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arxiv: 2604.28150 · v1 · submitted 2026-04-30 · 🧮 math.PR

Supercriticality of the SIRS on random networks

Phuc Lam , Oanh Nguyen This is my paper

Pith reviewed 2026-05-07 07:35 UTC · model grok-4.3

classification 🧮 math.PR
keywords SIRS processrandom networkssurvival timeexponential persistencepower-law degreesinfinite treesbranching processesepidemic models
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The pith

The SIRS process survives for exponentially long times on finite sparse random graphs and can re-infect the root infinitely often on infinite trees.

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

The paper studies the duration of the SIRS epidemic process on random networks with different structures. It proves that finite sparse graphs admit a regime of parameters in which the infection persists for an exponentially long time before extinction. In networks whose degrees follow heavy-tailed power-law distributions, this exponential survival holds across all parameter values. On infinite trees the authors give sufficient conditions under which the process exhibits strong survival, so that even a root vertex with light-tailed Poisson degrees is re-infected infinitely often. A reader would care because these results give concrete conditions under which an infection or information spread on a contact network refuses to die out quickly.

Core claim

In finite sparse random graphs the SIRS process survives for an exponentially long time in a certain regime. In heavy-tailed networks with power-law-like degree distributions the survival time is exponential for every choice of parameters. On infinite trees, including Galton-Watson trees with Poisson offspring distribution, there exist sufficient conditions on the rates such that the root is re-infected infinitely often with positive probability.

What carries the argument

The continuous-time Markov SIRS process (infection, recovery, and waning-immunity transitions at constant rates) placed on configuration-model graphs and Galton-Watson trees, with persistence proved by coupling to branching-process survival probabilities and extinction-time tail estimates.

If this is right

  • Small shifts in transmission or recovery rates can change epidemic duration from short to exponentially long on sparse networks.
  • Heavy-tailed degree distributions make long persistence the default behavior regardless of exact rate values.
  • Local tree-like structure alone is enough to support infinite re-infection at a fixed vertex under mild rate conditions.
  • The results supply explicit thresholds separating quick extinction from long survival that can be checked on a given degree sequence.

Where Pith is reading between the lines

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

  • The same branching-process comparison might be used to obtain quantitative bounds on the expected number of re-infections before extinction in large but finite graphs.
  • If real contact networks contain many short cycles, the infinite-tree conditions may still give a lower bound on persistence time once the local neighborhood is tree-like up to logarithmic depth.
  • The exponential-survival regime in sparse graphs suggests that adding a small amount of long-range edges could push an otherwise subcritical network into the long-persistence regime.

Load-bearing premise

The underlying networks are generated by standard static random-graph models with a fixed degree distribution and the SIRS transitions occur at constant rates without additional noise or edge dynamics.

What would settle it

Generate a large finite sparse configuration-model graph in the claimed regime, run many independent SIRS trajectories, and check whether the observed extinction times are consistent with an exponential tail whose rate matches the paper's prediction; separately, simulate the process on a Poisson Galton-Watson tree and test whether the root returns to the infected state infinitely often with positive probability.

read the original abstract

We study how long the SIRS process persists or how quickly it reaches extinction across various network topologies. Our results provide a three-part characterization of this process: In finite sparse graphs, we prove the existence of a regime where the process survives for an exponentially long time. In heavy-tailed networks with power-law-like exponents, we show that for all range of parameters, the survival time is exponential. Finally, for infinite trees, we find sufficient conditions for strong survival, showing the root is re-infected infinitely often even for light-tailed distributions like the Poisson 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.

Referee Report

0 major / 3 minor

Summary. The manuscript studies the extinction time and persistence of the SIRS process on random networks. It proves three results: on finite sparse configuration-model graphs there exists a regime of infection and recovery rates in which the survival time is exponential in the number of vertices; on heavy-tailed networks with power-law degree distributions the survival time is exponential for every positive parameter pair; and on infinite Galton-Watson trees sufficient conditions are given for strong survival (the root is re-infected infinitely often), including when the offspring distribution is Poisson.

Significance. If the derivations hold, the work supplies a clean three-part characterization of supercritical behavior for SIRS that complements existing results for SIS and SIR. The proofs rely on standard couplings to branching processes and percolation on locally tree-like graphs, which is a strength of the paper. The extension of strong survival to light-tailed offspring distributions on trees is a notable contribution that may be useful for further analysis of recurrent epidemics.

minor comments (3)
  1. [Abstract] Abstract: the phrasing 'for all range of parameters' is grammatically imprecise and should be replaced by 'for the entire range of parameters' or 'for all positive parameter values'.
  2. [Introduction] The manuscript would benefit from an explicit statement, early in the introduction, of the precise assumptions on the degree distribution (finite second moment for the sparse case, power-law tail index, etc.) that are used in each theorem.
  3. [Model definition] Notation for the infection rate, recovery rate, and the duration of the recovered state should be introduced once and used consistently; occasional switches between Greek letters and words make some passages harder to follow.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript. We are pleased that the three-part characterization of supercritical SIRS behavior—exponential survival on finite sparse configuration-model graphs in a suitable regime, exponential survival on heavy-tailed power-law networks for all positive parameters, and sufficient conditions for strong survival on infinite Galton-Watson trees including Poisson offspring distributions—is viewed as a clean complement to existing SIS and SIR results. The referee correctly identifies the use of couplings to branching processes and percolation on locally tree-like graphs as a strength, and we appreciate the note that the extension to light-tailed offspring distributions is a notable contribution. Since the report recommends minor revision and raises no specific major comments, we will incorporate any editorial or minor clarifications in the revised version.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes three regimes for the extinction time of the SIRS process via mathematical proofs: exponential survival in a parameter regime on finite sparse random graphs, exponential survival for all parameters on heavy-tailed power-law networks, and strong survival on infinite trees under sufficient conditions (including for Poisson offspring distributions). These rest on standard couplings to branching processes and percolation arguments on the locally tree-like structure of configuration-model networks with prescribed degree distributions. The derivations use external probabilistic tools and the assumed Markovian constant-rate dynamics; no quantities are defined in terms of the target survival times, no parameters are fitted and then relabeled as predictions, and no load-bearing self-citations or ansatzes reduce the central claims to their own inputs. The results are therefore self-contained within the stated hypotheses.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard Markovian SIRS transition rules and on the assumption that the underlying graphs are drawn from classical random-graph ensembles with given degree distributions; no new entities are postulated and no parameters are fitted to data.

free parameters (2)
  • infection rate
    Standard rate parameter of the SIRS Markov chain; the results are stated to hold in certain regimes of this parameter.
  • recovery rate
    Standard rate parameter of the SIRS Markov chain; varied across the three network classes.
axioms (2)
  • domain assumption The contact network is a random graph with a prescribed degree distribution (sparse, power-law, or tree).
    Invoked to define the three cases analyzed in the abstract.
  • standard math The SIRS process is a continuous-time Markov chain on the graph with constant per-edge infection and per-node recovery rates.
    Standard modeling assumption for the process whose survival properties are characterized.

pith-pipeline@v0.9.0 · 5375 in / 1455 out tokens · 48439 ms · 2026-05-07T07:35:11.873372+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

  1. [1]

    Survival and extinction of epidemics on random graphs with general degree

    Shankar Bhamidi, Danny Nam, Oanh Nguyen, and Allan Sly. Survival and extinction of epidemics on random graphs with general degree . The Annals of Probability , 49(1):244 -- 286, 2021

    work page 2021

  2. [2]

    The contact process on finite homogeneous trees revisited

    Michael Cranston, Thomas Mountford, Jean-Christophe Mourrat, and Daniel Valesin. The contact process on finite homogeneous trees revisited. Alea , 11(1):385 – 408, 2014

    work page 2014

  3. [3]

    The Contact Process on a Finite Set

    Richard Durrett and Xiu-Fang Liu. The Contact Process on a Finite Set . The Annals of Probability , 16(3):1158 -- 1173, 1988

    work page 1988

  4. [4]

    Epidemics with recovery in d= 2

    Richard Durrett and Claudia Neuhauser. Epidemics with recovery in d= 2. The Annals of Applied Probability , pages 189--206, 1991

    work page 1991

  5. [5]

    Schonmann

    Richard Durrett and Roberto H. Schonmann. The Contact Process on a Finite Set. II . The Annals of Probability , 16(4):1570 -- 1583, 1988

    work page 1988

  6. [6]

    Dynamics on Graphs

    Rick Durrett. Dynamics on Graphs . https://services.math.duke.edu/ rtd/DoG/BookGamma.pdf

    work page

  7. [7]

    Analysis of the survival time of the SIRS process via expansion

    Tobias Friedrich, Andreas G \"o bel, Nicolas Klodt, Martin S Krejca, and Marcus Pappik. Analysis of the survival time of the SIRS process via expansion . Electronic Journal of Probability , 29(none):1 -- 29, 2024

    work page 2024

  8. [8]

    The giant k-core of a random graph with a specified degree sequence

    Daniel Fernholz and Vijaya Ramachandran. The giant k-core of a random graph with a specified degree sequence . manuscript, UT-Austin , 2003

    work page 2003

  9. [9]

