Stochastic processes and applications in biology and medicine

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Stochastic processes and applications in biology and medicine

M. Iosifescu, P. Tăutu

（Biomathematics, v.3-4）

Editura Academiei , Springer, 1973

  • v. 1. Theory
  • v. 2. Models

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Note

Based on Procese stohastice și aplicaţii in biologie și medicină published Burcurest, 1968

Includes bibliographies

Description and Table of Contents

Volume

v. 1. Theory ISBN 9783540062707

Description

This volume is a revised and enlarged version of Chapters 1 and 2 of a book with the same title, published in Romanian in 1968. The revision resulted in a new book which has been divided into two parts because of the large amount of new material. The present part is intended to introduce mathematicians and biologists with a strong mathematical and probabilistic background to the study of stochastic processes. We hope some readers will be able to discover by themselves the new features of our treatment such as the inclusion of some unusual topics, the special attention paid to some usual topics, and the grouping of the material. We draw the reader's attention to the numbering, because there are structural differences between the two parts. In Part I there are Chapters, Sections, Subsections, Paragraphs and Subparagraphs. Thus the numbering a. b. c. d. e refers to Subparagraph e of Paragraph d of Subsection c of Section b of Chapter a. Definitions, theorems lemmas and propositions are numbered a. b. n, n = 1,2, ..., where a indicates the chapter and b the section. In Part II there are Sections, Subsections, Paragraphs, and SUbparagraphs. Thus the numbering a. b. c. d refers to Subparagraph d of Paragraph c of Subsection b of Section a. Theorems and lemmas are numbered a. n, n = 1, 2, ..., where a indicates the section.

