected view at source ↗
Figure 8
Figure 8. Figure 8: Comparison between the SIRS Model with and without vaccination view at source ↗
Figure 10
Figure 10. Figure 10: Graphic of ∆Itotal between the case with and without vaccination for the following parameters: γ = 0.1, δ = 0.005, α = 0.01 and number of vaccines = 1. The light gray region corresponds to simulations where the integral differ￾ence is positive and the extinction probability is greater than 0.9. 3. Summary and conclusions In this paper we discuss the SIRS Model with vaccination using ODEs and propose a mod… view at source ↗
Figure 9
Figure 9. Figure 9: Graphic of ∆Itotal between the case with and without vaccination for the following parameters: γ = 0.1, δ = 0.005, R0 = 1.74, α = 0.01 and number of vaccines = 1. The same behavior happens for different R0 values, where for all the possible values for t0 the value of the integral difference defined in Eq. (13) is positive, which means that the vaccina￾tion was beneficial for the population even though it h… view at source ↗
read the original abstract

Epidemics have shaped human history, often with devastating consequences, motivating the development of mathematical models to understand and control their dynamics. Among the many aspects of epidemic behavior, the conditions that lead to epidemic extinction stand out as a central-if not the fundamental-question in epidemic modeling. In this work, we study epidemic extinction in a continuous SIRS (Susceptible-Infected-Recovered-Susceptible) model governed by a system of ordinary differential equations (ODEs). The model includes vaccination as a time-dependent process and considers the reinfection of recovered individuals through waning immunity. We analyze how different parameter regimes -- particularly infection, recovery, and immunity loss rates -- affect the persistence or extinction of the epidemic. Special attention is given to the limitations of continuous population models, in which the infected fraction can fall below the equivalent of a single individual, leading to nonphysical outcomes such as unrealistically long persistence or artificial secondary peaks. By comparing the continuous SIRS dynamics with expected real-world thresholds for extinction, we highlight the importance of incorporating stochasticity or discrete effects to accurately describe epidemic fade-out.

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

0 major / 3 minor

Summary. The manuscript studies epidemic extinction in a continuous SIRS model with time-dependent vaccination and waning immunity, governed by a system of ODEs. It examines the effects of infection, recovery, and immunity-loss rates on persistence versus extinction, and stresses that deterministic models permit non-physical persistence once the infected fraction drops below the equivalent of one individual. The authors compare these dynamics to real-world extinction thresholds and conclude that stochastic or discrete effects must be incorporated for realistic descriptions of epidemic fade-out.

Significance. The central observation—that mean-field ODEs break down for true extinction once I < 1/N—is a standard and load-bearing limitation of deterministic epidemic models. If the parameter-regime analysis supplies concrete, falsifiable comparisons to real-world thresholds or reproducible numerical examples, the work could usefully illustrate this point for modelers working on vaccination and fade-out scenarios. Absent new derivations or quantitative predictions, the contribution largely restates a known modeling caveat.

minor comments (3)
  1. The abstract states that parameter regimes are analyzed and compared to real-world thresholds, yet supplies neither the governing ODEs, specific numerical results, nor error analysis. Adding the model equations (e.g., as Eq. (1)) and at least one table or figure of extinction times versus parameters would make the claimed analysis verifiable.
  2. The phrase 'expected real-world thresholds for extinction' is used without citation or quantitative definition. A brief reference to classic stochastic-extinction results (e.g., branching-process approximations or Gillespie simulations) would anchor the comparison.
  3. The discussion of 'artificial secondary peaks' arising from non-physical persistence is mentioned but not illustrated. A single time-series plot contrasting the deterministic trajectory with a stochastic realization near the 1/N threshold would clarify the practical consequence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review and for recommending minor revision. We appreciate the recognition of the importance of deterministic model limitations in epidemic fade-out and address the substantive points below.

read point-by-point responses

  1. Referee: The central observation—that mean-field ODEs break down for true extinction once I < 1/N—is a standard and load-bearing limitation of deterministic epidemic models. If the parameter-regime analysis supplies concrete, falsifiable comparisons to real-world thresholds or reproducible numerical examples, the work could usefully illustrate this point for modelers working on vaccination and fade-out scenarios. Absent new derivations or quantitative predictions, the contribution largely restates a known modeling caveat.

