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Hindawi Publishing CorporationInternational Journal of Stochastic AnalysisVolume 2013, Article ID 790625, 2 pageshttp://dx.doi.org/10.1155/2013/790625

EditorialApplications of Stochastic Processes in Biology and Medicine

Charles J. Mode,1 Rick Durrett,2 Fima Klebaner,3 and Peter Olofsson4

1 Department of Mathematics, Drexel University, Philadelphia, PA 19104, USA2Department of Mathematics, Duke University, Durham, NC, USA3 School of Mathematical Sciences, Monash University, Melbourne, VIC, Australia4Department of Mathematics, Trinity University, San Antonio, TX, USA

Correspondence should be addressed to Charles J. Mode; [email protected]

Received 27 February 2013; Accepted 27 February 2013

Copyright © 2013 Charles J. Mode et al.This is an open access article distributed under the Creative CommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Even though the title of this special edition suggests thata wide class of stochastic processes may be applicable inbiology and medicine, all the six papers that were acceptedfor publication in this edition were based, either directly orindirectly, on a class of stochastic processes called branchingprocesses. Within all the literature on probability, statisticsand stochastic processes, papers, and books on branchingprocesses occupy a relatively small but significant niche, eventhough their history spans and period of more than onehundred years. In particular, the papers that constitute thisedition may be classified into three types; namely, models ofepidemics of infectious diseases, genetics and evolution.

In a paper by N. Champagnat et al. on birth anddeath process with neutral mutation, classes of branchingprocesses are used to study mutations characterized in termsof splitting trees. It is interesting to note that mutations werenot described in terms of nucleotide substitutions that arewidely used in the classification of human haplotypes basedon mitochondrial DNA. Like the analysis used in papersand books on branching processes, limit theorems played abasic role in the paper and the time scale of evolution wasevolutionary time rather than generations.

In a paper by A. Denes et al. Galton-Watson multitypebranching processes in discrete time with immigration wereused to model risks of infectious disease by imported caseswith applications to the European Football Championshipsof 2012. Among the many tools used by these authors in theiranalysis of the model was probability generating functions.

In the paper of S. C. Penisson and Jacob, the decay phaseof the model for of an infectious disease caused by someinfectious agent was approximated by a branching process.

Limit theorems again played a basic role in the analysis of themodel. There is also an interesting application of the modelto BSE, mad cow disease, epidemic in Great Britain.

A paper by V. Vatutin et al. is devoted to decomposablebranching processes for the two-type case in, among otherthings, random environments. In particular, it is assumedthat the population has two types. One type lives on oneisland with variable environments, and the other type liveson another island with a constant environment. Moreover,it is assumed that migration occurs only from a randomenvironment to a constant environment. The main focus ofthe paper is to apply limit theorems to study probabilities ofextinction in the critical and subcritical cases.

In a paper by C. J. Mode et al. the focus of attention isto formulate and partially analyze a model of the emergenceof mutations and their subsequent evolution in an age-structured self-regulating stochastic process with two sexes.From the genetic point of view, only one autosomal locuswith two alleles is considered. The working paradigm ofthe paper differs from that of other papers in the editionin that it is based on computer intensive methods, and, inparticular, Monte Carlo simulation methods. The process isalso self-regulating in the sense that the subpopulation ofeach genotype will attain a population size determined bythe carrying capacity of the environment for that genotype.Another feature of this model is that, in contrast to mostmodels of classical branching processes, a set of nonlineardeterministic equations is embedded in the stochastic processand the analysis of this deterministic system is part of theanalysis of the model as a whole in computer simulationexperiments. Interestingly, at some points in the parameter

2 International Journal of Stochastic Analysis

space of the process, the embedded deterministic model willexhibit chaotic behavior, but examples with chaos were notconsidered in the paper.

Lastly in a paper by X. Wu and M. Kimmel, neutralevolution is modeled as an infinite allele Markov branchingprocesses. The main focus of attention in this paper is theproportion of alleles having a given number of copies atsome point in time, which is referred to as the frequencyspectrum by the authors. A formula for the variance ofthe frequency spectrum is also derived by the authors thatis useful in interval estimation and tests of hypotheses forparameter values. It is also shown that the mean of frequencyspectrum has an explicit form expressed in terms of ahyper-geometric function. A formula for the asymptotic rateof convergence is also derived. A technique used in thederivation of the formulas involves probability generatingfunctions. Simulations are also used to illustrate some of theresults that were derived from the birth and death process.

Charles J. ModeRick Durrett

Fima KlebanerPeter Olofsson

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