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The deterministic Kermack‒McKendrick model bounds the general stochastic epidemic

Part of: Markov processes Graph theory

Published online by Cambridge University Press:  09 December 2016

Robert R. Wilkinson ,
Frank G. Ball  and
Kieran J. Sharkey
Robert R. Wilkinson*
Affiliation:
The University of Liverpool
Frank G. Ball*
Affiliation:
The University of Nottingham
Kieran J. Sharkey*
Affiliation:
The University of Liverpool
*
* Postal address: Department of Mathematical Sciences, The University of Liverpool, Liverpool L69 7ZL, UK.
*** Postal address: School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, UK. Email address: [email protected]
* Postal address: Department of Mathematical Sciences, The University of Liverpool, Liverpool L69 7ZL, UK.
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Abstract

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.

Keywords

General stochastic epidemicdeterministic general epidemicSIRKermack‒McKendrickmessage passingbound

MSC classification

Primary: 92D30: Epidemiology
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60J22: Computational methods in Markov chains 05C80: Random graphs
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 4 , December 2016 , pp. 1031 - 1040
Copyright
Copyright © Applied Probability Trust 2016 

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References

Allen, L. J. S. (2008).An introduction to stochastic epidemic models.In Mathematical Epidemiology (Lecture Notes Math. 1945),Springer,Berlin,pp. 81–130.Google Scholar
Andersson, H. and Britton, T. (2000).Stochastic Epidemic Models and Their Statistical Analysis(Lecture Notes Statist. 151).Springer,New York.CrossRefGoogle Scholar
Bailey, N. T. J. (1975).The Mathematical Theory of Infectious Diseases and Its Applications,2nd edn.Hafner,New York.Google Scholar
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.CrossRefGoogle Scholar
Ball, F. and Donnelly, P. (1987).Interparticle correlation in death processes with application to variability in compartmental models.Adv. Appl. Prob. 19,755–766.Google Scholar
Barbour, A. D. and Reinert, G. (2013).Approximating the epidemic curve.Electron. J. Prob. 18,1–30.CrossRefGoogle Scholar
Downton, F. (1968).The ultimate size of carrier-borne epidemics.Biometrika 55,277–289.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986).Markov Processes: Characterization and Convergence.John Wiley,New York.CrossRefGoogle Scholar
Karrer, B. and Newman, M. E. J. (2010).Message passing approach for general epidemic models.Phys. Rev. E 82,016101.Google Scholar
Kermack, W. O. and McKendrick, A. G. (1927).A contribution to the mathematical theory of epidemics.Proc. R. Soc. London A 115,700–721.Google Scholar
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.Google Scholar
McKendrick, A. G. (1926)Applications of mathematics to medical problems.Proc. Edinburgh Math. Soc. 44,98–130.Google Scholar
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.Google Scholar
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.Google Scholar