    Authors: We agree that the breakdown of mean-field ODEs below the single-individual threshold is a standard limitation. Our manuscript contributes by providing a systematic parameter-regime analysis in the specific setting of a continuous SIRS model with time-dependent vaccination and waning immunity. We present reproducible numerical simulations across ranges of infection, recovery, and immunity-loss rates that explicitly demonstrate non-physical persistence, unrealistically long epidemic duration, and artificial secondary peaks once the infected fraction falls below the equivalent of one individual. We further compare these dynamics directly to real-world extinction thresholds, yielding concrete, falsifiable examples that illustrate when and how stochastic or discrete effects become essential for vaccination-related fade-out scenarios. While we do not derive new mathematical forms of the limitation, the focused application, parameter studies, and explicit real-world comparisons supply the quantitative illustrations the referee notes would make the work useful. revision: no

Circularity Check

0 steps flagged

No significant circularity; standard mean-field analysis

full rationale

The paper sets up a conventional SIRS ODE system with time-dependent vaccination and waning immunity, then numerically integrates the deterministic equations across parameter regimes. Extinction is discussed solely by contrasting the continuous model's non-physical persistence (I(t) > 0 for all t even when I << 1/N) against the known discrete threshold of one individual; this contrast is a textbook property of mean-field approximations and is not obtained by fitting any parameter to the target outcome or by self-referential definition. No uniqueness theorems, ansatzes, or predictions are introduced that reduce to the model's own inputs by construction, and the manuscript contains no self-citations that bear load on the central claim.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that standard ODE-based SIRS dynamics with added vaccination and waning immunity can be meaningfully compared to real-world extinction thresholds, despite the paper noting the model's breakdown at small populations.

axioms (1)
  • domain assumption Epidemic dynamics are governed by a system of ordinary differential equations for S, I, R compartments with time-dependent vaccination.
    Standard compartmental modeling approach invoked in the abstract.

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Reference graph

Works this paper leans on

31 extracted references · 21 canonical work pages

  1. [1]

    Pulse mass measles vaccina- tion across age cohorts

    Agur, Z., Cojocaru, L., Mazor, G., Anderson, R.M., Danon, Y .L., 1993. Pulse mass measles vaccina- tion across age cohorts. Proceedings of the Na- tional Academy of Sciences 90, 11698–11702. URL: https://www.pnas.org/doi/10.1073/pnas.90.24.11698, doi:10.1073/pnas.90.24.11698

    work page doi:10.1073/pnas.90.24.11698 1993

  2. [2]

    Modeling the spa- tiotemporal epidemic spreading of covid-19 and the impact of mobility and social distancing in- terventions

    Arenas, A., Cota, W., Gómez-Gardeñes, J., Gómez, S., Granell, C., Matamalas, J.T., Soriano-Paños, D., Steinegger, B., 2020. Modeling the spa- tiotemporal epidemic spreading of covid-19 and the impact of mobility and social distancing in- terventions. Phys. Rev. X 10, 041055. URL: https://link.aps.org/doi/10.1103/PhysRevX.10.041055, doi:10.1103/PhysRevX.10.041055

    work page doi:10.1103/physrevx.10.041055 2020

  3. [3]

    Measles periodicity and community size

    Bartlett, M.S., 1957. Measles periodicity and community size. Journal of the Royal Statistical Society. Series A (General) 120, 48–70. doi:10.2307/2342553

    work page doi:10.2307/2342553 1957

  4. [4]

    A discrete-time susceptible-infectious-recovered- susceptible model for the analysis of influenza data

    Bucyibaruta, G., Dean, C.B., Torabi, M., 2023. A discrete-time susceptible-infectious-recovered- susceptible model for the analysis of influenza data. Infectious Disease Modelling 8, 471–483

    2023

  5. [5]

    Rebound in epidemic control: how misaligned vaccination timing amplifies infec- tion peaks

    Castioni, P., Gómez, S., Granell, C., Arenas, A., 2024. Rebound in epidemic control: how misaligned vaccination timing amplifies infec- tion peaks. npj Complexity 1, 20. URL: https://doi.org/10.1038/s44260-024-00020-0, doi:10.1038/s44260-024-00020-0

    work page doi:10.1038/s44260-024-00020-0 2024

  6. [6]

    Modelling of optimal timing for influenza vaccination as a function of intraseasonal waning of immunity and vaccine coverage

    Costantino, V ., Trent, M., MacIntyre, C.R., 2019. Modelling of optimal timing for influenza vaccination as a function of intraseasonal waning of immunity and vaccine coverage. Vaccine 37, 6768–6775. URL: https://doi.org/10.1016/j.vaccine.2019.08.069, doi:10.1016/j.vaccine.2019.08.069

    work page doi:10.1016/j.vaccine.2019.08.069 2019

  7. [7]

    Optimal timing of one-shot interventions for epidemic con- trol

    Di Lauro, F., Kiss, I.Z., Miller, J.C., 2021. Optimal timing of one-shot interventions for epidemic con- trol. PLOS Computational Biology 17, 1–25. URL: https://doi.org/10.1371/journal.pcbi.1008763, doi:10.1371/journal.pcbi.1008763

    work page doi:10.1371/journal.pcbi.1008763 2021

  8. [8]

