Stochastic Population and Epidemic Models

Persistence and Extinction
Author: Linda J. S. Allen
Publisher: Springer
ISBN: 331921554X
Category: Mathematics
Page: 47
View: 1893
This monograph provides a summary of the basic theory of branching processes for single-type and multi-type processes. Classic examples of population and epidemic models illustrate the probability of population or epidemic extinction obtained from the theory of branching processes. The first chapter develops the branching process theory, while in the second chapter two applications to population and epidemic processes of single-type branching process theory are explored. The last two chapters present multi-type branching process applications to epidemic models, and then continuous-time and continuous-state branching processes with applications. In addition, several MATLAB programs for simulating stochastic sample paths are provided in an Appendix. These notes originated as part of a lecture series on Stochastics in Biological Systems at the Mathematical Biosciences Institute in Ohio, USA. Professor Linda Allen is a Paul Whitfield Horn Professor of Mathematics in the Department of Mathematics and Statistics at Texas Tech University, USA.

Controlled Branching Processes

Author: Miguel González Velasco,Inés MarÃa Del Puerto GarcÃa,George Petrov Yanev
Publisher: John Wiley & Sons
ISBN: 1119484642
Category: Mathematics
Page: 232
View: 637
The purpose of this book is to provide a comprehensive discussion of the available results for discrete time branching processes with random control functions. The independence of individuals’ reproduction is a fundamental assumption in the classical branching processes. Alternatively, the controlled branching processes (CBPs) allow the number of reproductive individuals in one generation to decrease or increase depending on the size of the previous generation. Generating a wide range of behaviors, the CBPs have been successfully used as modeling tools in diverse areas of applications.

Mathematical Models in Population Biology and Epidemiology

Author: Fred Brauer,Dawn Bies
Publisher: Springer Science & Business Media
ISBN: 1475735162
Category: Science
Page: 417
View: 5216
The goal of this book is to search for a balance between simple and analyzable models and unsolvable models which are capable of addressing important questions on population biology. Part I focusses on single species simple models including those which have been used to predict the growth of human and animal population in the past. Single population models are, in some sense, the building blocks of more realistic models -- the subject of Part II. Their role is fundamental to the study of ecological and demographic processes including the role of population structure and spatial heterogeneity -- the subject of Part III. This book, which will include both examples and exercises, is of use to practitioners, graduate students, and scientists working in the field.

Branching Process Models of Cancer

Author: Richard Durrett
Publisher: Springer
ISBN: 3319160656
Category: Mathematics
Page: 63
View: 8101
This volume develops results on continuous time branching processes and applies them to study rate of tumor growth, extending classic work on the Luria-Delbruck distribution. As a consequence, the author calculate the probability that mutations that confer resistance to treatment are present at detection and quantify the extent of tumor heterogeneity. As applications, the author evaluate ovarian cancer screening strategies and give rigorous proofs for results of Heano and Michor concerning tumor metastasis. These notes should be accessible to students who are familiar with Poisson processes and continuous time Markov chains. Richard Durrett is a mathematics professor at Duke University, USA. He is the author of 8 books, over 200 journal articles, and has supervised more than 40 Ph.D students. Most of his current research concerns the applications of probability to biology: ecology, genetics and most recently cancer.

Modeling Biomolecular Networks in Cells

Structures and Dynamics
Author: Luonan Chen,Ruiqi Wang,Chunguang Li,Kazuyuki Aihara
Publisher: Springer Science & Business Media
ISBN: 9781849962148
Category: Technology & Engineering
Page: 343
View: 1286
Modeling Biomolecular Networks in Cells shows how the interaction between the molecular components of basic living organisms can be modelled mathematically and the models used to create artificial biological entities within cells. Such forward engineering is a difficult task but the nonlinear dynamical methods espoused in this book simplify the biology so that it can be successfully understood and the synthesis of simple biological oscillators and rhythm-generators made feasible. Such simple units can then be co-ordinated using intercellular signal biomolecules. The formation of such man-made multicellular networks with a view to the production of biosensors, logic gates, new forms of integrated circuitry based on "gene-chips" and even biological computers is an important step in the design of faster and more flexible "electronics". The book also provides theoretical frameworks and tools with which to analyze the nonlinear dynamical phenomena which arise from the connection of building units in a biomolecular network.

Stochastic Models for Structured Populations

Scaling Limits and Long Time Behavior
Author: Sylvie Meleard,Vincent Bansaye
Publisher: Springer
ISBN: 3319217119
Category: Mathematics
Page: 107
View: 3150
In this contribution, several probabilistic tools to study population dynamics are developed. The focus is on scaling limits of qualitatively different stochastic individual based models and the long time behavior of some classes of limiting processes. Structured population dynamics are modeled by measure-valued processes describing the individual behaviors and taking into account the demographic and mutational parameters, and possible interactions between individuals. Many quantitative parameters appear in these models and several relevant normalizations are considered, leading to infinite-dimensional deterministic or stochastic large-population approximations. Biologically relevant questions are considered, such as extinction criteria, the effect of large birth events, the impact of environmental catastrophes, the mutation-selection trade-off, recovery criteria in parasite infections, genealogical properties of a sample of individuals. These notes originated from a lecture series on Structured Population Dynamics at Ecole polytechnique (France). Vincent Bansaye and Sylvie Méléard are Professors at Ecole Polytechnique (France). They are a specialists of branching processes and random particle systems in biology. Most of their research concerns the applications of probability to biodiversity, ecology and evolution.

