Journal of Applied Probability

The deterministic Kermack‒McKendrick model bounds the general stochastic epidemic

Robert R. Wilkinson, Frank G. Ball, and Kieran J. Sharkey

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We prove that, for Poisson transmission and recovery processes, the classic susceptible→infected→recovered (SIR) epidemic model of Kermack and McKendrick provides, for any given time t>0, a strict lower bound on the expected number of susceptibles and a strict upper bound on the expected number of recoveries in the general stochastic SIR epidemic. The proof is based on the recent message passing representation of SIR epidemics applied to a complete graph.

Article information

J. Appl. Probab., Volume 53, Number 4 (2016), 1031-1040.

First available in Project Euclid: 7 December 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 92D30: Epidemiology
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60J22: Computational methods in Markov chains [See also 65C40] 05C80: Random graphs [See also 60B20]

General stochastic epidemic deterministic general epidemic SIR Kermack‒McKendrick message passing bound


Wilkinson, Robert R.; Ball, Frank G.; Sharkey, Kieran J. The deterministic Kermack‒McKendrick model bounds the general stochastic epidemic. J. Appl. Probab. 53 (2016), no. 4, 1031--1040.

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  • Allen, L. J. S. (2008). An introduction to stochastic epidemic models. In Mathematical Epidemiology (Lecture Notes Math. 1945), Springer, Berlin, pp. 81–130.
  • Andersson, H. and Britton, T. (2000). Stochastic Epidemic Models and Their Statistical Analysis (Lecture Notes Statist. 151). Springer, New York.
  • Bailey, N. T. J. (1975). The Mathematical Theory of Infectious Diseases and Its Applications, 2nd edn. Hafner, New York.
  • Ball, F. (1990). A new look at Downton's carrier-borne epidemic model. In Stochastic Processes in Epidemic Theory (Lecture Notes Biomath. 86), Springer, Berlin, pp. 71–85.
  • Ball, F. and Donnelly, P. (1987). Interparticle correlation in death processes with application to variability in compartmental models. Adv. Appl. Prob. 19, 755–766.
  • Barbour, A. D. and Reinert, G. (2013). Approximating the epidemic curve. Electron. J. Prob. 18, 1–30.
  • Downton, F. (1968). The ultimate size of carrier-borne epidemics. Biometrika 55, 277–289.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York.
  • Karrer, B. and Newman, M. E. J. (2010). Message passing approach for general epidemic models. Phys. Rev. E 82, 016101.
  • Kermack, W. O. and McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proc. R. Soc. London A 115, 700–721.
  • McKendrick, A. G. (1914). Studies on the theory of continuous probabilities, with special reference to its bearing on natural phenomena of a progressive nature. Proc. London Math. Soc. 2 13, 401–416.
  • McKendrick, A. G. (1926) Applications of mathematics to medical problems. Proc. Edinburgh Math. Soc. 44, 98–130.
  • Simon, P. L. and Kiss, I. Z. (2013). From exact stochastic to mean-field ODE models: a new approach to prove convergence results. IMA J. Appl. Math. 78, 945–964.
  • Wilkinson, R. R. and Sharkey, K. J. (2014). Message passing and moment closure for susceptible-infected-recovered epidemics on finite networks. Phys. Rev. E 89, 022808. \endharvreferences