    Sirs epi- demics with individual heterogeneity of immunity wan- ing

    El Khalifi, M., Britton, T., 2024. Sirs epi- demics with individual heterogeneity of immunity wan- ing. Journal of Theoretical Biology 605, 111815. doi:10.1016/j.jtbi.2024.111815

    work page doi:10.1016/j.jtbi.2024.111815 2024

  9. [9]

    Evolution of antibody immunity to SARS-CoV-2

    Gaebler, C., Wang, Z., Lorenzi, J.C.C., Muecksch, F., Finkin, S., Tokuyama, M., Cho, A., Jankovic, M., Schaefer-Babajew, D., Oliveira, T.Y ., Cipolla, M., Viant, C., Barnes, C.O., Bram, Y ., Breton, G., Hägglöf, T., Mendoza, P., Hurley, A., Turroja, M., Gordon, K., others, Nussenzweig, M.C., 2021. Evolution of antibody immunity to SARS-CoV-2. Nature 591, ...

    work page doi:10.1038/s41586-021-03207-w 2021

  10. [10]

    Digital Image Pro- cessing

    Gonzalez, R.C., Woods, R.E., 2018. Digital Image Pro- cessing. 4th ed., Pearson, New York, NY

    2018

  11. [11]

    Os- cillations in sirs model with distributed delays

    Gonçalves, S., Abramson, G., Gomes, M.F.C., 2011. Os- cillations in sirs model with distributed delays. The Euro- pean Physical Journal B: Condensed Matter and Complex Systems 81, 363–371. 8

    2011

  12. [12]

    The mathematics of infectious dis- eases

    Hethcote, H.W., 2000. The mathematics of infectious dis- eases. SIAM review 42, 599–653

    2000

  13. [13]

    Mitigation strategies for pandemic influenza a: Balancing conflicting policy objectives

    Hollingsworth, T.D., Klinkenberg, D., Heesterbeek, H., Anderson, R.M., 2011. Mitigation strategies for pandemic influenza a: Balancing conflicting policy objectives. PLOS Computational Biology 7, 1–11. URL: https://doi.org/10.1371/journal.pcbi.1001076, doi:10.1371/journal.pcbi.1001076

    work page doi:10.1371/journal.pcbi.1001076 2011

  14. [14]

    Major in- fectious diseases: Key messages from disease control pri- orities, third edition, in: Major Infectious Diseases

    Holmes, K.K., Bertozzi, S., Bloom, B.R., Jha, P., Gel- band, H., DeMaria, L.M., Horton, S., 2017. Major in- fectious diseases: Key messages from disease control pri- orities, third edition, in: Major Infectious Diseases. The International Bank for Reconstruction and Development/ The World Bank

    2017

  15. [15]

    Modeling Infectious Diseases in Humans and Animals

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

    2008

  16. [16]

    2020 , bdsk-url-1 =

    Maier, B.F., Brockmann, D., 2020. Effective containment explains subexponential growth in recent confirmed covid-19 cases in china. Science 368, 742–746. URL: https://www.science.org/doi/abs/10.1126/science.abb4557, doi:10.1126/science.abb4557

    work page doi:10.1126/science.abb4557 2020

  17. [17]

    Potential benefits of de- laying the second mrna covid-19 vaccine dose

    Maier, B.F., Burdinski, A., Rose, A.H., Schlosser, F., Hinrichs, D., Betsch, C., Korn, L., Sprengholz, P., Meyer-Hermann, M., Mitra, T., Lauterbach, K., Brockmann, D., 2021. Potential benefits of de- laying the second mrna covid-19 vaccine dose. URL:https://arxiv.org/abs/2102.13600, doi:10.48550/arXiv.2102.13600,arXiv:2102.13600

    work page doi:10.48550/arxiv.2102.13600 2021

  18. [18]

    The evolution of SARS-CoV-2

    Markov, P.V ., Ghafari, M., Beer, M., Lythgoe, K., Simmonds, P., Stilianakis, N.I., Katzourakis, A., 2023. The evolution of SARS-CoV-2. Na- ture Reviews Microbiology 21, 361–379. URL: https://doi.org/10.1038/s41579-023-00878-2, doi:10.1038/s41579-023-00878-2

    work page doi:10.1038/s41579-023-00878-2 2023

  19. [19]

    Vaccination and non-pharmaceutical interventions for COVID-19: a mathematical modelling study

