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REVIEW 4 major objections 4 minor 54 references

Extreme outbreaks in non-Markovian epidemics on complex networks

T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A single per-edge transmission probability governs the full outbreak-size distribution of non-Markovian SIR epidemics on networks, including rare large outbreaks.

desk verdict A useful empirical mapping for non-Markovian outbreak tails, but the 'single transmissibility' equivalence is oversold; the paper's own simulations only show approximate collapse. read the letter →

arxiv 2607.24615 v1 pith:C23RHRAL submitted 2026-07-27 physics.soc-ph cond-mat.stat-mechphysics.comp-phq-bio.PE

Ami Taitelbaum , Michael Assaf This is my paper · ORCID
classification physics.soc-phcond-mat.stat-mechphysics.comp-phq-bio.PE
keywords non-Markovianepidemicsoutbreak-sizedistributionextremeoutbreakriskedgetransmissibilitybondpercolationlargedeviationsWKBcomplexnetworkswaiting-timedistributions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Extreme epidemic risk lives in the right tail of the outbreak-size distribution, but that tail is hard to compute when infection and recovery are not memoryless. The paper's central claim is that for SIR dynamics on networks, arbitrary infection and recovery waiting-time distributions can be folded into a single per-edge transmissibility $T$, which turns the non-Markovian process into an effective Markovian one with the same full outbreak-size statistics. On weakly heterogeneous networks this gives a universal well-mixed semiclassical curve parameterized by the effective reproduction number $R_0^{\mathrm{eff}} = -\ln(1-x_r^*)/x_r^*$; on strongly heterogeneous and empirical networks it gives Markovian dynamics on the same network at $R_0^{\mathrm{net}} = T/(1-T)\,\lambda_c^{-1}$. If the claim is right, a measured generation interval and contact pattern are enough to predict the probability of an unusually large outbreak, without simulating the full non-Markovian dynamics.

What carries the argument

The load-bearing object is the per-edge transmissibility $T = \int_0^\infty f_i(t)\Phi_r(t)\,dt$, the probability that infection along one susceptible–infected edge fires before recovery. For gamma-distributed infection and recovery it takes the closed form $T = I_z(\alpha_\mathrm{inf}, \alpha_\mathrm{rec})$ with $z = \alpha_\mathrm{inf}\beta/(\alpha_\mathrm{inf}\beta + \alpha_\mathrm{rec}\gamma)$. This one number converts the final state into a bond-percolation configuration, and the mapping then works in two directions: invert the percolation final size $x_r^*$ to get $R_0^{\mathrm{eff}}$ for the well-mixed WKB large-deviation action, or invert the Markovian relation to get $R_0^{\mathrm{net}}$ for Markovian dynamics on the same network. The WKB action and its Gaussian width, taken from the well-mixed theory, supply the shape of the universal curve.

What would settle it

Run stochastic SIR simulations on the same weakly heterogeneous network with two different waiting-time distributions—for example gamma with $\alpha_\mathrm{inf}<1$ and a log-normal distribution—tuned so that Eq. (2) gives the same $T$ and hence the same deterministic final size $x_r^*$; if the normalized outbreak-size distributions do not collapse onto the WKB curve at $R_0^{\mathrm{eff}}$ beyond finite-size fluctuations, the universal-collapse claim is falsified.

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Extended reading notes

Core claim

On its own terms, the paper demonstrates that the final state of a non-Markovian SIR epidemic is a bond-percolation configuration with per-edge occupation probability $T$, and that this single number controls not just the mean outbreak fraction but the whole distribution of outcomes. For weakly heterogeneous degree distributions, inverting the deterministic final-size relation gives $R_0^{\mathrm{eff}}$, and the rescaled outbreak-size distributions measured across regular, Erdős–Rényi, and gamma-distributed networks with gamma and log-normal waiting times collapse onto the well-mixed WKB prediction. For strongly heterogeneous or correlated networks, the same transmissibility defines $R_0^{\mathrm{net}}$, and Markovian simulations on the actual network reproduce the non-Markovian conditional mean, standard deviation, and full distribution, including on the empirical Hamsterster network with assortative mixing. The qualitative asymmetry between infection- and recovery-time shapes follows directly from $T$: broader-than-exponential infections enlarge $T$ and drive larger outbreaks, while broader-than-exponential recoveries shrink it.

