Revista de la
Unión Matemática Argentina
Extinction time of an epidemic with infection-age-dependent infectivity
Anicet Mougabe-Peurkor, Ibrahima Dramé, Modeste N'zi, and Étienne Pardoux

Volume 67, no. 2 (2024), pp. 417–443    

Published online: September 12, 2024

https://doi.org/10.33044/revuma.4032

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Abstract

This paper studies the distribution function of the time of extinction of a subcritical epidemic, when a large enough proportion of the population has been immunized and/or the infectivity of the infectious individuals has been reduced, so that the effective reproduction number is less than one. We do that for a SIR/SEIR model, where infectious individuals have an infection-age-dependent infectivity, as in the model introduced in Kermack and McKendrick's seminal 1927 paper. Our main conclusion is that simplifying the model as an ODE SIR model, as it is largely done in the epidemics literature, introduces a bias toward shorter extinction time.

References

  1. K. B. Athreya and P. E. Ney, Branching processes, Die Grundlehren der mathematischen Wissenschaften, Band 196, Springer-Verlag, New York-Heidelberg, 1972.  DOI  MR  Zbl
  2. P. Billingsley, Convergence of probability measures, second ed., Wiley Series in Probability and Statistics, Wiley, New York, 1999.  DOI  MR  Zbl
  3. T. Britton and E. Pardoux (eds.), Stochastic epidemic models with inference, Lecture Notes in Mathematics 2255, Springer, Cham, 2019.  DOI  MR  Zbl
  4. K. S. Crump and C. J. Mode, A general age-dependent branching process. I, J. Math. Anal. Appl. 24 (1968), 494–508.  DOI  MR  Zbl
  5. K. S. Crump and C. J. Mode, A general age-dependent branching process. II, J. Math. Anal. Appl. 25 (1969), 8–17.  DOI  MR  Zbl
  6. R. Forien, G. Pang, and E. Pardoux, Epidemic models with varying infectivity, SIAM J. Appl. Math. 81 no. 5 (2021), 1893–1930.  DOI  MR  Zbl
  7. R. Forien, G. Pang, and E. Pardoux, Multi-patch multi-group epidemic model with varying infectivity, Probab. Uncertain. Quant. Risk 7 no. 4 (2022), 333–364.  DOI  MR  Zbl
  8. Q. Griette, Z. Liu, P. Magal, and R. N. Thompson, Real-time prediction of the end of an epidemic wave: COVID-19 in China as a case-study, in Mathematics of public health, Fields Inst. Commun. 85, Springer, Cham, 2022, pp. 173–195.  DOI  MR  Zbl
  9. X. He, E. H. Y. Lau, P. Wu, X. Deng, J. Wang, X. Hao, Y. C. Lau, J. Y. Wong, Y. Guan, X. Tan, X. Mo, Y. Chen, B. Liao, W. Chen, F. Hu, Q. Zhang, M. Zhong, Y. Wu, L. Zhao, F. Zhang, B. J. Cowling, F. Li, and G. M. Leung, Temporal dynamics in viral shedding and transmissibility of COVID-19, Nat. Med. 26 (2020), 672–675.  DOI
  10. Y. Iwasa, M. A. Nowak, and F. Michor, Evolution of resistance during clonal expansion, Genetics 172 no. 4 (2006), 2557–2566.  DOI
  11. W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. London Ser. A 115 (1927), 700–721.  DOI
  12. E. Pardoux, Markov processes and applications: algorithms, networks, genome and finance, Wiley Series in Probability and Statistics, John Wiley & Sons, Chichester; Dunod, Paris, 2008.  DOI  MR  Zbl