Table of Contents

1 Discrete parameter stochastic processes.- 1.1. Denumerable Markov chains.- 1.1.1. Preliminaries.- 1.1.1.1. Definition of a Markov chain.- 1.1.1.2. The existence theorem.- 1.1.1.3. The n-step transition probabilities.- 1.1.1.4. Strong Markov property.- 1.1.2. Classification of states.- 1.1.2.1. Return states and recurrent states.- 1.1.2.2. Regenerative phenomena.- 1.1.2.3. Positive states and limit theorems.- 1.1.2.4. Geometric ergodicity.- 1.1.2.5. Essential states and classes of states.- 1.1.2.6. Conditional motion.- 1.1.3. Taboo and stationarity.- 1.1.3.1. Taboo transition probabilities.- 1.1.3.2. Ratio limit theorems.- 1.1.3.3. Stationary distributions.- 1.1.3.4. Stationary measures.- 1.1.3.5. Integral representations.- 1.1.4. Finite state Markov chains.- 1.1.4.1. Specific properties.- 1.1.4.2. The matrix method.- 1.1.4.3. The Ehrenfest model.- 1.2. Noteworthy classes of denumerable Markov chains.- 1.2.1. Random walk.- 1.2.1.1. Homogeneous random walk.- 1.2.1.2. Some important special cases.- 1.2.1.3. Barriers.- 1.2.1.4. Generalizations.- 1.2.1.5. Integral representations for certain nonhomogeneous random walks.- 1.2.2. Galton-Watson chains.- 1.2.2.1. Basic properties.- 1.2.2.2. Extinction probability.- 1.2.2.3. Time to extinction.- 1.2.2.4. Stationary distributions and stationary measures.- 1.2.2.5. Spectral theory of Galton-Watson chains.- 1.2.2.6. Asymptotic properties.- 1.2.2.7. Multitype Galton-Watson chains.- 1.2.3. Markov chains occurring in queueing theory.- 1.2.3.1. Preliminaries.- 1.2.3.2. Queueing systems.- 1.2.3.3. An imbedded Markov chain in the M/G/1 queue.- 1.2.3.4. An imbedded Markov chain in the GI/M/1 queue.- 1.3. Markov chains with arbitrary state space.- 1.3.1. Preliminaries.- 1.3.1.1. Definition and existence theorem.- 1.3.1.2. Generalized n-step transition functions.- 1.3.2. Uniform ergodicity.- 1.3.2.1. Uniform strong ergodicity.- 1.3.2.2. Uniform weak ergodicity.- 1.3.3. The coefficient of ergodicity.- 1.3.3.1. Definition and properties.- 1.3.3.2. Application to uniform ergodicity.- 1.3.3.3. Some limit theorems.- 1.3.4. Compact Markov chains.- 1.3.4.1. Definition and properties.- 1.3.4.2. Random systems with complete connections.- References.- 2 Continuous parameter stochastic processes.- 2.1. Some general problems.- 2.1.1. Preliminaries.- 2.1.1.1. Definition of a stochastic process.- 2.1.1.2. Finite dimensional distributions.- 2.1.2. Basic concepts.- 2.1.2.1. Separability.- 2.1.2.2. Stochastic continuity and measurability.- 2.1.3. Trajectories.- 2.1.3.1. Generalities.- 2.1.3.2. Continuous trajectories.- 2.1.3.3. Trajectories without discontinuities of the second kind.- 2.1.4. Convergence of stochastic processes.- 2.1.4.1. Weak convergence of processes.- 2.1.4.2. The Prohorov theorem.- 2.2. Processes with independent increments.- 2.2.1. Preliminaries.- 2.2.1.1. Definition and existence theorem.- 2.2.1.2. Stochastic continuity.- 2.2.2. Basic processes with independent increments.- 2.2.2.1. The Poisson process.- 2.2.2.2. The Wiener process.- 2.2.2.3. Brownian motion.- 2.2.3. General properties.- 2.2.3.1. Integral decomposition.- 2.2.3.2. The three parts decomposition.- 2.3. Markov processes.- 2.3.1. Preliminaries.- 2.3.1.1. Transition functions.- 2.3.1.2. Definition and existence theorem.- 2.3.1.3. The strong Markov property.- 2.3.1.4. The semi-group approach to homogeneous Markov processes.- 2.3.2. Markov jump processes. I. General theory.- 2.3.2.1. Transition intensity functions.- 2.3.2.2. The Kolmogorov-Feller equations.- 2.3.2.3. Determining a transition function from its intensity.- 2.3.2.4. The minimal process.- 2.3.3. Markov jump processes. II. Discrete state space.- 2.3.3.1. The case of a finite state space.- 2.3.3.2. The case of a denumerable state space.- 2.3.3.3. Poisson processes as Markov jump processes.- 2.3.3.4. Markov branching processes.- 2.3.4. Homogeneous Markov jump processes with discrete state space.- 2.3.4.1. Preliminaries.- 2.3.4.2. Continuity and differentiability properties.- 2.3.4.3. The Kolmogorov differential equations.- 2.3.4.4. Continuous parameter regenerative phenomena.- 2.3.4.5. Properties of trajectories.- 2.3.4.6. Discrete skeletons and classification of states.- 2.3.4.7. Birth-and-death processes.- 2.3.5. Markov diffusion processes.- 2.3.5.1. Classical diffusion processes.- 2.3.5.2. The Kolmogorov equations.- 2.3.5.3. Approximations.- 2.3.5.4. Boundaries.- 2.3.5.5. Brownian motion as diffusion process.- 2.3.6. Extensions of Markov processes.- 2.3.6.1. Semi-Markov processes.- 2.3.6.2. Renewal processes.- References.- Notation index.- Author index.
Volume

v. 2. Models ISBN 9783540062714

Description

This volume is a revised and enlarged version of Chapter 3 of. a book with the same title, published in Romanian in 1968. The revision resulted in a new book which has been divided into two of the large amount of new material. The whole book parts because is intended to introduce mathematicians and biologists with a strong mathematical background to the study of stochastic processes and their applications in biological sciences. It is meant to serve both as a textbook and a survey of recent developments. Biology studies complex situations and therefore needs skilful methods of abstraction. Stochastic models, being both vigorous in their specification and flexible in their manipulation, are the most suitable tools for studying such situations. This circumstance deter- mined the writing of this volume which represents a comprehensive cross section of modern biological problems on the theory of stochastic processes. Because of the way some specific problems have been treat- ed, this volume may also be useful to research scientists in any other field of science, interested in the possibilities and results of stochastic modelling. To understand the material presented, the reader needs to be acquainted with probability theory, as given in a sound introductory course, and be capable of abstraction.

