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A Note on the Derivation of Epidemic Final Sizes

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Abstract

Final size relations are known for many epidemic models. The derivations are often tedious and difficult, involving indirect methods to solve a system of integro-differential equations. Often when the details of the disease or population change, the final size relation does not. An alternate approach to deriving final sizes has been suggested. This approach directly considers the underlying stochastic process of the epidemic rather than the approximating deterministic equations and gives insight into why the relations hold. It has not been widely used. We suspect that this is because it appears to be less rigorous. In this article, we investigate this approach more fully and show that under very weak assumptions (which are satisfied in all conditions we are aware of for which final size relations exist) it can be made rigorous. In particular, the assumptions must hold whenever integro-differential equations exist, but they may also hold in cases without such equations. Thus, the use of integro-differential equations to find a final size relation is unnecessary and a simpler, more general method can be applied.

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Notes

  1. This is the probability the first M do not transmit minus the probability that the first M+1 do not transmit.

  2. It is possible that a stub is left over at the end, or an edge is repeated, or an individual has an edge to itself. The proportion of individuals involved in such events is negligibly small as N→∞.

  3. If transmission and recovery occur as Poisson processes, \(\hat{T}=T\), otherwise \(\hat{T}\) will be altered to account for the fact that the initial infections are part-way through their infectious period.

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Acknowledgements

J.C.M. was supported by (1) the RAPIDD program of the Science and Technology Directorate, Department of Homeland Security and the Fogarty International Center, National Institutes of Health and (2) the Center for Communicable Disease Dynamics, Department of Epidemiology, Harvard School of Public Health under Award Number U54GM088558 from the National Institute Of General Medical Sciences. The content is solely the responsibility of the author and does not necessarily represent the official views of the National Institute Of General Medical Sciences or the National Institutes of Health.

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  1. Depts of Mathematics and Biology, The Pennsylvania State University, University Park, PA, 16802, USA

    Joel C. Miller

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Correspondence to Joel C. Miller.

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Miller, J.C. A Note on the Derivation of Epidemic Final Sizes. Bull Math Biol 74, 2125–2141 (2012). https://doi.org/10.1007/s11538-012-9749-6

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  • DOI: https://doi.org/10.1007/s11538-012-9749-6

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