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Branching Processes: Their Role in Epidemiology
National Agricultural Research Institute, UR341, Department of Applied Mathematics and Informatics, F-78352 Jouy-en-Josas, France
Received: 11 January 2010; in revised form: 1 March 2010 / Accepted: 16 March 2010 / Published: 19 March 2010
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.
Keywords: branching process; age-dependence; population-dependence; extinction time; epidemic size
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MDPI and ACS Style
Jacob, C. Branching Processes: Their Role in Epidemiology. Int. J. Environ. Res. Public Health 2010, 7, 1186-1204.
Jacob C. Branching Processes: Their Role in Epidemiology. International Journal of Environmental Research and Public Health. 2010; 7(3):1186-1204.
Jacob, Christine. 2010. "Branching Processes: Their Role in Epidemiology." Int. J. Environ. Res. Public Health 7, no. 3: 1186-1204.