Machine generated contents note: -- 1.
Review of Probability Theory and an Introduction to Stochastic Processes -- -- 1.1.
Introduction -- -- 1.2.
Brief Review of Probability Theory -- -- 1.2.1.
Basic Probability Concepts -- -- 1.2.2.
Probability Distributions -- -- 1.2.3.
Expectation -- -- 1.2.4.
Multivariate Distributions -- -- 1.3.
Generating Functions -- -- 1.4.
Central Limit Theorem -- -- 1.5.
Introduction to Stochastic Processes -- -- 1.6.
An Introductory Example: A Simple Birth Process -- -- 1.7.
Exercises for Chapter 1 -- -- 1.8.
References for Chapter 1 -- -- 1.9.
Appendix for Chapter 1 -- -- 1.9.1.
Probability Distributions -- -- 1.9.2.
MATLAB® and FORTRAN Programs -- -- 1.9.3.
Interevent Time -- -- 2.
Discrete-Time Markov Chains -- -- 2.1.
Introduction -- -- 2.2.
Definitions and Notation -- -- 2.3.
Classification of States -- -- 2.4.
First Passage Time -- -- 2.5.
Basic Theorems for Markov Chains -- -- 2.6.
Stationary Probability Distribution
2.7.
Finite Markov Chains -- -- 2.7.1.
Mean First Passage Time -- -- 2.8.
An Example: Genetics Inbreeding Problem -- -- 2.9.
Monte Carlo Simulation -- -- 2.10.
Unrestricted Random Walk in Higher Dimensions -- -- 2.10.1.
Two Dimensions -- -- 2.10.2.
Three Dimensions -- -- 2.11.
Exercises for Chapter 2 -- -- 2.12.
References for Chapter 2 -- -- 2.13.
Appendix for Chapter 2 -- -- 2.13.1.
Proofs of Theorems 2.5 and 2.6 -- -- 2.13.2.
Perron and Frobenius Theorems -- -- 2.13.3.
The n-Step Transition Matrix -- -- 2.13.4.
Genetics Inbreeding Problem -- -- 3.
Biological Applications of Discrete-Time Markov Chains -- -- 3.1.
Introduction -- -- 3.2.
Proliferating Epithelial Cells -- -- 3.3.
Restricted Random Walk Models -- -- 3.4.
Random Walk with Absorbing Boundaries -- -- 3.4.1.
Probability of Absorption -- -- 3.4.2.
Expected Time until Absorption -- -- 3.4.3.
Probability Distribution for Absorption -- -- 3.5.
Random Walk on a Semi-Infinite Domain -- -- 3.6.
General Birth and Death Process -- -- 3.6.1.
Expected Time to Extinction
3.7.
Logistic Growth Process -- -- 3.8.
Quasistationary Probability Distribution -- -- 3.9.
SIS Epidemic Model -- -- 3.9.1.
Deterministic Model -- -- 3.9.2.
Stochastic Model -- -- 3.10.
Chain Binomial Epidemic Models -- -- 3.10.1.
Greenwood Model -- -- 3.10.2.
Reed-Frost Model -- -- 3.10.3.
Duration and Size -- -- 3.11.
Exercises for Chapter 3 -- -- 3.12.
References for Chapter 3 -- -- 3.13.
Appendix for Chapter 3 -- -- 3.13.1.
MATLAB® Programs -- -- 3.13.2.
Maple™ Program -- -- 4.
Discrete-Time Branching Processes -- -- 4.1.
Introduction -- -- 4.2.
Definitions and Notation -- -- 4.3.
Probability Generating Function of Xn -- -- 4.4.
Probability of Population Extinction -- -- 4.5.
Mean and Variance of Xn -- -- 4.6.
Environmental Variation -- -- 4.7.
Multitype Branching Processes -- -- 4.7.1.
An Example: Age-Structured Model -- -- 4.7.2.
Environmental Variation -- -- 4.8.
Exercises for Chapter 4 -- -- 4.9.
References for Chapter 4 -- -- 5.
Continuous-Time Markov Chains -- -- 5.1.
Introduction
5.2.
Definitions and Notation -- -- 5.3.
The Poisson Process -- -- 5.4.
Generator Matrix Q -- -- 5.5.
Embedded Markov Chain and Classification of States -- -- 5.6.
Kolmogorov Differential Equations -- -- 5.7.
Stationary Probability Distribution -- -- 5.8.
Finite Markov Chains -- -- 5.9.
Generating Function Technique -- -- 5.10.
Interevent Time and Stochastic Realizations -- -- 5.11.
Review of Method of Characteristics -- -- 5.12.
Exercises for Chapter 5 -- -- 5.13.
References for Chapter 5 -- -- 5.14.
Appendix for Chapter 5 -- -- 5.14.1.