Table of Contents

0. Prolegomenon.- 1. Preliminary considerations.- 1.1. Stochastic and deterministic models in biology.- 1.1.1. Comparisons with deterministic models.- 1.1.2. Equipollent and conjunct models.- 1.2. The structure of biological populations.- 1.2.1. Elements of bio-logic.- 1.2.2. Distinguishability and indistinguishability.- 1.2.3. Equivalence relations and graphs.- 1.2.4. Descendants, generations and families.- 1.2.5. The temporal structure of populations and biological objects.- 1.2.6. Characteristic distributions for some biological populations.- 2. Population growth models.- 2.1. Stochastic population processes.- 2.1.1. Point processes as models of stochastic populations.- 2.1.2. Homogeneous birth-and-death processes.- 2.1.3. A random walk example.- 2.1.4. Birth, death and diffusion processes.- 2.2. Population processes in Euclidean space.- 2.2.1. Homogeneous spatial models.- 2.2.2. Birth, death and migration processes in R2 [and R3.- 2.3. Intrinsic processes.- 2.3.1. Multiple-phase processes.- 2.3.2. The life cycle process.- 2.3.3. The birth, death and marks transmission process.- 2.3.4. Interdependent (self)-replicating process.- 2.3.5. Point mutation processes.- 2.4. Stochastic demographic models.- 2.4.1. The discrete time model.- 2.4.2. A model related to human populations.- 2.4.3. Population growth of the sexes.- 2.4.4. The reproductive process.- 2.5. Other growth models and derived processes.- 2.5.1. The cumulative process.- 2.5.2. The Prendiville (logistic) process.- 2.5.3. Some problems of survival and extinction.- 3. Population dynamics processes.- 3.1. Some multi-dimensional Markov jump processes.- 3.1.1. Consael processes.- 3.1.2. A generalized w-dimensional (n ? 2) linear growth process.- 3.2. Immigration-emigration processes.- 3.2.1. The Kendall process.- 3.2.2. Immigration and emigration processes.- 3.2.3. Intermigration and colonization.- 3.2.4. Taxis and kinesis as dispersion processes.- 3.3. Competition processes.- 3.3.1. Some introductory remarks.- 3.3.2. Stochastic competition processes.- 3.3.3. Quasi-competition processes.- 3.3.4. Population excess and cannibalism.- 3.4. Poikilopoiesis models.- 3.4.1. A stochastic approach to embryogenesis.- 3.4.2. Hematopoiesis models.- 4. Evolutionary processes.- 4.1. Basic problems, models and methods.- 4.1.1. Paradigm for the stochastic evolutionary processes.- 4.1.2. Classical genetic stochastic models.- 4.1.3. Tendency to homozygosity.- 4.1.4. Diffusion approximations.- 4.1.5. Non-Mendelian situations.- 4.1.6. Direct product Galton-Watson chains.- 4.2. Random drift and systematic evolutionary processes.- 4.2.1. Random drift.- 4.2.2. Selection.- 4.2.3. Mutation.- 4.3. Problems of molecular genetics.- 4.3.1. Growing point of donor DNA attachment model.- 4.3.2. An example of a random system with complete connections.- 5. Models in physiology and pathology.- 5.1. Stochastic models in physiology.- 5.1.1. Models of the appearance and the transmission of the neural flux.- 5.1.2. Chemical mediation processes.- 5.1.3. Some problems of stochastic networks.- 5.1.4. A model of muscle contraction.- 5.1.5. Renewal processes in pharmacology.- 5.1.6. Multicompartment systems.- 5.2. Models in pathology.- 5.2.1. The process of infection.- 5.2.2. The clinical process.- 5.2.3. Stochastic models for tumour growth.- 5.2.4. Some stochastic aspects of chemotherapy.- 5.2.5. Competing risks of illness.- 5.2.6. Control of biological processes.- 5.3. Epidemic processes.- 5.3.1. The classical models.- 5.3.2. A general approach to epidemics.- 5.3.3. Epidemic Markov chains.- References.- Notation index.- Author index.

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