Mathematical Epidemiology

Author: Fred Brauer,Pauline van den Driessche,J. Wu
Publisher: Springer Science & Business Media
ISBN: 3540789103
Category: Medical
Page: 414
View: 4203
Based on lecture notes of two summer schools with a mixed audience from mathematical sciences, epidemiology and public health, this volume offers a comprehensive introduction to basic ideas and techniques in modeling infectious diseases, for the comparison of strategies to plan for an anticipated epidemic or pandemic, and to deal with a disease outbreak in real time. It covers detailed case studies for diseases including pandemic influenza, West Nile virus, and childhood diseases. Models for other diseases including Severe Acute Respiratory Syndrome, fox rabies, and sexually transmitted infections are included as applications. Its chapters are coherent and complementary independent units. In order to accustom students to look at the current literature and to experience different perspectives, no attempt has been made to achieve united writing style or unified notation. Notes on some mathematical background (calculus, matrix algebra, differential equations, and probability) have been prepared and may be downloaded at the web site of the Centre for Disease Modeling (

An Introduction to Mathematical Biology

Author: Linda J. S. Allen
Publisher: Prentice Hall
ISBN: 9780130352163
Category: Mathematics
Page: 348
View: 2209
KEY BENEFIT: This reference introduces a variety of mathematical models for biological systems, and presents the mathematical theory and techniques useful in analyzing those models. Material is organized according to the mathematical theory rather than the biological application. Contains applications of mathematical theory to biological examples in each chapter. Focuses on deterministic mathematical models with an emphasis on predicting the qualitative solution behavior over time. Discusses classical mathematical models from population , including the Leslie matrix model, the Nicholson-Bailey model, and the Lotka-Volterra predator-prey model. Also discusses more recent models, such as a model for the Human Immunodeficiency Virus - HIV and a model for flour beetles. KEY MARKET: Readers seeking a solid background in the mathematics behind modeling in biology and exposure to a wide variety of mathematical models in biology.

Probabilistic Models of Population Evolution

Scaling Limits, Genealogies and Interactions
Author: Étienne Pardoux
Publisher: Springer
ISBN: 3319303287
Category: Mathematics
Page: 125
View: 4402
This expository book presents the mathematical description of evolutionary models of populations subject to interactions (e.g. competition) within the population. The author includes both models of finite populations, and limiting models as the size of the population tends to infinity. The size of the population is described as a random function of time and of the initial population (the ancestors at time 0). The genealogical tree of such a population is given. Most models imply that the population is bound to go extinct in finite time. It is explained when the interaction is strong enough so that the extinction time remains finite, when the ancestral population at time 0 goes to infinity. The material could be used for teaching stochastic processes, together with their applications. Étienne Pardoux is Professor at Aix-Marseille University, working in the field of Stochastic Analysis, stochastic partial differential equations, and probabilistic models in evolutionary biology and population genetics. He obtained his PhD in 1975 at University of Paris-Sud.

Mathematics in Population Biology

Author: Horst R. Thieme
Publisher: Princeton University Press
ISBN: 0691187657
Category: Science
Page: N.A
View: 8535

Mathematics of Epidemics on Networks

From Exact to Approximate Models
Author: István Z. Kiss,Joel C. Miller,Péter L. Simon
Publisher: Springer
ISBN: 3319508067
Category: Mathematics
Page: 413
View: 1630
This textbook provides an exciting new addition to the area of network science featuring a stronger and more methodical link of models to their mathematical origin and explains how these relate to each other with special focus on epidemic spread on networks. The content of the book is at the interface of graph theory, stochastic processes and dynamical systems. The authors set out to make a significant contribution to closing the gap between model development and the supporting mathematics. This is done by: Summarising and presenting the state-of-the-art in modeling epidemics on networks with results and readily usable models signposted throughout the book; Presenting different mathematical approaches to formulate exact and solvable models; Identifying the concrete links between approximate models and their rigorous mathematical representation; Presenting a model hierarchy and clearly highlighting the links between model assumptions and model complexity; Providing a reference source for advanced undergraduate students, as well as doctoral students, postdoctoral researchers and academic experts who are engaged in modeling stochastic processes on networks; Providing software that can solve differential equation models or directly simulate epidemics on networks. Replete with numerous diagrams, examples, instructive exercises, and online access to simulation algorithms and readily usable code, this book will appeal to a wide spectrum of readers from different backgrounds and academic levels. Appropriate for students with or without a strong background in mathematics, this textbook can form the basis of an advanced undergraduate or graduate course in both mathematics and other departments alike.

