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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 Ito's formula
The inclusion of examples and exercises from cellular and molecular biology
Double the number of exercises and MATLAB (R) 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. "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 Ito formula."
-Hongyu Miao, Mathematical Reviews, Issue 2011m

An Introduction to Stochastic Processes with Applications to Biology

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

Ito Stochastic Integral

Ito 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 Ito SDEs

Scalar Ito 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.