Branching Processes in Biology

Abstract

Problems in engineering, computational science, and the physical and biological sciences are using increasingly sophisticated mathematical techniques. Thus, the bridge between the mathematical sciences and other disciplines is heavily traveled. The correspondingly increased dialog between the disciplines has led to the establishment of the series: Interdisciplinary Applied Mathematics.The purpose of this series is to meet the current and future needs for the interaction between various science and technology areas on the one hand and mathematics on the other. This is done, firstly, by encouraging the ways that mathematics may be applied in traditional areas, as well as point towards new and innovative areas of applications; and secondly, by encouraging other scientific disciplines to engage in a dialog with mathematicians outlining their problems to both access new methods as well as to suggest innovative developments within mathematics itself.The series will consist of monographs and high-level texts from researchers working on the interplay between mathematics and other fields of science and technology.More information about this series at

A stochastic microscopic model for the dynamics of antigenic variation

“…The generalizations of the model presented above lead to changes in the structure of critical, subcritical and supercritical regions in the space of parameters. As it happens also in the simpler version of the model described in Section 2, the branching process shows instability in the asymptotic limit, meaning that the process explodes exponentially fast (see for example Kimmel and Axelrod, 2002) or it becomes extinct supposedly at an exponential or subexponential rate -as suggested by the bounds obtained for percolation and spatial birth-and-death processes in random environment at subcritical regime (see Fernández et al, 2005 and references therein). All states are transient, excepting when the population becomes extinct, and consequently any kind of stable oscillatory regime are extremely unlikely.…”

“…type-2) cell. The following result is adapted from Kimmel and Axelrod (2002), Jagers (1975); Haccou et al (2005):…”

Fast maximum likelihood estimation of mutation rates using a birth–death process

“…That is, the parent cell mutates with probability p before dividing into two progeny cells. Other mutation schemes described in the literature (Zheng, 1999;Kimmel and Axelrod, 2010) may also be used which lead to different pgf's for the offspring distribution. For example, "post-division mutation" refers to the scheme that after each cell division, the two progeny cells mutate independently with probability p. This yields an offspring distribution pgf ½p þð1 À pÞs 2 .…”

Toy models for macroevolutionary patterns and trends

“…While directed networks called phylogenies (trees of life) are often used to represent evolutionary relationships, there are other implications of branching (or GaltonWatson) processes (Kimmel and Axelrod, 2002). Branching processes are often used to characterize discrete generations .…”