Load-bearing premise

For weakly heterogeneous networks (degree scatter up to about half the mean degree), the paper assumes that the whole rescaled outbreak-size distribution is fixed by one effective reproduction number $R_0^{\mathrm{eff}}$ and is insensitive to the detailed shape of the waiting-time distributions and the degree distribution; this is supported by simulation collapse rather than derived from the underlying dynamics.

Editorial extensions

If this is right

  • Given measured generation intervals and infectious-period distributions, practitioners can compute $T$ and then $R_0^{\mathrm{eff}}$ or $R_0^{\mathrm{net}}$ to obtain the full outbreak-size distribution without simulating non-Markovian network dynamics.
  • On weakly heterogeneous networks the right tail is analytic: the well-mixed WKB action gives the probability of an outbreak of any fractional size, including events far above the mean.
  • On strongly heterogeneous or correlated networks, the equivalence reduces the problem to Markovian simulation on the existing network, which is substantially cheaper than simulating arbitrary waiting-time clocks.
  • Changing the shape of the infection-time distribution has opposite and stronger effects than changing the recovery-time distribution, and both are captured quantitatively by the transmissibility formula.
  • Extreme-outbreak risk quantified this way is sizable: in the paper's examples, about one-fifth of the probability mass of extensive outbreaks lies more than 20% above the mean, on both an Erdős–Rényi and the empirical Hamsterster network.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Inference: the reduction predicts that any two waiting-time distributions producing the same transmissibility $T$ and the same deterministic final size should yield identical normalized outbreak-size distributions; running this matched-transmissibility comparison for Weibull versus gamma or log-normal timings is a direct test the paper does not perform.
  • Inference: the breakdown of the well-mixed collapse around degree coefficient of variation 0.5 suggests a crossover regime rather than a sharp threshold; a natural extension would be to add a degree-dispersion correction to the WKB action and check whether it interpolates between the two regimes.
  • Inference: for real-time epidemic risk assessment, the mapping implies that empirically measured contact patterns and inter-event-time distributions can be converted directly into tail-risk probabilities, such as the probability of exceeding hospital capacity, which is the practical payoff the paper points to without developing a forecasting procedure.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes a mapping from non-Markovian SIR dynamics on networks to an effective Markovian description, parameterized by a single per-edge transmissibility T defined in Eq. (2). The authors claim that this mapping reproduces the full outbreak-size distribution: for weakly heterogeneous networks, the distribution collapses onto a universal well-mixed WKB curve governed by R0^eff from Eq. (8), and for highly heterogeneous or empirical networks, Markovian simulations on the same network at R_net^0 from Eq. (9) capture the distribution. The claims are supported by extensive simulations on regular, Erdős–Rényi, gamma-distributed, and Hamsterster networks with gamma and log-normal waiting-time distributions.

Significance. If the central equivalence held, the paper would provide a practical and valuable route from measured waiting-time distributions to extreme-outbreak risk on networks. The manuscript contains useful ingredients: an explicit transmissibility formula for gamma-distributed infection and recovery, a generalized bond-percolation treatment for degree-correlated networks, and an unusually broad simulation campaign. The paper is also clearly written and carefully describes its simulation methodology. However, the central claim that a single edge transmissibility determines the full outbreak-size distribution is not correct in general, and the empirical evidence for the universal collapse is partly circular. Because the advertised result is the full-distribution equivalence, the error is load-bearing.

