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  • Letter
  • Open Access

Heterogeneity can markedly increase final outbreak size in the SIR model of epidemics

Alexander Leibenzon and Michael Assaf

  • Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel

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Phys. Rev. Research 6, L012010 – Published 16 January, 2024Erratum Phys. Rev. Research 6, 039004 (2024)

DOI: https://doi.org/10.1103/PhysRevResearch.6.L012010

Abstract

We study the susceptible-infected-recovered (SIR) model of epidemics on positively correlated heterogeneous networks with population variability, and explore the dependence of the final outbreak size on the network heterogeneity strength and basic reproduction number R0—the ratio between the infection and recovery rates per individual. We reveal a critical value R0c, above which the maximal outbreak size is obtained at zero heterogeneity, but below which the maximum is obtained at finite heterogeneity strength. This second-order phase transition, universal for all network distributions with finite standardized moments, indicates that network heterogeneity can greatly increase the final outbreak size. We also show that this effect can be enhanced by adding population heterogeneity, in the form of varying interindividual susceptibility and infectiousness. Our results provide key insight as to the predictability of the well-mixed SIR model for the final outbreak size, in realistic scenarios.

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Erratum

Erratum: Heterogeneity can markedly increase final outbreak size in the SIR model of epidemics [Phys. Rev. Research 6, 012010 (2024)]

Alexander Leibenzon and Michael Assaf
Phys. Rev. Research 6, 039004 (2024)

Article Text

References (45)

