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Publisher:

Chapman & Hall/CRC

Publication Date:

2011

Number of Pages:

466

Format:

Hardcover

Edition:

2

Price:

89.95

ISBN:

9781439818824

Category:

Textbook

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by , on ]

Kathy Temple

01/17/2012

*An Introduction to Stochastic Processes with Applications to Biology *offers a fairly standard treatment of non-measure-theoretic stochastic processes, with a substantial number of applications to biology. The topics covered include the standard material on discrete and continuous-time Markov chains, as well as two chapters on diffusions and stochastic differential equations. The chief contribution of the book is its large selection of biological examples worked-out and discussed in the text itself and in the exercises.

I can see two possible audiences for this book: the suggested usage, which is as a text in an advanced undergraduate/beginning graduate course for students in mathematics and related fields, and those who know something about stochastic processes but would like to expand their acquaintance with biological examples. For my own set of undergraduates, the book would be a bit terse as a stand-alone text (it covers far more material than I could cover in a one-term course). Motivated undergraduates and beginning graduate students would find it accessible, however.

Although there is a quick review of probability theory in the first chapter, the book is probably best suited to students who already have some probability background. The chapters on diffusions and stochastic differential equations are noticeably more difficult than the earlier material, although that is probably more the fault of the subject than the author — stochastic calculus is a bit mysterious under the best of circumstances. The treatment is heuristic rather than rigorous, since the book doesn’t assume a familiarity with measure theory.

Instructors who are already teaching a stochastic processes course and want to introduce biological examples will find this book to be a gold mine of useful material. In addition to the examples presented along with the exposition of the theory, chapters three and seven consist entirely of biological examples. A warning, however, to those who would like to use the book in this way: the transition matrices used for discrete-time Markov chains are the transposes of the standard matrices, and the notation for the one-step transition probabilities reverses the originating and destination states. According to the author, this brings the notation more in line with that used for deterministic matrix models. If you are just reading the examples, perhaps being forewarned will save you a few minutes of confusion!

The warning in the previous paragraph hints at my one small comment on the book: there are sections where material is presented in what may be the most logical way for someone used to discrete models, but is not the “natural” method for probabilists. For instance, the derivation of the mean and variance of a discrete-time branching process are done via a differential equation for the cumulant generating function, rather than the standard conditioning argument (which is done afterwards for means but not for variances). Brownian motion is introduced by deriving the forward Kolmogorov differential equation from random walks, rather than constructing it pathwise from random walks. The path properties of Brownian motion are discussed, but not until a couple of sections later.

These are very small quibbles, however. In general, the book will be a useful addition to the library of anyone interested in stochastic processes who wants to learn more about their biological applications. I certainly learned a great deal from it!

Kathy Temple teaches probability and statistics to future actuaries at Central Washington University. She knows a fair bit about stochastic processes, but her last formal biology course was in high school.

**Review of Probability Theory and an Introduction to Stochastic Processes **

Introduction

Brief Review of Probability Theory

Generating Functions

Central Limit Theorem

Introduction to Stochastic Processes

An Introductory Example: A Simple Birth Process

**Discrete-Time Markov Chains **

Introduction

Definitions and Notation

Classification of States

First Passage Time

Basic Theorems for Markov Chains

Stationary Probability Distribution

Finite Markov Chains

An Example: Genetics Inbreeding Problem

Monte Carlo Simulation

Unrestricted Random Walk in Higher Dimensions

**Biological Applications of Discrete-Time Markov Chains**

Introduction

Proliferating Epithelial Cells

Restricted Random Walk Models

Random Walk with Absorbing Boundaries

Random Walk on a Semi-Infinite Domain

General Birth and Death Process

Logistic Growth Process

Quasistationary Probability Distribution

SIS Epidemic Model

Chain Binomial Epidemic Models

**Discrete-Time Branching Processes **

Introduction

Definitions and Notation

Probability Generating Function of *X*_{n }

Probability of Population Extinction

Mean and Variance of *X _{n}*

Environmental Variation

Multitype Branching Processes

**Continuous-Time Markov Chains **

Introduction

Definitions and Notation

The Poisson Process

Generator Matrix *Q *

Embedded Markov Chain and Classification of States

Kolmogorov Differential Equations

Stationary Probability Distribution

Finite Markov Chains

Generating Function Technique

Interevent Time and Stochastic Realizations

Review of Method of Characteristics

**Continuous-Time Birth and Death Chains **

Introduction

General Birth and Death Process

Stationary Probability Distribution

Simple Birth and Death Processes

Queueing Process

Population Extinction

First Passage Time

Logistic Growth Process

Quasistationary Probability Distribution

An Explosive Birth Process

Nonhomogeneous Birth and Death Process

**Biological Applications of Continuous-Time Markov Chains **

Introduction

Continuous-Time Branching Processes

SI and SIS Epidemic Processes

Multivariate Processes

Enzyme Kinetics

SIR Epidemic Process

Competition Process

Predator-Prey Process

**Diffusion Processes and Stochastic Differential Equations **

Introduction

Definitions and Notation

Random Walk and Brownian Motion

Diffusion Process

Kolmogorov Differential Equations

Wiener Process

Itô Stochastic Integral

Itô Stochastic Differential Equation (SDE)

First Passage Time

Numerical Methods for SDEs

An Example: Drug Kinetics

**Biological Applications of Stochastic Differential Equations **

Introduction

Multivariate Processes

Derivation of Itô SDEs

Scalar Itô SDEs for Populations

Enzyme Kinetics

SIR Epidemic Process

Competition Process

Predator-Prey Process

Population Genetics Process

**Appendix: Hints and Solutions to Selected Exercises**

**Index**

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