major comments (4)
  1. [Eq. (2) and the paragraph following Eq. (9)] The transmissibility T is only the marginal probability that a given S–I edge transmits before recovery. The joint law of transmissions from a single infected node is not determined by T. If an infected node has recovery time ρ and k susceptible neighbors, the number of transmissions has generating function E_ρ[(1 - F_i(ρ) + F_i(ρ) s)^k], which depends on the whole infection CDF F_i, not only on T = E[F_i(ρ)]. A concrete counterexample is a star with a single infected center and k leaves: the final outbreak size is exactly this number of transmissions, so two infection waiting-time distributions with identical T but different F_i give different final-size distributions. Equation (9) constructs an effective Markovian process with the same T, which predicts a Binomial(k, T) offspring distribution; this cannot reproduce both non-Markovian distributions. Thus the abstract claim that arbitrary waiting-time statistics can be incorporated through T alone and that the full outbreak-size statistics are reproduced is not supported.
  2. [Appendix D and Fig. S3] For the Hamsterster network, R_net^0 is defined as the value that reproduces the simulated mean outbreak fraction x*_r in a Markovian simulation on the same network (last paragraph of Appendix D). The agreement shown in Figs. 4(c,d) and S3 is therefore not an independent test of the mapping: the single free parameter is calibrated to the first moment of the simulated distribution, and the comparison only checks whether the remaining shape is captured. To validate the predictive claim, R_net^0 should be computed directly from T and the network structure (e.g., through the correlated percolation equations S1–S2) without reference to the simulated x*_r, or the authors should provide an out-of-sample test on a network not used for calibration.
  3. [Eq. (8) and Fig. 2] The universal collapse onto the well-mixed WKB action, Eq. (6), is presented as an empirical observation. No derivation shows that the large-deviation rate function for the network SIR final size equals the well-mixed action as a function of R0^eff alone. Since R0^eff is computed from the deterministic final size via Eq. (8), it carries information only about the first moment; the right tail and the fluctuation scale could in principle depend on higher moments of the offspring distribution. The rescaling in Fig. 2(a) by the empirical mode and conditional standard deviation can mask such dependencies. The authors should either derive the claimed universality or test it directly by fixing T and varying the waiting-time shape while holding the network and mean final size fixed; the present simulations vary T and shape simultaneously.
  4. [Figs. 2–4 and the effective-Markovian claim] The numerical evidence does not directly probe the degenerate case that the theory is built on. In all reported simulations, the infection shape parameter α_inf is varied together with T, so T and the higher-order structure of the waiting-time distribution change at the same time. The claim that T alone controls the distribution requires comparing two non-Markovian processes with the same T but different waiting-time shapes. Such a comparison is absent, and the star counterexample above shows that the two processes will generally differ. At minimum, the manuscript needs to clearly state that the equivalence is approximate and holds only within the tested range of shapes and network parameters, and to add the fixed-T comparison.
minor comments (4)
  1. [Appendix A] The conditioning rule for 'extensive outbreaks' is not fully specified: 'the minimum separating the two modes of the bimodal outbreak-size distribution' is ambiguous for parameter sets where the two modes are not well separated. A precise algorithmic definition and a sensitivity check would be helpful.
  2. [Risk assessment paragraph] The probability categories 'up to 20% below the mean', 'up to 20% above the mean', and 'more than 20% above the mean' should be defined as fractions of the conditional mean x*_r, and the interval endpoints should be stated explicitly, since the quoted percentages will depend on the conditioning event.
  3. [Fig. 1 caption and text after Eq. (5)] The statement that 'similar trends are observed for Weibull and log-normal WT distributions' is not shown; the authors should either include the figure or label the statement as a remark based on simulations.
  4. [Appendix A, simulation validation] The validation of the first-reaction method against the modified next-reaction method of Ref. [49] is mentioned without quantitative results. A brief description of the validation (e.g., maximum relative error in the quantities reported later) would strengthen confidence in the numerics.

Circularity Check

1 steps flagged · score 6.0 of 10

Partially circular: the Hamsterster effective-R0 is calibrated to the target mean, so that mean comparison is forced by construction; the synthetic-network and weak-heterogeneity predictions remain independent.

  1. fitted input called prediction [Appendix D (correlated Hamsterster, after Eq. S2; used in Figs. 4(c,d) and S3)]
    "For the correlated Hamsterster network, we solve Eq. (S1) numerically and obtain x∗r from Eq. (S2); the effective Markovian Rnet0 from Eq. (9) is then defined as the value reproducing this x∗r in a Markovian simulation on the same network."

