Skip to main page content
U.S. flag

An official website of the United States government

Dot gov

The .gov means it’s official.
Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.

Https

The site is secure.
The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

Access keys NCBI Homepage MyNCBI Homepage Main Content Main Navigation
. 2012 Sep;74(9):2125-41.
doi: 10.1007/s11538-012-9749-6. Epub 2012 Jul 25.

A note on the derivation of epidemic final sizes

Affiliations

A note on the derivation of epidemic final sizes

Joel C Miller. Bull Math Biol. 2012 Sep.

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.

PubMed Disclaimer

References

    1. Andreasen Viggo. Dynamics of annual influenza A epidemics with immuno-selection. Journal of Mathematical Biology. 2003;46:504–536. - PubMed
    1. Andreasen Viggo. The final size of an epidemic and its relation to the basic reproduction number. Bulletin of Mathematical Biology. 2011;73(10):2305–2321. - PMC - PubMed
    1. Arino J, Brauer F, van den Driessche P, Watmough J, Wu J. A final size relation for epidemic models. Mathematical Biosciences and Engineering. 2007;4(2):159. - PubMed
    1. Ball F, Clancy D. The final outcome of an epidemic model with several different types of infective in a large population. Journal of applied probability. 1995:579–590.
    1. Bollobás B, Janson S, Riordan O. The phase transition in inhomogeneous random graphs. Random Structures and Algorithms. 2007;31(1):3.

Publication types

LinkOut - more resources