An Introduction to Stochastic Processes with Applications to Biology, Second Edition / Edition 2

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Overview

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.

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Editorial Reviews

From the Publisher
"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

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Product Details

  • ISBN-13: 9781439818824
  • Publisher: Taylor & Francis
  • Publication date: 12/7/2010
  • Edition description: New Edition
  • Edition number: 2
  • Pages: 490
  • Product dimensions: 6.20 (w) x 9.40 (h) x 1.40 (d)

Meet the Author

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.

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Table of Contents

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.

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