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Branching Processes: Their Role in Epidemiology

Author

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  • Christine Jacob

    (National Agricultural Research Institute, UR341, Department of Applied Mathematics and Informatics, F-78352 Jouy-en-Josas, France)

Abstract

Branching processes are stochastic individual-based processes leading consequently to a bottom-up approach. In addition, since the state variables are random integer variables (representing population sizes), the extinction occurs at random finite time on the extinction set, thus leading to fine and realistic predictions. Starting from the simplest and well-known single-type Bienaymé-Galton-Watson branching process that was used by several authors for approximating the beginning of an epidemic, we then present a general branching model with age and population dependent individual transitions. However contrary to the classical Bienaymé-Galton-Watson or asymptotically Bienaymé-Galton-Watson setting, where the asymptotic behavior of the process, as time tends to infinity, is well understood, the asymptotic behavior of this general process is a new question. Here we give some solutions for dealing with this problem depending on whether the initial population size is large or small, and whether the disease is rare or non-rare when the initial population size is large.

Suggested Citation

  • Christine Jacob, 2010. "Branching Processes: Their Role in Epidemiology," IJERPH, MDPI, vol. 7(3), pages 1-19, March.
    • Handle: RePEc:gam:jijerp:v:7:y:2010:i:3:p:1186-1204:d:7528
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    References listed on IDEAS

    as
    1. Christine Jacob & Pierre Magal, 2007. "Influence of Routine Slaughtering on the Evolution of BSE: Example of British and French Slaughterings," Risk Analysis, John Wiley & Sons, vol. 27(5), pages 1151-1167, October.
    2. J. A. P. Heesterbeek & K. Dietz, 1996. "The concept of Ro in epidemic theory," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 50(1), pages 89-110, March.
    3. Klebaner, F. C. & Nerman, O., 1994. "Autoregressive approximation in branching processes with a threshold," Stochastic Processes and their Applications, Elsevier, vol. 51(1), pages 1-7, June.
    4. Klebaner, Fima C., 1993. "Population-dependent branching processes with a threshold," Stochastic Processes and their Applications, Elsevier, vol. 46(1), pages 115-127, May.
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