Electronic Journal of Probability

Approximating the epidemic curve

Andrew Barbour and Gesine Reinert

Full-text: Open access

Abstract

Many models of epidemic spread have a common qualitative structure.  The numbers of infected individuals during the initial stages of an epidemic can be well approximated by a branching process, after which the proportion of individuals that are susceptible follows a more or less deterministic course.  In this paper, we show that both of these features are consequences of assuming a locally branching structure in the models, and that the deterministic course can itself be determined from the distribution of the limiting random variable associated with the backward, susceptibility branching process.  Examples considered includea stochastic version of the Kermack & McKendrick model, the Reed-Frost model, and the Volz configuration model.

Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 54, 30 pp.

Dates
Accepted: 16 May 2013
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465064279

Digital Object Identifier
doi:10.1214/EJP.v18-2557

Mathematical Reviews number (MathSciNet)
MR3065864

Zentralblatt MATH identifier
1301.92072

Subjects
Primary: 92H30
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J85: Applications of branching processes [See also 92Dxx]

Keywords
Epidemics Reed--Frost configuration model deterministic approximation branching processes

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Barbour, Andrew; Reinert, Gesine. Approximating the epidemic curve. Electron. J. Probab. 18 (2013), paper no. 54, 30 pp. doi:10.1214/EJP.v18-2557. https://projecteuclid.org/euclid.ejp/1465064279


Export citation

References

  • Aldous, David. On simulating a Markov chain stationary distribution when transition probabilities are unknown. Discrete probability and algorithms (Minneapolis, MN, 1993), 1–9, IMA Vol. Math. Appl., 72, Springer, New York, 1995.
  • Ball, Frank. The threshold behaviour of epidemic models. J. Appl. Probab. 20 (1983), no. 2, 227–241.
  • Ball, Frank; Donnelly, Peter. Strong approximations for epidemic models. Stochastic Process. Appl. 55 (1995), no. 1, 1–21.
  • Barbour, A. D.; Holst, Lars; Janson, Svante. Poisson approximation. Oxford Studies in Probability, 2. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1992. x+277 pp. ISBN: 0-19-852235-5
  • Barbour, A. D. and Reinert, G.: Asymptotic behaviour of gossip processes and small world networks. Appl. Probab. (to appear), ARXIV1202.5895.
  • Bhamidi, S., van der Hofstad, R. and G. Hooghiemstra, G.: Universality for first passage percolation on sparse random graphs, ARXIV1210.6839.
  • Brauer, Fred. The Kermack-McKendrick epidemic model revisited. Math. Biosci. 198 (2005), no. 2, 119–131.
  • Brauer, Fred; Castillo-Chavez, Carlos. Mathematical models in population biology and epidemiology. Second edition. Texts in Applied Mathematics, 40. Springer, New York, 2012. xxiv+508 pp. ISBN: 978-1-4614-1685-2; 978-1-4614-1686-9
  • Decreusefond, Laurent; Dhersin, Jean-Stéphane; Moyal, Pascal; Tran, Viet Chi. Large graph limit for an SIR process in random network with heterogeneous connectivity. Ann. Appl. Probab. 22 (2012), no. 2, 541–575.
  • Diekmann, O. Limiting behaviour in an epidemic model. Nonlinear Anal. 1 (1976/77), no. 5, 459–470.
  • Diekmann, Odo; Heesterbeek, J. A. P. Mathematical epidemiology of infectious diseases. Model building, analysis and interpretation. Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2000. xvi+303 pp. ISBN: 0-471-49241-8
  • Gupta, S.D., Lal, V., Jain, R. and Gupta, O. P.: Modeling of H1N1 Outbreak in Rajasthan: Methods and Approaches. Indian J. Community Med. 36, (2011), 36–38.
  • Jagers, Peter. Branching processes with biological applications. Wiley Series in Probability and Mathematical Statistics—Applied Probability and Statistics. Wiley-Interscience [John Wiley & Sons], London-New York-Sydney, 1975. xiii+268 pp. ISBN: 0-471-43652-6
  • Jagers, Peter. General branching processes as Markov fields. Stochastic Process. Appl. 32 (1989), no. 2, 183–212.
  • Kendall, David G. Deterministic and stochastic epidemics in closed populations. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. IV, pp. 149–165. University of California Press, Berkeley and Los Angeles, 1956.
  • Kermack, W. O. and McKendrick, A. G.: Contributions to the mathematical theory of epidemics, part I. Proc. Roy. Soc. Edin. A/ 115, (1927), 700–721; Bull. Math. Biol./ 53, (1991), 33–55.
  • Massart, P. The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. Ann. Probab. 18 (1990), no. 3, 1269–1283.
  • McDiarmid, Colin. Concentration. Probabilistic methods for algorithmic discrete mathematics, 195–248, Algorithms Combin., 16, Springer, Berlin, 1998.
  • Metz, J. A. J.: The epidemic in a closed population with all susceptibles equally vulnerable; some results for large susceptible populations and small initial infections. Acta Biotheor. 27, (1978), 75–123.
  • Miller, Joel C. A note on a paper by Erik Volz: SIR dynamics in random networks [ ]. J. Math. Biol. 62 (2011), no. 3, 349–358.
  • Moore, C. and Newman, M. E. J.: Epidemics and percolation in small-world networks. Phys. Rev. E 61, (2000), 5678–5682.
  • Nerman, Olle. On the convergence of supercritical general (C-M-J) branching processes. Z. Wahrsch. verw. Gebiete 57 (1981), no. 3, 365–395.
  • Roos, Bero. On the rate of multivariate Poisson convergence. J. Multivariate Anal. 69 (1999), no. 1, 120–134.
  • Volz, Erik. SIR dynamics in random networks with heterogeneous connectivity. J. Math. Biol. 56 (2008), no. 3, 293–310.
  • Whittle, P. The outcome of a stochastic epidemic—a note on Bailey's paper. Biometrika 42, (1955). 116–122.