- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Large deviations in non-Markovian stochastic epidemics

Matan Shmunik and Michael Assaf*

  • *Contact author: michael.assaf@mail.huji.ac.il
PDF

Phys. Rev. Research 8, L022038 – Published 1 June, 2026

DOI: https://doi.org/10.1103/bswp-wrfc

Abstract

We develop a framework for non-Markovian, well-mixed susceptible-infected-recovered (SIR) and susceptible-infected-susceptible (SIS) models beyond mean field, utilizing the continuous-time random walk formalism. Using a gamma distribution for the infection and recovery interevent times as a test case, we derive asymptotical late-time master equations with effective memory kernels and obtain analytical predictions for the final outbreak-size distribution in the SIR model, and quasistationary distribution and disease lifetime in the SIS model. In particular, we show that varying the width of the interevent time distribution can greatly alter the outbreak-size distribution and disease lifetime. Overall, our analysis, confirmed against numerical simulations, paves the way for studying large deviations in structured populations on degree-heterogeneous networks.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (75)

  1. R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control (Oxford University Press, New York, 1991).
  2. W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. London Ser. A 115, 700 (1927).
  3. L. J. S. Allen, Some discrete-time SI, SIR, and SIS epidemic models, Math. Biosci. 124, 83 (1994).
  4. D. Mollison, Epidemic models: Their Structure and Relation to Data (Cambridge University Press, 1995).
  5. D. J. Daley and J. M. Gani, Epidemic Modelling: An Introduction (Cambridge University Press, Cambridge, UK, 1999).
  6. H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev. 42, 599 (2000).
  7. M. J. Keeling and K. T. Eames, Networks and epidemic models, J. R. Soc. Interface 2, 295 (2005).
  8. J. Neipel, J. Bauermann, S. Bo, T. Harmon, and F. Jülicher, Power-law population heterogeneity governs epidemic waves, PLoS One 15, e0239678 (2020).
  9. W. Yang, D. Zhang, L. Peng, C. Zhuge, and L. Hong, Rational evaluation of various epidemic models based on the COVID-19 data of China, Epidemics 37, 100501 (2021).
  10. R. C. Reiner, Jr., R. M. Barber, J. K. Collins, P. Zheng, C. Adolph, J. Albright, C. M. Antony, A. Y. Aravkin, S. D. Bachmeier, B. BangJensen, et al., Modeling COVID-19 scenarios for the United States, Nat. Med. 27, 94 (2021).
  11. H. W. Hethcote, Three basic epidemiological models, in Applied Mathematical Ecology, edited by S. A. Levin, T. G. Hallam, and L. J. Gross (Springer, Berlin, 1989), pp. 119–144.
  12. J. L. Aron and I. B. Schwartz, Seasonality and period-doubling bifurcations in an epidemic model, J. Theor. Biol. 110, 665 (1984).
  13. L. B. Shaw and I. B. Schwartz, Fluctuating epidemics on adaptive networks, Phys. Rev. E 77, 066101 (2008).
  14. R. Pastor-Satorras, C. Castellano, P. van Mieghem, and A. Vespignani, Epidemic processes in complex networks, Rev. Mod. Phys. 87, 925 (2015).
  15. B. Karrer and M. E. Newman, Message passing approach for general epidemic models, Phys. Rev. E 82, 016101 (2010).
  16. J. Lindquist, J. Ma, P. van den Driessche, and F. H. Willeboordse, Effective degree network disease models, J. Math. Biol. 62, 143 (2011).
  17. J. C. Miller, A. C. Slim, and E. M. Volz, Edge-based compartmental modelling for infectious disease spread, J. R. Soc. Interface 9, 890 (2012).
  18. M. J. Keeling, The effects of local spatial structure on epidemiological invasions, Proc. R. Soc. London B 266, 859 (1999).
  19. T. House and M. J. Keeling, Insights from unifying modern approximations to infections on networks, J. R. Soc. Interface 8, 67 (2011).
  20. C.-R. Cai, Z.-X. Wu, M. Z. Chen, P. Holme, and J.-Y. Guan, Solving the dynamic correlation problem of the susceptible-infected-susceptible model on networks, Phys. Rev. Lett. 116, 258301 (2016).
  21. O. Ovaskainen, The quasistationary distribution of the stochastic logistic model, J. Appl. Probab. 38, 898 (2001).
  22. M. I. Dykman, I. B. Schwartz, and A. S. Landsman, Disease extinction in the presence of random vaccination, Phys. Rev. Lett. 101, 078101 (2008).
  23. O. Ovaskainen and B. Meerson, Stochastic models of population extinction, Trends Ecol. Evol. 25, 643 (2010).
  24. M. Assaf and B. Meerson, Extinction of metastable stochastic populations, Phys. Rev. E 81, 021116 (2010).
