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An Introduction to Stochastic Processes with Applications to Biology

Chapman & Hall/CRC
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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.

Date Received: 
Saturday, January 8, 2011
Include In BLL Rating: 
Linda J. S. Allen
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Kathy Temple
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Review of Probability Theory and an Introduction to Stochastic Processes
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
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
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
Definitions and Notation
Probability Generating Function of Xn
Probability of Population Extinction
Mean and Variance of Xn
Environmental Variation
Multitype Branching Processes

Continuous-Time Markov Chains
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
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
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
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
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


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Thursday, May 31, 2012