    Rnet0 is not computed independently for the correlated network; it is chosen so that a Markovian simulation on the same network has the same mean final size x∗r as the percolation equations. Since x∗r is the first-order statistic of the non-Markovian outbreak distribution that the mapping claims to capture, the mean agreement between the green Markovian and black non-Markovian simulations in Fig. 4(c) and Fig. S3 is enforced by construction rather than predicted. The standard deviation and full tail shape remain independent content, but the central parameter for the empirical-network application is calibrated to the target, making the claim that the mapping 'captures the Hamsterster outbreak statistics' partially circular.

full rationale

The main derivation chain is self-contained. R0^eff from Eq. (8) is computed from the transmissibility T and the network percolation relation Eq. (3), not fitted to the simulated outbreak distribution, so the WKB comparison for weakly heterogeneous networks in Figs. 2–3 is an independent test. The WKB formulas in Eqs. (6)–(7) are prior published results [10,13,14]; although the authors overlap, they are external, parameter-free inputs rather than a self-citation chain invoked to forbid alternatives. For synthetic heterogeneous networks, Rnet0 from Eq. (9) is derived from T and the known Markovian threshold λc, again without fitting to non-Markovian simulation data. The single genuinely circular step is the Hamsterster calibration in Appendix D: Rnet0 is defined as the value reproducing the percolation mean x∗r, so the mean is imposed by construction. The variance and full distribution shape on the Hamsterster network still provide independent evidence, and the weakly heterogeneous universal-collapse result is unaffected by this partial, application-specific circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard percolation mapping, the well-mixed WKB large-deviation theory, and an empirical universal-collapse assumption. The only fitted parameter is R_net^0 for the empirical Hamsterster network, plus a data-dependent conditioning cutoff for extensive outbreaks. No new physical entities are introduced.

free parameters (2)
  • R_net^0 for Hamsterster network = reproduces x*_r in Markovian simulation on the same network
    In Appendix D, the effective Markovian R_net^0 is defined as the value reproducing the simulated mean outbreak x*_r, rather than being derived purely from the transmissibility and network structure. This is a fitted value for the empirical network analysis.
  • Extensive-outbreak conditioning cutoff = minimum separating the two modes of the bimodal distribution
    Outbreak statistics are conditioned on realizations above the minimum separating the two modes. This is a data-dependent choice that affects the conditional distribution and the reported tail probabilities, though it is a standard conditioning in bimodal outbreak analysis.
assumptions (4)
  • domain assumption The final state of non-Markovian SIR is a bond-percolation configuration with independent edge occupation probability T (Eq. 2).
    The paper relies on this standard percolation mapping from Refs. [15,16] without proving it for general waiting-time distributions. Edge transmission events are independent only conditional on a node's recovery time; the unconditional independence used in percolation is an approximation on loopy networks.
  • domain assumption Networks are locally tree-like and uncorrelated for Eq. (3), with degree distribution p_k and mean degree kbar.
    The bond-percolation self-consistency equations (3) assume a locally tree-like, uncorrelated configuration model. The paper later extends to correlated networks via Appendix D, but the universal weak-heterogeneity curve relies on this assumption.
  • domain assumption The well-mixed Markovian WKB action S(x*_s) from Ref. [10] (Eq. 6) describes large deviations of the network epidemic after substituting R0 -> R0^eff.
    The paper assumes that the large-deviation action for a weakly heterogeneous network with non-Markovian waiting times is the same as for a well-mixed Markovian SIR with an effective reproduction number. This is the core universal-collapse assumption and is not derived from the network dynamics.
  • domain assumption Gamma and log-normal waiting-time distributions with means matched to 1/beta and 1/gamma are representative of empirical generation intervals.
    The paper tests gamma distributions primarily and log-normal for one case. The claim of arbitrariness of waiting-time distributions is extrapolated from these families.

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Pith. "Pith review of Extreme outbreaks in non-Markovian epidemics on complex networks." pith.science (2026). https://pith.science/paper/C23RHRAL

@misc{pith2026260724615,
  author       = {Pith},
  title        = {Pith review of: Extreme outbreaks in non-Markovian epidemics on complex networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C23RHRAL}},
  note         = {Machine review of arXiv:2607.24615}
}
read the original abstract

Extreme epidemic risk is controlled by the right tail of the outbreak-size distribution, but this distribution is generally unknown for non-Markovian spreading on networks. Here we determine this distribution by mapping non-Markovian SIR dynamics to an effective Markovian description. We show that arbitrary infection and recovery time statistics can be incorporated through a single edge transmissibility, yielding an effective Markovian process that reproduces the full outbreak-size statistics. For weakly heterogeneous networks, the reduction yields a universal well-mixed semiclassical theory governed by the bond-percolation reproductive number. Outbreak statistics across diverse waiting-time distributions and topologies collapse onto one predictive curve. For highly heterogeneous and empirical networks, the corresponding effective Markovian dynamics on the network captures the complete distribution. Our results provide a direct route from measured waiting-time distributions to quantitative predictions of network-level extreme-outbreak risk.

