Bulletin of Mathematical Biology

, Volume 74, Issue 9, pp 2125–2141

# A Note on the Derivation of Epidemic Final Sizes

• Joel C. Miller
Original Article

## 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.

## Keywords

Epidemics SIR Final size relation

## Notes

### 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.

## References

1. Andreasen, V. (2003). Dynamics of annual influenza A epidemics with immuno-selection. Journal of Mathematical Biology, 46, 504–536.
2. Andreasen, V. (2011). The final size of an epidemic and its relation to the basic reproduction number. Bulletin of Mathematical Biology, 73(10), 2305–2321.
3. Arino, J., Brauer, F., van den Driessche, P., Watmough, J., & Wu, J. (2007). A final size relation for epidemic models. Mathematical Biosciences and Engineering, 4(2), 159.
4. Ball, F., & Clancy, D. (1995). The final outcome of an epidemic model with several different types of infective in a large population. Journal of Applied Probability, 32, 579–590.
5. Bollobás, B., Janson, S., & Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures & Algorithms, 31(1), 3.
6. Brauer, F. (2008). Age-of-infection and the final size relation. Mathematical Biosciences and Engineering, 5(4), 681.
7. Britton, T., Deijfen, M., & Martin-Löf, A. (2006). Generating simple random graphs with prescribed degree distribution. Journal of Statistical Physics, 124(6), 1377–1397.
8. Chung, F., & Lu, L. (2002). Connected components in random graphs with given expected degree sequences. Annals of Combinatorics, 6(2), 125–145.
9. Diekmann, O., & Heesterbeek, J. A. P. (2000). Mathematical epidemiology of infectious diseases. Chichester: Wiley. Google Scholar
10. Diekmann, O., Heesterbeek, J. A. P., & Metz, J. A. J. (1990). On the definition and the computation of the basic reproduction ratio $$\mathcal{R}_{0}$$ in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology, 28, 365–382.
11. Feld, S. L. (1991). Why your friends have more friends than you do. American Journal of Sociology, 96(6), 1464–1477.
12. van der Hofstad, R. (2012). Critical behavior in inhomogeneous random graphs. Random Struct. Algorithms. To be published arXiv:0902.0216v2 [math.PR].
13. Katriel, G. (2012). The size of epidemics in populations with heterogeneous susceptibility. Journal of Mathematical Biology, 65(2), 237–262. doi:.
14. Kenah, E., & Robins, J. M. (2007). Second look at the spread of epidemics on networks. Physical Review E, 76(3), 036113.
15. Kermack, W. O., & McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 115, 700–721.
16. Ma, J. J., & Earn, D. J. D. (2006). Generality of the final size formula for an epidemic of a newly invading infectious disease. Bulletin of Mathematical Biology, 68(3), 679–702.
17. Meester, R., & Trapman, P. (2011). Bounding basic characteristics of spatial epidemics with a new percolation model. Advances in Applied Probability, 43(2), 335–347.
18. Miller, J. C. (2007). Epidemic size and probability in populations with heterogeneous infectivity and susceptibility. Physical Review E, 76(1), 010101(R).
19. Miller, J. C. (2008). Bounding the size and probability of epidemics on networks. Journal of Applied Probability, 45, 498–512.
20. Miller, J. C., Slim, A. C., & Volz, E. M. (2012). Edge-based compartmental modelling for infectious disease spread. Journal of the Royal Society Interface, 9(70), 890–906.
21. Molloy, M., & Reed, B. (1995). A critical point for random graphs with a given degree sequence. Random Structures & Algorithms, 6(2), 161–179.
22. Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Review, 45, 167–256.
23. Norros, I., & Reittu, H. (2006). On a conditionally Poissonian graph process. Advances in Applied Probability, 38(1), 59–75.