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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 thedynamics 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 walk in two and three dimensions are optional.
Numerous applications of stochastic processes important in areas outside of biology, including finance, economics, physics, chemistry, and engineering, can be found in the references. This book stresses biological applications and therefore, some topics in stochastic processes important to these other areas are omitted or discussed very briefly. For example, martingales are not discussed and queueing theory is only briefly discussed in Chapter 6.
Throughout this book, the emphasis is placed on Markov processes due to their rich structure and the numerous biological models satisfying the Markov property. However, there are also many biological applications where the Markov restriction does not apply. A stochastic process is discussed in Section 7.2, which is a non-Markovian, age-dependent process belonging to a class of stochastic processes known as regenerative processes. It is important to note that in some applications the Markov restriction is not necessary, e.g., first passage time in Chapter 2 and the waiting time distribution in Chapter 5. This latter theory can be discussed in the more general context of renewal theory.
In writing this book, I received much help and advice from colleagues and friends and I would like to acknowledge their contributions. First, I thank my husband, Edward Allen, for his careful proofreading of many drafts of this book, especially for his help with Chapter 8, and for his continuous encouragement throughout the long process of writing and rewriting. I thank Robert Paige, Texas Tech University, for reviewing Chapter 1 and Thomas Gard, University of Georgia, for reviewing Chapter 8 and for their numerous suggestions on these chapters. I am grateful to my graduate students, Nadarajah Kirupaharan and Keith Emmert for their help in checking for errors. Also, I am grateful to the students in the Biomathematics classes at Texas Tech University during the Spring Semesters of 2000 and 2002 for their feedback on the exercises. I thank Texas Tech University for granting me a leave of absence during the Spring Semester of 2001. During that time I organized my notes and wrote a preliminary draft of this book. In addition, I thank Prentice Hall Editor George Lobell for his advice, Adam Lewenberg for his assistance in formatting and setting the final page layout for this book, and the Prentice Hall reviewers for their many helpful comments and suggestions: Wei-Min Huang, Lehigh University, John Borkowski, Montana State University, Michael Neubauer, California State University at Northridge, and Aparna Huzurbazar, The University of New Mexico, reviewers for Chapters 1-3, Andrea Brose, UCLA, reviewer for Chapters 1-5, and Xiuli Chao, North Carolina State University, Bozenna Pasik-Duncan, University of Kansas, Magda Peligrad, University of Cincinnati, and Andre Adler, Illinois Institute of Technology, reviewers for Chapters 1-8. Many books, monographs, and articles were sources of reference in writing this book. These sources of reference are too numerous to mention here but are acknowledged in the list of references at the end of each chapter. Finally, and most importantly, I thank God for His constant support and guidance. After countless revisions based on suggestions from knowledgeable colleagues and friends, I assume full responsibility for any omissions and errors in this final draft.
Linda J. S. Allen