Figures

Figures reproduced from arXiv: 2607.24615 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Per-edge transmissibility [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Universal collapse of outbreak statistics for non [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. ER network with [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Works this paper leans on

54 extracted references · 44 canonical work pages

  1. [1]

    W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A115, 700 (1927)

    work page 1927

  2. [2]

    R. M. Anderson and R. M. May,Infectious Diseases of Humans: Dynamics and Control(Oxford University Press, Oxford, 1991)

    work page 1991

  3. [3]

    M. J. Keeling and P. Rohani,Modeling Infectious Dis- eases in Humans and Animals(Princeton University Press, Princeton, 2008)

    work page 2008

  4. [4]

    Britton, Stochastic epidemic models: A survey, Math

    T. Britton, Stochastic epidemic models: A survey, Math. Biosci.225, 24 (2010)

    work page 2010

  5. [5]

    House, J

    T. House, J. V. Ross, and D. Sirl, How big is an outbreak likely to be? methods for epidemic final-size calculation, Proc. R. Soc. A469, 20120436 (2013)

    work page 2013

  6. [6]

    Cirillo and N

    P. Cirillo and N. N. Taleb, Tail risk of contagious diseases, Nat. Phys.16, 606 (2020)

    work page 2020

  7. [7]

    Parnass, O

    G. Parnass, O. Levtzion-Korach, R. Peres, and M. As- saf, Estimating emergency department crowding with stochastic population models, PLoS ONE18, e0295130 (2023)

    work page 2023

  8. [8]

    Clancy, SIR epidemic models with general infectious period distribution, Statistics & Probability Letters85, 1 (2014)

    D. Clancy, SIR epidemic models with general infectious period distribution, Statistics & Probability Letters85, 1 (2014)

    work page 2014

Show all 54 references
  1. [9]

    J. C. Miller, Distribution of outbreak sizes for SIR disease in finite populations (2019), arXiv:1907.05138 [q-bio.PE]

    2019 arXiv

  2. [10]

    Hindes, M

    J. Hindes, M. Assaf, and I. B. Schwartz, Outbreak size distribution in stochastic epidemic models, Phys. Rev. Lett.128, 078301 (2022)

    2022

  3. [11]

    Hindes, L

    J. Hindes, L. Mier-y Teran-Romero, I. B. Schwartz, and M. Assaf, Outbreak-size distributions under fluctuating rates, Phys. Rev. Research5, 043264 (2023)

    2023

  4. [12]

    M. I. Dykman, E. Mori, J. Ross, and P. M. Hunt, Large fluctuations and optimal paths in chemical kinetics, J. Chem. Phys.100, 5735 (1994)

    1994

  5. [13]

    Assaf and B

    M. Assaf and B. Meerson, Extinction of metastable stochastic populations, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics81, 021116 (2010)

    2010

  6. [14]

    Assaf and B

    M. Assaf and B. Meerson, WKB theory of large devia- tions in stochastic populations, J. Phys. A: Math. Theor. 50, 263001 (2017)

    2017

  7. [15]

    Grassberger, On the critical behavior of the gen- eral epidemic process and dynamical percolation, Math

    P. Grassberger, On the critical behavior of the gen- eral epidemic process and dynamical percolation, Math. Biosci.63, 157 (1983)

    1983

  8. [16]

    M. E. J. Newman, Spread of epidemic disease on net- works, Phys. Rev. E66, 016128 (2002)

    2002

  9. [17]

    J. C. Miller, A note on a paper by Erik Volz: SIR dynam- ics in random networks, J. Math. Biol.62, 349 (2011)

    2011

  10. [18]

    Ball, Central limit theorems for SIR epidemics and percolation on configuration model random graphs, Ann

    F. Ball, Central limit theorems for SIR epidemics and percolation on configuration model random graphs, Ann. Appl. Probab.31, 2091 (2021)

