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The limiting conditional probability distribution in a stochastic model of T cell repertoire maintenance

By Emily R. Stirk, Grant Lythe, Hugo van den Berg, Gareth A. D. Hurst and Carmen Molina-Paris

Abstract

The limiting conditional probability distribution (LCD) has been much studied in the field of mathematical biology, particularly in the context of epidemiology and the persistence of epidemics However, it has not yet been applied to the immune system. One of the characteristic features of the T cell repertoire is its diversity This diversity declines in old age, whence the concepts of extinction and persistence are also relevant to the immune system In this paper we model T cell repertoire maintenance by means of a continuous-rime birth and death process on the positive integers, where the origin is an absorbing state We show that eventual extinction is guaranteed. The late-time behaviour of the process before extinction takes place is modelled by the LCD, which we prove always exists for the process studied here In most cases, analytic expressions for the LCD cannot be computed but the probability distribution may be approximated by means of the stationary probability distributions of two related processes We show how these approximations are related to the LCD of the original process and use them to study the LCD in two special cases We also make use of the large N expansion to derive a further approximation to the LCD The accuracy of the various approximations is then analysed (C) 2009 Elsevier Inc. All rights reserved

Topics: QH301
Publisher: Elsevier Science Inc.
Year: 2010
DOI identifier: 10.1016/j.mbs.2009.12.004
OAI identifier: oai:wrap.warwick.ac.uk:6105
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