Allen An Introduction to Stochastic Processes with Applications to Biology, Second Edition

2. Auflage 2011
ISBN: 978-1-4398-9468-2
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 496 Seiten

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.

Zielgruppe

Graduate students and professionals in mathematics, engineering, and biology; electrical, mechanical, and bio engineers.

Autoren/Hrsg.

Allen, Linda J. S.

Fachgebiete

• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
• Mathematik | Informatik Mathematik Stochastik Bayesianische Inferenz

Inhaltsverzeichnis

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.

Über Autor(innen)

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.