My goal in writing this book is to provide an introduction to the basic theory of stochastic processes and to some of the many biological applications of stochastic processes. The mathematical and biological background required is kept to a minimum so that the topics are accessible to students and scientists in biology, mathematics, and engineering. Many of the biological applications are from the areas of population dynamics and epidemiology due to personal preference and expertise and because these applications can be readily understood. The topics in this book are covered in a one-semester graduate course offered by the Department of Mathematics and Statistics at Texas Tech University. This book is intended for an introductory course in stochastic processes. The targeted audiences for this book are advanced undergraduate students and beginning graduate students in mathematics, statistics, biology, or engineering. The level of material in this book requires only a basic background in probability theory, linear algebra, and analysis. Measure theory is not required.
Exercises at the end of each chapter help reinforce concepts discussed in each chapter. To better visualize and understand the dynamics of various stochastic processes, students are encouraged to use the MAT'LAB programs provided in the Appendices. These programs can be modified for other types of processes or adapted to other programming languages. In addition, research on current stochastic biological models in the literature can be assigned as individual or group research projects. The book is organized according to the following three types of stochastic processes: discrete time Markov chains, continuous time Markov chains and continuous time and state Markov processes. Because many biological phenomena can be modeled by one or more of these three modeling approaches, there may be different stochastic models for the same biological phenomena, e.g. logistic growth and epidemics.
Biological applications are presented in each chapter. Some chapters and sections are devoted entirely to the discussion of biological applications and their analysis (e.g., Chapter 7). In Chapter 1, topics from probability theory are briefly reviewed which are particularly relevant to stochastic processes. In Chapters 2 and 3, the theory and biological applications of discrete time Markov chains are discussed, including the classical gambler's ruin problem, birth and death processes and epidemic processes. In Chapter 4, the topic of branching process is discussed, a discrete time Markov chain important to applications in biology and medicine. An application to an age-structured population is discussed in Chapter 4.
Chapters 5, 6, and 7 present the theory and biological applications of continuous time Markov chains. Chapter 6 concentrates on birth and death processes and in Chapter 7 there are applications to epidemic, competition, predation and population genetics processes. The last chapter, Chapter 8, is a brief introduction to continuous time and continuous state, Markov processes; that is, diffusion processes and stochastic differential equations. Chapter 8 is a non-measure theoretic introduction to stochastic differential equations. These eight chapters can be covered in a one-semester course. One may be selective about the particular applications covered, particularly in Chapters 3, 7, and 8. In addition, Section 1.6 on the simple birth process and Section 2.
10 on the random wa.