Calculation of the Matrix Exponential -- -- 5.14.2.
MATLAB® Programs -- -- 6.
Continuous-Time Birth and Death Chains -- -- 6.1.
Introduction -- -- 6.2.
General Birth and Death Process -- -- 6.3.
Stationary Probability Distribution -- -- 6.4.
Simple Birth and Death Processes -- -- 6.4.1.
Simple Birth -- -- 6.4.2.
Simple Death -- -- 6.4.3.
Simple Birth and Death -- -- 6.4.4.
Simple Birth and Death with Immigration -- -- 6.5.
Queueing Process -- -- 6.6.
Population Extinction
6.7.
First Passage Time -- -- 6.7.1.
Definition and Computation -- -- 6.7.2.
Summary of First Passage Time -- -- 6.8.
Logistic Growth Process -- -- 6.9.
Quasistationary Probability Distribution -- -- 6.10.
An Explosive Birth Process -- -- 6.11.
Nonhomogeneous Birth and Death Process -- -- 6.12.
Exercises for Chapter 6 -- -- 6.13.
References for Chapter 6 -- -- 6.14.
Appendix for Chapter 6 -- -- 6.14.1.
Generating Functions for the Simple Birth and Death Process -- -- 6.14.2.
Proofs of Theorems 6.2 and 6.3 -- -- 6.14.3.
Comparison Theorem -- -- 7.
Biological Applications of Continuous-Time Markov Chains -- -- 7.1.
Introduction -- -- 7.2.
Continuous-Time Branching Processes -- -- 7.3.
SI and SIS Epidemic Processes -- -- 7.3.1.
Stochastic SI Model -- -- 7.3.2.
Stochastic SIS Model -- -- 7.4.
Multivariate Processes -- -- 7.5.
Enzyme Kinetics -- -- 7.5.1.
Deterministic Model -- -- 7.5.2.
Stochastic Model -- -- 7.6.
SIR Epidemic Process -- -- 7.6.1.
Deterministic Model -- -- 7.6.2.
Stochastic Model -- -- 7.6.3.
Final Size
7.6.4.
Duration -- -- 7.7.
Competition Process -- -- 7.7.1.
Deterministic Model -- -- 7.7.2.
Stochastic Model -- -- 7.8.
Predator-Prey Process -- -- 7.8.1.
Deterministic Model -- -- 7.8.2.
Stochastic Model -- -- 7.9.
Exercises for Chapter 7 -- -- 7.10.
References for Chapter 7 -- -- 7.11.
Appendix for Chapter 7 -- -- 7.11.1.
MATLAB® Programs -- -- 8.
Diffusion Processes and Stochastic Differential Equations -- -- 8.1.
Introduction -- -- 8.2.
Definitions and Notation -- -- 8.3.
Random Walk and Brownian Motion -- -- 8.4.
Diffusion Process -- -- 8.5.
Kolmogorov Differential Equations -- -- 8.6.
Wiener Process -- -- 8.7.
Ito Stochastic Integral -- -- 8.8.
Ito Stochastic Differential Equation -- -- 8.9.
First Passage Time -- -- 8.10.
Numerical Methods for SDEs -- -- 8.11.
An Example: Drug Kinetics -- -- 8.12.
Exercises for Chapter 8 -- -- 8.13.
References for Chapter 8 -- -- 8.14.
Appendix for Chapter 8 -- -- 8.14.1.
Derivation of Kolmogorov Equations -- -- 8.14.2.
MATLAB® Program -- -- 9.
Biological Applications of Stochastic Differential Equations
9.1.
Introduction -- -- 9.2.
Multivariate Processes -- -- 9.3.
Derivation of Ito SDEs -- -- 9.4.
Scalar Ito SDEs for Populations -- -- 9.4.1.
Simple Birth and Death with Immigration -- -- 9.4.2.
Logistic Growth -- -- 9.4.3.
Quasistationary Density Function -- -- 9.5.
Enzyme Kinetics -- -- 9.6.
SIR Epidemic Process -- -- 9.7.
Competition Process -- -- 9.8.
Predator-Prey Process -- -- 9.9.
Population Genetics Process -- -- 9.10.
Exercises for Chapter 9 -- -- 9.11.
References for Chapter 9 -- -- 9.12.
Appendix for Chapter 9 -- -- 9.12.1.
MATLAB® Programs -- -- Appendix A
Hints and Solutions to Selected Exercises -- -- A.1.
Chapter 1 -- -- A.2.
Chapter 2 -- -- A.3.
Chapter 3 -- -- A.4.
Chapter 4 -- -- A.5.
Chapter 5 -- -- A.6.
Chapter 6 -- -- A.7.
Chapter 7 -- -- A.8.
Chapter 8 -- -- A.9.
Chapter 9.