2nd Edition

An Introduction to Stochastic Processes with Applications to Biology

By Linda J. S. Allen Copyright 2010
    496 Pages 80 B/W Illustrations
    by Chapman & Hall

    An Introduction to Stochastic Processes with Applications to Biology, Second Edition presents the basic theory of stochastic processes necessary in understanding and applying stochastic methods to biological problems in areas such as population growth and extinction, drug kinetics, two-species competition and predation, the spread of epidemics, and the genetics of inbreeding. Because of their rich structure, the text focuses on discrete and continuous time Markov chains and continuous time and state Markov processes.

    New to the Second Edition

    • A new chapter on stochastic differential equations that extends the basic theory to multivariate processes, including multivariate forward and backward Kolmogorov differential equations and the multivariate Itô’s formula
    • The inclusion of examples and exercises from cellular and molecular biology
    • Double the number of exercises and MATLAB® programs at the end of each chapter
    • Answers and hints to selected exercises in the appendix
    • Additional references from the literature

    This edition continues to provide an excellent introduction to the fundamental theory of stochastic processes, along with a wide range of applications from the biological sciences. To better visualize the dynamics of stochastic processes, MATLAB programs are provided in the chapter appendices.

    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 Xn
    Probability of Population Extinction
    Mean and Variance of Xn
    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

    Exercises and References appear at the end of each chapter.

    Biography

    Linda J.S. Allen is a Paul Whitfield Horn Professor in the Department of Mathematics and Statistics at Texas Tech University. Dr. Allen has served on the editorial boards of the Journal of Biological Dynamics, SIAM Journal of Applied Mathematics, Journal of Difference Equations and Applications, Journal of Theoretical Biology, and Mathematical Biosciences. Her research interests encompass mathematical population biology, epidemiology, and immunology.

    "This book provides an excellent introduction to the basic theory of stochastic processes with regard to applications in biology. … In this edition a new chapter on stochastic differential equations was added."
    —Franziska Wandtner, Zentralblatt MATH 1263

    "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. … 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, MAA Reviews, January 2012

    "… a good introductory textbook for junior graduate students who are interested in mathematical biology. … First, this book is written in plain language so students with a basic probability background can easily grasp the material. … the author obviously understands well the level of knowledge of junior graduate students so the depth of concepts is finely controlled. Second, this book covers a rich set of selected topics with a clear focus on Markov-type processes. … Third, it must be mentioned that the author has made a great effort to encourage the use of stochastic models in practice by providing many pieces of MATLAB codes, which are usually unavailable in other books on stochastic processes. Finally, compared with the previous edition, this newly released version particularly extends the stochastic differential equation part by including the multivariate Kolmogorov equations and the Itô formula."
    —Hongyu Miao, Mathematical Reviews, Issue 2011m