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
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
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.
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 ProcessDiscrete-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 DimensionsBiological Applications of Discrete-Time Markov ChainsIntroduction Proliferating Epithelial Cells Restricted Random Walk Models Random Walk with Absorbing BoundariesRandom Walk on a Semi-Infinite Domain General Birth and Death Process Logistic Growth Process Quasistationary Probability Distribution SIS Epidemic ModelChain Binomial Epidemic ModelsDiscrete-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 ProcessesContinuous-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 CharacteristicsContinuous-Time Birth and Death Chains Introduction General Birth and Death Process Stationary Probability Distribution Simple Birth and Death ProcessesQueueing Process Population Extinction First Passage TimeLogistic Growth Process Quasistationary Probability Distribution An Explosive Birth Process Nonhomogeneous Birth and Death ProcessBiological Applications of Continuous-Time Markov Chains Introduction Continuous-Time Branching Processes SI and SIS Epidemic ProcessesMultivariate Processes Enzyme KineticsSIR Epidemic ProcessCompetition ProcessPredator-Prey ProcessDiffusion Processes and Stochastic Differential Equations Introduction Definitions and Notation Random Walk and Brownian Motion Diffusion Process Kolmogorov Differential Equations Wiener Process It� Stochastic Integral It� Stochastic Differential Equation (SDE)First Passage Time Numerical Methods for SDEs An Example: Drug KineticsBiological Applications of Stochastic Differential Equations Introduction Multivariate Processes Derivation of It� SDEs Scalar It� SDEs for Populations Enzyme Kinetics SIR Epidemic Process Competition Process Predator-Prey Process Population Genetics ProcessAppendix: Hints and Solutions to Selected ExercisesIndex