  1. W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A 115, 700 (1927).
  2. R. Anderson and R. May, Infectious Diseases of Humans: Dynamics and Control (OUP Oxford, 1992).
  3. H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev. 42, 599 (2000).
  4. G. Chowell, L. Sattenspiel, S. Bansal, and C. Viboud, Mathematical models to characterize early epidemic growth: A review, Phys. Life Rev. 18, 66 (2016).
  5. 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).
  6. R. Pastor-Satorras, C. Castellano, P. Van Mieghem, and A. Vespignani, Epidemic processes in complex networks, Rev. Mod. Phys. 87, 925 (2015).
  7. M. Saeedian, M. Khalighi, N. Azimi-Tafreshi, G. R. Jafari, and M. Ausloos, Memory effects on epidemic evolution: The susceptible-infected-recovered epidemic model, Phys. Rev. E 95, 022409 (2017).
  8. M. Bohner, S. Streipert, and D. F. Torres, Exact solution to a dynamic sir model, Nonlinear Anal. Hybri. 32, 228 (2019).
  9. J. C. Miller, A note on a paper by Erik Volz: SIR dynamics in random networks, J. Math. Biol. 62, 349 (2011).
  10. E. Volz, SIR dynamics in random networks with heterogeneous connectivity, J. Math. Biol. 56, 293 (2008).
  11. M. E. J. Newman, Spread of epidemic disease on networks, Phys. Rev. E 66, 016128 (2002).
  12. E. Kenah and J. M. Robins, Second look at the spread of epidemics on networks, Phys. Rev. E 76, 036113 (2007).
  13. P.-A. Noël, B. Davoudi, R. C. Brunham, L. J. Dubé, and B. Pourbohloul, Time evolution of epidemic disease on finite and infinite networks, Phys. Rev. E 79, 026101 (2009).
  14. L. A. Meyers, B. Pourbohloul, M. E. J. Newman, D. M. Skowronski, and R. C. Brunham, Network theory and sars: Predicting outbreak diversity, J. Theor. Biol. 232, 71 (2005).
  15. 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).
  16. F. Ball and D. Clancy, The final size and severity of a generalised stochastic multitype epidemic model, Adv. Appl. Probab. 25, 721 (1993).
  17. M. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals (Princeton University Press, 2008).
  18. T. House, J. V. Ross, and D. Sirl, How big is an outbreak likely to be? Methods for epidemic final-size calculation, Proc. Math. Phys. Eng. Sci. 469, 20120436 (2013).
  19. L. J. Allen, A primer on stochastic epidemic models: Formulation, numerical simulation, and analysis, Infect. Dis. Model. 2, 128 (2017).
  20. J. C. Miller, Distribution of outbreak sizes for SIR disease in finite populations, arXiv:1907.05138.
  21. J. Hindes, M. Assaf, and I. B. Schwartz, Outbreak size distribution in stochastic epidemic models, Phys. Rev. Lett. 128, 078301 (2022).
  22. H. W. Hethcote, An immunization model for a heterogeneous population, Theor. Popul. Biol 14, 338 (1978).
  23. N. Becker and P. Yip, Analysis of variations in an infection rate, Aust. J. Stat. 31, 42 (1989).
  24. A. S. Novozhilov, On the spread of epidemics in a closed heterogeneous population, Math. Biosci. 215, 177 (2008).
  25. M. G. M. Gomes, M. U. Ferreira, R. M. Corder, J. G. King, C. Souto-Maior, C. Penha-Gonçalves, G. Gonçalves, M. Chikina, W. Pegden, and R. Aguas, Individual variation in susceptibility or exposure to SARS-CoV-2 lowers the herd immunity threshold, J. Theor. Biol. 540, 111063 (2022).
  26. J. O. Lloyd-Smith, S. J. Schreiber, P. E. Kopp, and W. M. Getz, Superspreading and the effect of individual variation on disease emergence, Nature (London) 438, 355 (2005).
  27. J. Neipel, J. Bauermann, S. Bo, T. Harmon, and F. Jülicher, Power-law population heterogeneity governs epidemic waves, PLoS ONE 15, e0239678 (2020).
  28. A. V. Tkachenko, S. Maslov, A. Elbanna, G. N. Wong, Z. J. Weiner, and N. Goldenfeld, Time-dependent heterogeneity leads to transient suppression of the COVID-19 epidemic, not herd immunity, Proc. Natl. Acad. Sci. USA 118, e2015972118 (2021).
  29. M. Assaf and B. Meerson, Extinction of metastable stochastic populations, Phys. Rev. E 81, 021116 (2010).
  30. M. Assaf and B. Meerson, WKB theory of large deviations in stochastic populations, J. Phys. A: Math. Theor. 50, 263001 (2017).
  31. Y. Moreno, J. B. Gómez, and A. F. Pacheco, Epidemic incidence in correlated complex networks, Phys. Rev. E 68, 035103(R) (2003).
  32. Here, we recover the result of random networks for α=0 [45]. Notably, we account for correlations between nearest neighbors only, and neglect long-range correlations. Yet, the model still adequately describes correlated networks [31].
  33. For k0=O(1) our derivation is invalid since ε=O(1), and in addition, stochastic effects become dominant, such that R0c rapidly grows as k0 is decreased; see Fig. 1.
  34. D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem. C 81, 2340 (1977).
  35. M. Molloy and B. Reed, A critical point for random graphs with a given degree sequence, Random Struct. Algorithms 6, 161 (1995).
  36. M. E. J. Newman, Assortative mixing in networks, Phys. Rev. Lett. 89, 208701 (2002).
  37. The maximal relative error in Fig. 2 between the numerical and approximated values of R∞ was <0.2%.
  38. Naturally, other distributions of susceptibility are possible; yet, these do not qualitatively change our results.
  39. R. Gani and S. Leach, Transmission potential of smallpox in contemporary populations, Nature (London) 414, 748 (2001).
  40. R. Grant, L.-B. L. Nguyen, and R. Breban, Modelling human-to-human transmission of monkeypox, Bull. World Health Organ. 98, 638 (2020).
  41. S. A. Truelove, L. T. Keegan, W. J. Moss, L. H. Chaisson, E. Macher, A. S. Azman, and J. Lessler, Clinical and epidemiological aspects of diphtheria: A systematic review and pooled analysis, Clin. Infect. Dis. 71, 89 (2020).
  42. M. A. Billah, M. M. Miah, and M. N. Khan, Reproductive number of coronavirus: A systematic review and meta-analysis based on global level evidence, PLOS ONE 15, e0242128 (2020).
  43. Y. Liu and J. Rocklöv, The effective reproductive number of the omicron variant of SARS-CoV-2 is several times relative to delta, J. Travel Med. 29, taac037 (2022).
  44. M. Biggerstaff, S. Cauchemez, C. Reed, M. Gambhir, and L. Finelli, Estimates of the reproduction number for seasonal, pandemic, and zoonotic influenza: a systematic review of the literature, BMC Infectious Diseases 14, 480 (2014).
  45. S. L. Feld, Why your friends have more friends than you do, Am. J. Sociol. 96, 1464 (1991).

Outline

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Phys. Rev. Research 6, L012010– Published 16 January, 2024
  • Received 1 March 2023
  • Revised 13 August 2023
  • Accepted 14 December 2023
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