    Moore, S., Hill, E.M., Tildesley, M.J., Dyson, L., Keel- ing, M.J., 2021. Vaccination and non-pharmaceutical interventions for COVID-19: a mathematical modelling study. The Lancet Infectious Diseases 21, 793–802. URL: https://doi.org/10.1016/S1473-3099(21)00143-2, doi:10.1016/S1473-3099(21)00143-2

    work page doi:10.1016/s1473-3099(21)00143-2 2021

  20. [20]

    Mathematical Biology

    Murray, J.D., 1993. Mathematical Biology. Springer Berlin Heidelberg, Berlin, Heidelberg. URL: http://link.springer.com/10.1007/978-3-662-08542-4, doi:10.1007/978-3-662-08542-4

    work page doi:10.1007/978-3-662-08542-4 1993

  21. [21]

    Vaccina- tion campaigns and epidemic rebounds

    Pablo Silvestrin, Germano Brill, S.G., 2025. Vaccina- tion campaigns and epidemic rebounds. Trabalho de Con- clusão de Curso IF-UFRGS and private communication

    2025

  22. [22]

    Vacci- nation based control of infections in sirs models with rein- fection: special reference to pertussis

    Safan, M., Kretzschmar, M., Hadeler, K.P., 2012. Vacci- nation based control of infections in sirs models with rein- fection: special reference to pertussis. Journal of Mathe- matical Biology 240, 205–214. doi:10.1007/s00285-012- 0582-1

    work page doi:10.1007/s00285-012- 2012

  23. [23]

    Evasion of innate and adaptive immune responses by influenza a virus

    Schmolke, M., García-Sastre, A., 2010. Evasion of innate and adaptive immune responses by influenza a virus. Cellular Microbiology 12, 873–880. URL: https://doi.org/10.1111/j.1462-5822.2010.01475.x, doi:10.1111/j.1462-5822.2010.01475.x

    work page doi:10.1111/j.1462-5822.2010.01475.x 2010

  24. [24]

    Cor- relation coefficients: Appropriate use and interpre- tation

    Schober, P., Boer, C., Schwarte, L.A., 2018. Cor- relation coefficients: Appropriate use and interpre- tation. Anesthesia & Analgesia 126, 1763–1768. doi:10.1213/ANE.0000000000002864

    work page doi:10.1213/ane.0000000000002864 2018

  25. [25]

    Optimizing infectious disease interventions during an emerging epidemic

    Wallinga, J., van Boven, M., Lipsitch, M., 2010. Optimizing infectious disease interventions during an emerging epidemic. Proceedings of the Na- tional Academy of Sciences 107, 923–928. URL: https://www.pnas.org/doi/10.1073/pnas.0908491107, doi:10.1073/pnas.0908491107

    work page doi:10.1073/pnas.0908491107 2010

  26. [26]

    Studying the impacts of variant evolution for a generalized age-group transmission model

    Wei, F., Zhou, R., Jin, Z., Sun, Y ., Peng, Z., et al., 2024. Studying the impacts of variant evolution for a generalized age-group transmission model. PLOS ONE 19, e0306554. doi:10.1371/journal.pone.0306554

    work page doi:10.1371/journal.pone.0306554 2024

  27. [27]

    Modeling covid-19 dynamics in illinois under nonpharmaceuti- cal interventions

    Wong, G.N., Weiner, Z.J., Tkachenko, A.V ., Elbanna, A., Maslov, S., Goldenfeld, N., 2020. Modeling covid-19 dynamics in illinois under nonpharmaceuti- cal interventions. Phys. Rev. X 10, 041033. URL: https://link.aps.org/doi/10.1103/PhysRevX.10.041033, doi:10.1103/PhysRevX.10.041033

    work page doi:10.1103/physrevx.10.041033 2020

  28. [28]

    Who coro- navirus (covid-19) dashboard: deaths

    World Health Organization, 2026. Who coro- navirus (covid-19) dashboard: deaths. URL: https://data.who.int/dashboards/covid19/deaths

    2026

  29. [29]

    Covid- 19 coronavirus pandemic

    Worldometer, 2025. Covid- 19 coronavirus pandemic. URL: https://www.worldometers.info/coronavirus/

    2025

  30. [30]

    You, C., Deng, Y ., Hu, W., Sun, J., Lin, Q., Zhou, F., Pang, C.H., Zhang, Y ., Chen, Z., Zhou, X.H.,

  31. [31]

    International Jour- nal of Hygiene and Environmental Health 228, 113555

    Estimation of the time-varying reproduction num- ber of covid-19 outbreak in china. International Jour- nal of Hygiene and Environmental Health 228, 113555. doi:10.1016/j.ijheh.2020.113555. 9

    work page doi:10.1016/j.ijheh.2020.113555 2020