  25. J. Hindes and I. B. Schwartz, Epidemic extinction and control in heterogeneous networks, Phys. Rev. Lett. 117, 028302 (2016).
  26. M. Assaf and B. Meerson, WKB theory of large deviations in stochastic populations, J. Phys. A: Math. Theor. 50, 263001 (2017).
  27. J. Hindes and M. Assaf, Degree dispersion increases the rate of rare events in population networks, Phys. Rev. Lett. 123, 068301 (2019).
  28. J. Hindes, M. Assaf, and I. B. Schwartz, Outbreak size distribution in stochastic epidemic models, Phys. Rev. Lett. 128, 078301 (2022).
  29. J. Hindes, L. Mier-y Teran-Romero, I. B. Schwartz, and M. Assaf, Outbreak-size distributions under fluctuating rates, Phys. Rev. Res. 5, 043264 (2023).
  30. E. Korngut, O. Vilk, and M. Assaf, Weighted-ensemble network simulations of the susceptible-infected-susceptible model of epidemics, Phys. Rev. E 111, 014146 (2025).
  31. E. Korngut and M. Assaf, Impact of network assortativity on disease lifetime in the SIS model of epidemics, Phys. Rev. E 112, 024302 (2025).
  32. E. Volz, SIR dynamics in random networks with heterogeneous connectivity, J. Math. Biol. 56, 293 (2008).
  33. N. Sherborne, J. C. Miller, K. B. Blyuss, and I. Z. Kiss, Mean-field models for non-Markovian epidemics on networks, J. Math. Biol. 76, 755 (2018).
  34. N. T. Bailey, A statistical method of estimating the periods of incubation and infection of an infectious disease, Nature (London) 174, 139 (1954).
  35. K. Gough, The estimation of latent and infectious periods, Biometrika 64, 559 (1977).
  36. H. J. Wearing, P. Rohani, and M. J. Keeling, Appropriate models for the management of infectious diseases, PLoS Med. 2, e174 (2005).
  37. E. Cator, R. van de Bovenkamp, and P. van Mieghem, Susceptible-infected-susceptible epidemics on networks with general infection and cure times, Phys. Rev. E 87, 062816 (2013).
  38. 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).
  39. M. Boguñá, L. F. Lafuerza, R. Toral, and M. Á. Serrano, Simulating non-Markovian stochastic processes, Phys. Rev. E 90, 042108 (2014).
  40. I. Z. Kiss, G. Röst, and Z. Vizi, Generalization of pairwise models to non-Markovian epidemics on networks, Phys. Rev. Lett. 115, 078701 (2015).
  41. G. Röst, Z. Vizi, and I. Z. Kiss, Impact of non-Markovian recovery on network epidemics, in Biomat 2015 (World Scientific, Singapore, 2015), pp. 40–53.
  42. I. Z. Kiss, J. C. Miller, and P. L. Simon, Non-Markovian epidemics, in Mathematics of Epidemics on Networks: From Exact to Approximate Models (Springer International Publishing, Cham, 2017), pp. 303–326.
  43. G. Röst, Z. Vizi, and I. Kiss, Pairwise approximation for SIR-type network epidemics with non-Markovian recovery, Proc. R. Soc. A 474, 20170695 (2018).
  44. N. Masuda and L. E. Rocha, A Gillespie algorithm for non-Markovian stochastic processes, SIAM Rev. 60, 95 (2018).
  45. M. Starnini, J. P. Gleeson, and M. Boguñá, Equivalence between non-Markovian and Markovian dynamics in epidemic spreading processes, Phys. Rev. Lett. 118, 128301 (2017).
  46. M. Feng, S.-M. Cai, M. Tang, and Y.-C. Lai, Equivalence and its invalidation between non-Markovian and Markovian spreading dynamics on complex networks, Nat. Commun. 10, 3748 (2019).
  47. F. Ball, A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models, Adv. Appl. Probab. 18, 289 (1986).
  48. A. N. Startsev, On the distribution of the size of an epidemicin a non-Markovian model, Theory Probab. Appl. 41, 730 (1997).
  49. D. Clancy, SIR epidemic models with general infectious period distribution, Stat. Probab. Lett. 85, 1 (2014).
  50. R. R. Wilkinson and K. J. Sharkey, Impact of the infectious period on epidemics, Phys. Rev. E 97, 052403 (2018).
  51. T. Aquino and M. Dentz, Chemical continuous time random walks, Phys. Rev. Lett. 119, 230601 (2017).
  52. O. Vilk, R. Metzler, and M. Assaf, Non-Markovian gene expression, Phys. Rev. Res. 6, L022026 (2024).
  53. O. Vilk and M. Assaf, Escape from a metastable state in non-Markovian population dynamics, Phys. Rev. E 110, 044132 (2024).
  54. O. Vilk, M. Mobilia, and M. Assaf, Non-Markovian rock-paper-scissors games, Phys. Rev. Res. 7, 023284 (2025).
  55. M. I. Dykman, E. Mori, J. Ross, and P. Hunt, Large fluctuations and optimal paths in chemical kinetics, J. Chem. Phys. 100, 5735 (1994).
  56. S. Redner, A Guide to First-Passage Processes (Cambridge University Press, Cambridge, UK, 2001).