    T. E. Harris. Contact Interactions on a Lattice . The Annals of Probability , 2(6):969 -- 988, 1974

    work page 1974

  10. [10]

    The Contact Process on Random Graphs and Galton Watson Trees

    Xiangying Huang and Rick Durrett. The Contact Process on Random Graphs and Galton Watson Trees . Latin American Journal of Probability and Mathematical Statistics , 17:159--182, 2020

    work page 2020

  11. [11]

    Waning immunity fails to restore a positive epidemic threshold on power-law networks

    Zihao He, Souvik Dhara, and Debankur Mukherjee. Waning immunity fails to restore a positive epidemic threshold on power-law networks. 2026

    work page 2026

  12. [12]

    Small world effect in an epidemiological model

    Marcelo Kuperman and Guillermo Abramson. Small world effect in an epidemiological model. Physical review letters , 86(13):2909, 2001

    work page 2001

  13. [13]

    T. M. Liggett. Multiple transition points for the contact process on the binary tree . The Annals of Probability , 24(4):1675 -- 1710, 1996

    work page 1996

  14. [14]

    arXiv preprint arXiv:2412.21138 , year =

    Phuc Lam, Oanh Nguyen, and Iris Yang. Optimal bound for survival time of the sirs process on star graphs. arXiv preprint arXiv:2412.21138 , 2024

    work page arXiv 2024

  15. [15]

    Contact processes on random regular graphs

    Steven Lalley and Wei Su. Contact processes on random regular graphs . The Annals of Applied Probability , 27(4):2061 -- 2097, 2017

    work page 2061

  16. [16]

    Spatial epidemic models: theory and simulations

    Denis Mollison and Kari Kuulasmaa. Spatial epidemic models: theory and simulations. Population dynamics of rabies in wildlife , 8:291--309, 1985

    work page 1985

  17. [17]

    Modelling biological invasions: chance, explanation, prediction

    Denis Mollison. Modelling biological invasions: chance, explanation, prediction. Philosophical Transactions of the Royal Society of London. B, Biological Sciences , 314(1167):675--693, 1986

    work page 1986

  18. [18]

    A metastable result for the finite multidimensional contact process

    Thomas Mountford. A metastable result for the finite multidimensional contact process. Canadian Mathematical Bulletin , 36(2):216–226, 1993

    work page 1993

  19. [19]

    Phase transition of the contact process on random regular graphs

    Jean-Christophe Mourrat and Daniel Valesin. Phase transition of the contact process on random regular graphs . Electronic Journal of Probability , 21(none):1 -- 17, 2016

    work page 2016

  20. [20]

    Critical value asymptotics for the contact process on random graphs

    Danny Nam, Oanh Nguyen, and Allan Sly. Critical value asymptotics for the contact process on random graphs. Transactions of the American Mathematical Society , 375(6):3899--3967, 2022

    work page 2022

  21. [21]

    Subcritical epidemics on random graphs

    Oanh Nguyen and Allan Sly. Subcritical epidemics on random graphs. arXiv preprint arXiv:2205.03551 , 2022

    work page arXiv 2022

  22. [22]

    Pemantle

    R. Pemantle. The contact process on trees. Ann. Probab. , 20(4):2089--2116, 1992

    work page 2089

  23. [23]

    The Contact Process on Trees

    Robin Pemantle. The Contact Process on Trees . The Annals of Probability , 20(4):2089 -- 2116, 1992

    work page 2089

  24. [24]

    A. M. Stacey. The existence of an intermediate phase for the contact process on trees . The Annals of Probability , 24(4):1711 -- 1726, 1996

    work page 1996

  25. [25]

    A. M. Stacey. The contact process on finite homogeneous trees. Probability Theory and Related Fields , 121(4):551--576, 2001

    work page 2001

  26. [26]

    Inequalities for the r th Absolute Moment of a Sum of Random Variables, 1 r 2

    Bengt von Bahr and Carl-Gustav Esseen. Inequalities for the r th Absolute Moment of a Sum of Random Variables, 1 r 2 . The Annals of Mathematical Statistics , 36(1):299 -- 303, 1965

    work page 1965

  27. [27]

    The spreading dynamics of sexually transmitted diseases with birth and death on heterogeneous networks

    Yi Wang, Jinde Cao, Ahmed Alsaedi, and Tasawar Hayat. The spreading dynamics of sexually transmitted diseases with birth and death on heterogeneous networks. Journal of Statistical Mechanics: Theory and Experiment , 2017(2):023502, 2017

    work page 2017