Approximating the epidemic curve

Andrew David Barbour (Universität Zürich)
Gesine Reinert (University of Oxford)


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.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-30

Publication Date: May 16, 2013

DOI: 10.1214/EJP.v18-2557


  • 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. MR1380517
  • Ball, Frank. The threshold behaviour of epidemic models. J. Appl. Probab. 20 (1983), no. 2, 227--241. MR0698527
  • Ball, Frank; Donnelly, Peter. Strong approximations for epidemic models. Stochastic Process. Appl. 55 (1995), no. 1, 1--21. MR1312145
  • 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 MR1163825
  • 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. MR2187870
  • 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 MR3024808
  • 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. MR2953563
  • Diekmann, O. Limiting behaviour in an epidemic model. Nonlinear Anal. 1 (1976/77), no. 5, 459--470. MR0624451
  • 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 MR1882991
  • 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 MR0488341
  • Jagers, Peter. General branching processes as Markov fields. Stochastic Process. Appl. 32 (1989), no. 2, 183--212. MR1014449
  • 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. MR0084936
  • 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. MR1062069
  • McDiarmid, Colin. Concentration. Probabilistic methods for algorithmic discrete mathematics, 195--248, Algorithms Combin., 16, Springer, Berlin, 1998. MR1678578
  • 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 [ MR2358436]. J. Math. Biol. 62 (2011), no. 3, 349--358. MR2771177
  • 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. MR0629532
  • Roos, Bero. On the rate of multivariate Poisson convergence. J. Multivariate Anal. 69 (1999), no. 1, 120--134. MR1701409
  • Volz, Erik. SIR dynamics in random networks with heterogeneous connectivity. J. Math. Biol. 56 (2008), no. 3, 293--310. MR2358436
  • Whittle, P. The outcome of a stochastic epidemic—a note on Bailey's paper. Biometrika 42, (1955). 116--122. MR0070099

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