  57. See Supplemental Material at http://link.aps.org/supplemental/10.1103/bswp-wrfc for more details on the model and additional results from the analysis and simulations.
  58. This is justified since for N≫1, the epidemic duration is still much longer than the system’s relaxation time [75].
  59. Note that the leading-order WKB approximation here is valid as long as the total action, NS, is large [24, 26, 55].
  60. In each realization, we sample the WTs for both infection and recovery processes from the given WT distributions, execute the reaction that occurs first, advance the system time accordingly, update the distributions’ mean rates, reset the timers (as done in Ref. [44] when the WT means are state dependent), and repeat until either disease extinction occurs or the maximum time is reached.
  61. D. F. Anderson, A modified next reaction method for simulating chemical systems with time-dependent propensities and delays, J. Chem. Phys. 127, 214107 (2007).
  62. In these simulations, we performed a sufficiently large number of realizations so that, even after excluding those with a final outbreak fraction <10−2, we still retained the desired number of samples (see Figs. 1 and 2). This allowed us to probe the distribution around the mean without the influence of near-immediate extinction events.
  63. Notably, the dependence of σ on xi* and α remains identical also when infection is exponential and recovery is gamma distributed, even though xi* changes in that case.
  64. M. Assaf and M. Mobilia, Fixation of a deleterious allele under mutation pressure and finite selection intensity, J. Theor. Biol. 275, 93 (2011).
  65. J. Zhang, M. Litvinova, W. Wang, Y. Wang, X. Deng, X. Chen, M. Li, W. Zheng, L. Yi, X. Chen, et al., Evolving epidemiology and transmission dynamics of coronavirus disease 2019 outside Hubei Province, China: A descriptive and modelling study, Lancet Infect. Dis. 20, 793 (2020).
  66. Q. Bi, Y. Wu, S. Mei, C. Ye, X. Zou, Z. Zhang, X. Liu, L. Wei, S. A. Truelove, T. Zhang, et al., Epidemiology and transmission of COVID-19 in 391 cases and 1286 of their close contacts in Shenzhen, China: A retrospective cohort study, Lancet Infect. Dis. 20, 911 (2020).
  67. N. M. Linton, T. Kobayashi, Y. Yang, K. Hayashi, A. R. Akhmetzhanov, S.-m. Jung, B. Yuan, R. Kinoshita, and H. Nishiura, Incubation period and other epidemiological characteristics of 2019 novel coronavirus infections with right truncation: A statistical analysis of publicly available case data, J. Clin. Med. 9, 538 (2020).
  68. X. He, E. H. Y. Lau, P. Wu, X. Deng, J. Wang, X. Hao, Y. C. Lau, J. Y. Wong, Y. Guan, X. Tan, et al., Temporal dynamics in viral shedding and transmissibility of COVID-19, Nat. Med. 26, 672 (2020).
  69. S. T. Ali, L. Wang, E. H. Lau, X.-K. Xu, Z. Du, Y. Wu, G. M. Leung, and B. J. Cowling, Serial interval of SARS-CoV-2 was shortened over time by nonpharmaceutical interventions, Science 369, 1106 (2020).
  70. S. W. Park, B. M. Bolker, S. Funk, C. J. E. Metcalf, J. S. Weitz, B. T. Grenfell, and J. Dushoff, Reconciling early-outbreak estimates of the basic reproductive number and its uncertainty: Framework and applications to the novel coronavirus (SARS-CoV-2) outbreak, J. R. Soc. Interface 17, 20200144 (2020).
  71. H. Lee, G. Lee, T. Kim, S. Kim, H. Kim, and S. Lee, Variability in the serial interval of COVID-19 in South Korea: A comprehensive analysis of age and regional influences, Front. Public Health 12, 1362909 (2024).
  72. F. Carrat, E. Vergu, N. M. Ferguson, M. Lemaitre, S. Cauchemez, S. Leach, and A.-J. Valleron, Time lines of infection and disease in human influenza: A review of volunteer challenge studies, Am. J. Epidemiol. 167, 775 (2008).
  73. A. W. Byrne, D. McEvoy, A. B. Collins, K. Hunt, M. Casey, A. Barber, F. Butler, J. Griffin, E. A. Lane, C. McAloon, et al., Inferred duration of infectious period of SARS-CoV-2: Rapid scoping review and analysis of available evidence for asymptomatic and symptomatic COVID-19 cases, BMJ Open 10, e039856 (2020).
  74. I. Voinsky, G. Baristaite, and D. Gurwitz, Effects of age and sex on recovery from COVID-19: Analysis of 5769 Israeli patients, J. Infect. 81, e102 (2020).
  75. M. Turkyilmazoglu, Explicit formulae for the peak time of an epidemic from the SIR model, Physica D 422, 132902 (2021).

Outline

Information

Phys. Rev. Research 8, L022038– Published 1 June, 2026
  • Received 11 September 2025
  • Accepted 25 April 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation