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Cell contamination and branching processes in a random environment with immigration

Part of: Markov processes Physiological, cellular and medical topics Special processes

Published online by Cambridge University Press:  01 July 2016

Vincent Bansaye
Vincent Bansaye*
Affiliation:
Université Pierre et Marie Curie et CNRS
*
∗ Current address: CMAP, École Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France. Email address: [email protected]
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Abstract

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We consider a branching model for a population of dividing cells infected by parasites. Each cell receives parasites by inheritance from its mother cell and independent contamination from outside the cell population. Parasites multiply randomly inside the cell and are shared randomly between the two daughter cells when the cell divides. The law governing the number of parasites which contaminate a given cell depends only on whether the cell is already infected or not. We first determine the asymptotic behavior of branching processes in a random environment with state-dependent immigration, which gives the convergence in distribution of the number of parasites in a cell line. We then derive a law of large numbers for the asymptotic proportions of cells with a given number of parasites. The main tools are branching processes in a random environment and laws of large numbers for a Markov tree.

Keywords

Branching processes in a random environment with immigrationMarkov chain indexed by a treeempirical measurerenewal theorem

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J85: Applications of branching processes 60K37: Processes in random environments 92C37: Cell biology 92D25: Population dynamics (general)
Secondary: 92D30: Epidemiology
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 4 , December 2009 , pp. 1059 - 1081
Copyright
Copyright © Applied Probability Trust 2009 

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