    2021

  11. [19]

    Hindes and M

    J. Hindes and M. Assaf, Degree dispersion increases the rate of rare events in population networks, Phys. Rev. Lett.123, 068301 (2019)

    2019

  12. [20]

    I. Z. Kiss, G. R¨ ost, and Z. Vizi, Generalization of pairwise models to non-Markovian epidemics on networks, Phys. Rev. Lett.115, 078701 (2015)

    2015

  13. [21]

    Di Domenico, E

    L. Di Domenico, E. Valdano, and V. Colizza, Limited data on infectious disease distribution exposes ambigu- ity in epidemic modeling choices, Phys. Rev. Research6, 023265 (2024)

    2024

  14. [22]

    Ascione and E

    C. Ascione and E. Valdano, Interplay of generation time and spatial structure in epidemic dynamics and the re- liability of reproduction ratio estimates, Phys. Rev. Re- search8, 013136 (2026)

    2026

  15. [23]

    A. L. Lloyd, Realistic distributions of infectious periods in epidemic models: Changing patterns of persistence and dynamics, Theor. Popul. Biol.60, 59 (2001)

    2001

  16. [24]

    H. J. Wearing, P. Rohani, and M. J. Keeling, Appropriate models for the management of infectious diseases, PLoS Med.2, e174 (2005)

    2005

  17. [25]

    Ferretti, C

    L. Ferretti, C. Wymant, M. Kendall, L. Zhao, A. Nur- tay, L. Abeler-D¨ orner, M. Parker, D. Bonsall, and C. Fraser, Quantifying SARS-CoV-2 transmission sug- gests epidemic control with digital contact tracing, Sci- ence368, eabb6936 (2020)

    2020

  18. [26]

    Karrer and M

    B. Karrer and M. E. J. Newman, Message passing ap- proach for general epidemic models, Phys. Rev. E82, 016101 (2010)

    2010

  19. [27]

    Van Mieghem and R

    P. Van Mieghem and R. van de Bovenkamp, Non- Markovian infection spread dramatically alters the susceptible-infected-susceptible epidemic threshold in networks, Phys. Rev. Lett.110, 108701 (2013)

    2013

  20. [28]

    Cator, R

    E. Cator, R. van de Bovenkamp, and P. Van Mieghem, Susceptible-infected-susceptible epidemics on networks with general infection and cure times, Phys. Rev. E87, 062816 (2013)

    2013

  21. [29]

    R¨ ost, Z

    G. R¨ ost, Z. Vizi, and I. Z. Kiss, Pairwise approximation for SIR-type network epidemics with non-Markovian re- covery, Proc. R. Soc. A474, 20170695 (2018)

    2018

  22. [30]

    Van Mieghem and Q

    P. Van Mieghem and Q. Liu, Explicit non-Markovian susceptible-infected-susceptible mean-field epidemic threshold for Weibull and Gamma infections but Markovian curing, Phys. Rev. E100, 022317 (2019)

    2019

  23. [31]

    Giuggioli and S

    L. Giuggioli and S. Sarvaharman, Spatio-temporal dy- namics of random transmission events: from information sharing to epidemic spread, J. Phys. A: Math. Theor.55, 375005 (2022)

    2022

  24. [32]

    Ma and D

    J. Ma and D. J. D. Earn, Generality of the final size formula for an epidemic of a newly invading infectious disease, Bull. Math. Biol.68, 679 (2006)

    2006

  25. [33]

    Starnini, J

    M. Starnini, J. P. Gleeson, and M. Bogu˜ n´ a, Equivalence between non-Markovian and Markovian dynamics in epi- demic spreading processes, Phys. Rev. Lett.118, 128301 (2017)

    2017

  26. [34]

    Feng, S.-M

    M. Feng, S.-M. Cai, M. Tang, and Y.-C. Lai, Equiva- lence and its invalidation between non-Markovian and Markovian spreading dynamics on complex networks, Nat. Commun.10, 3748 (2019)

    2019

  27. [35]

    Aquino and M

    T. Aquino and M. Dentz, Chemical continuous time ran- dom walks, Phys. Rev. Lett.119, 230601 (2017)

    2017

  28. [36]

