The Annals of Applied Probability

Proliferating parasites in dividing cells: Kimmel’s branching model revisited

Vincent Bansaye

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Abstract

We consider a branching model introduced by Kimmel for cell division with parasite infection. Cells contain proliferating parasites which are shared randomly between the two daughter cells when they divide. We determine the probability that the organism recovers, meaning that the asymptotic proportion of contaminated cells vanishes. We study the tree of contaminated cells, give the asymptotic number of contaminated cells and the asymptotic proportions of contaminated cells with a given number of parasites. This depends on domains inherited from the behavior of branching processes in random environment (BPRE) and given by the bivariate value of the means of parasite offsprings. In one of these domains, the convergence of proportions holds in probability, the limit is deterministic and given by the Yaglom quasistationary distribution. Moreover, we get an interpretation of the limit of the Q-process as the size-biased quasistationary distribution.

Article information

Source
Ann. Appl. Probab., Volume 18, Number 3 (2008), 967-996.

Dates
First available in Project Euclid: 26 May 2008

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1211819791

Digital Object Identifier
doi:10.1214/07-AAP465

Mathematical Reviews number (MathSciNet)
MR2418235

Zentralblatt MATH identifier
1142.60054

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J85: Applications of branching processes [See also 92Dxx] 60K37: Processes in random environments
Secondary: 92C37: Cell biology 92D25: Population dynamics (general) 92D30: Epidemiology

Keywords
Bienaymé Galton Watson process (BGW) branching processes in random environment (BPRE) Markov chain indexed by a tree quasistationary distribution empirical measures

Citation

Bansaye, Vincent. Proliferating parasites in dividing cells: Kimmel’s branching model revisited. Ann. Appl. Probab. 18 (2008), no. 3, 967--996. doi:10.1214/07-AAP465. https://projecteuclid.org/euclid.aoap/1211819791


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