    O. Vilk, R. Metzler, and M. Assaf, Non-Markovian gene expression, Phys. Rev. Research6, L022026 (2024)

    2024

  29. [37]

    Shmunik and M

    M. Shmunik and M. Assaf, Large deviations in non- Markovian stochastic epidemics, Phys. Rev. Research8, L022038 (2026)

    2026

  30. [38]

    Kunegis, KONECT – The Koblenz Network Collec- tion, inProc

    J. Kunegis, KONECT – The Koblenz Network Collec- tion, inProc. Int. Conf. on World Wide Web Companion (2013) pp. 1343–1350

    2013

  31. [39]

    M. E. J. Newman, Assortative mixing in networks, Phys. Rev. Lett.89, 208701 (2002)

    2002

  32. [40]

    Pastor-Satorras, C

    R. Pastor-Satorras, C. Castellano, P. Van Mieghem, and A. Vespignani, Epidemic processes in complex networks, Rev. Mod. Phys.87, 925 (2015)

    2015

  33. [41]

    M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Ran- dom graphs with arbitrary degree distributions and their 6 applications, Phys. Rev. E64, 026118 (2001)

    2001

  34. [42]

    Erd˝ os and A

    P. Erd˝ os and A. R´ enyi, On random graphs I, Publ. Math. Debrecen6, 290 (1959)

    1959

  35. [43]

    Only the extensive-outbreak peak is shown ( x∗r >0.3, where the bimodal structure is well-separated)

  36. [44]

    For networks withσ k =O( ¯k), as occurs in many realistic networks,R net 0 cannot replaceR eff 0 in the mapping to an effective well-mixed model

    When ¯k≫1 andσ k/¯k≪1,R eff 0 reduces toR net 0 . For networks withσ k =O( ¯k), as occurs in many realistic networks,R net 0 cannot replaceR eff 0 in the mapping to an effective well-mixed model

  37. [45]

    Z. Vizi, I. Z. Kiss, J. C. Miller, and G. R¨ ost, A mono- tonic relationship between the variability of the infectious period and final size in pairwise epidemic modelling, J. Math. Ind.9, 1 (2019)

    2019

  38. [46]

    Mossong, N

    J. Mossong, N. Hens, M. Jit, P. Beutels, K. Aura- nen, R. Mikolajczyk, M. Massari, S. Salmaso, G. Scalia Tomba, J. Wallinga, J. Heijne, M. Sadkowska-Todys, M. Rosinska, and W. J. Edmunds, Social contacts and mixing patterns relevant to the spread of infectious dis- eases, PLoS...

    2008

  39. [47]

    D. T. Gillespie, A general method for numerically simu- lating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys.22, 403 (1976)

    1976

  40. [48]

    D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem.81, 2340 (1977)

    1977

  41. [49]

    Bogu˜ n´ a, L

    M. Bogu˜ n´ a, L. F. Lafuerza, R. Toral, and M. A. Serrano, Simulating non-Markovian stochastic processes, Phys. Rev. E90, 042108 (2014)

    2014

  42. [50]

    D. F. Anderson, A modified next reaction method for simulating chemical systems with time dependent propensities and delays, J. Chem. Phys.127, 214107 (2007)

    2007

  43. [51]

    Bollob´ as, A probabilistic proof of an asymptotic for- mula for the number of labelled regular graphs, Eur

    B. Bollob´ as, A probabilistic proof of an asymptotic for- mula for the number of labelled regular graphs, Eur. J. Comb.1, 311 (1980)

    1980

  44. [52]

    Steger and N

    A. Steger and N. C. Wormald, Generating random reg- ular graphs quickly, Comb. Probab. Comput.8, 377 (1999)

    1999

  45. [53]

    Korngut and M

    E. Korngut and M. Assaf, Impact of network assorta- tivity on disease lifetime in the SIS model of epidemics, Phys. Rev. E112, 024302 (2025)

    2025

  46. [54]

    V´ azquez and Y

    A. V´ azquez and Y. Moreno, Resilience to damage of graphs with degree correlations, Phys. Rev. E67, 015101(R) (2003). APPENDIX A: SIMULA TION DET AILS This appendix describes the simulation methods, and the different networks and WT distributions used throughout the manuscrip...

    2003

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