Mathematical Biology

An Introduction with Maple and Matlab
Author: Ronald W. Shonkwiler,James Herod
Publisher: Springer Science & Business Media
ISBN: 0387709843
Category: Science
Page: 551
View: 8202
This text presents mathematical biology as a field with a unity of its own, rather than only the intrusion of one science into another. The book focuses on problems of contemporary interest, such as cancer, genetics, and the rapidly growing field of genomics.

Modeling with Itô Stochastic Differential Equations

Author: E. Allen
Publisher: Springer Science & Business Media
ISBN: 1402059531
Category: Mathematics
Page: 230
View: 2938
This book explains a procedure for constructing realistic stochastic differential equation models for randomly varying systems in biology, chemistry, physics, engineering, and finance. Introductory chapters present the fundamental concepts of random variables, stochastic processes, stochastic integration, and stochastic differential equations. These concepts are explained in a Hilbert space setting which unifies and simplifies the presentation.

Mathematical Demography

Selected Papers
Author: David P. Smith,Nathan Keyfitz
Publisher: Springer Science & Business Media
ISBN: 3642358586
Category: Social Science
Page: 335
View: 8000
Mathematical demography is the centerpiece of quantitative social science. The founding works of this field from Roman times to the late Twentieth Century are collected here, in a new edition of a classic work by David R. Smith and Nathan Keyfitz. Commentaries by Smith and Keyfitz have been brought up to date and extended by Kenneth Wachter and Hervé Le Bras, giving a synoptic picture of the leading achievements in formal population studies. Like the original collection, this new edition constitutes an indispensable source for students and scientists alike, and illustrates the deep roots and continuing vitality of mathematical demography.

Dynamical Systems and Population Persistence

Author: Hal L. Smith,Horst R. Thieme
Publisher: American Mathematical Soc.
ISBN: 082184945X
Category: Mathematics
Page: 405
View: 484
"The mathematical theory of persistence answers questions such as which species, in a mathematical model of interacting species, will survive over the long term. It applies to infinite-dimensional as well as to finite-dimensional dynamical systems, and to discrete-time as well as to continuous-time semiflows. This monograph provides a self-contained treatment of persistence theory that is accessible to graduate students. The key results for deterministic autonomous systems are proved in full detail such as the acyclicity theorem and the tripartition of a global compact attractor. Suitable conditions are given for persistence to imply strong persistence even for nonautonomous semiflows, and time-heterogeneous persistence results are developed using so-called 'average Lyapunov functions'. Applications play a large role in the monograph from the beginning. These include ODE models such as an SEIRS infectious disease in a meta-population and discrete-time nonlinear matrix models of demographic dynamics. Entire chapters are devoted to infinite-dimensional examples including an SI epidemic model with variable infectivity, microbial growth in a tubular bioreactor, and an age-structured model of cells growing in a chemostat."--Publisher's description.

Branching Processes

Variation, Growth, and Extinction of Populations
Author: Patsy Haccou,Peter Jagers,Vladimir A. Vatutin
Publisher: Cambridge University Press
ISBN: 9780521832205
Category: Mathematics
Page: 316
View: 5874
This book covers the mathematical idea of branching processes, and tailors it for a biological audience.

Epidemic Models

Their Structure and Relation to Data
Author: Denis Mollison
Publisher: Cambridge University Press
ISBN: 9780521475365
Category: Mathematics
Page: 424
View: 6669
The problems of understanding and controlling disease present a range of mathematical challenges, from broad theoretical issues to specific practical ones, making epidemiology one of the most vibrant branches of applied ecology. Progress in this field requires interdisciplinary collaboration; leading researchers with a wide range of mathematical expertise and close involvement in applied fields across the social, medical and biological sciences came together for a NATO Advanced Research Workshop marking the opening of a six-month programme on Epidemic Models at the Newton Institute in Cambridge in 1993. This volume is a result of that collaboration and surveys the state of epidemic modelling at the time in relation to basic aims such as understanding, prediction, and evaluation and implementation of control strategies.

Stochastic Geometry, Spatial Statistics and Random Fields

Models and Algorithms
Author: Volker Schmidt
Publisher: Springer
ISBN: 3319100645
Category: Mathematics
Page: 464
View: 7384
This volume is an attempt to provide a graduate level introduction to various aspects of stochastic geometry, spatial statistics and random fields, with special emphasis placed on fundamental classes of models and algorithms as well as on their applications, e.g. in materials science, biology and genetics. This book has a strong focus on simulations and includes extensive codes in Matlab and R which are widely used in the mathematical community. It can be seen as a continuation of the recent volume 2068 of Lecture Notes in Mathematics, where other issues of stochastic geometry, spatial statistics and random fields were considered with a focus on asymptotic methods.

An Introduction to Mathematical Epidemiology

Author: Maia Martcheva
Publisher: Springer
ISBN: 1489976124
Category: Mathematics
Page: 453
View: 2515
The book is a comprehensive, self-contained introduction to the mathematical modeling and analysis of infectious diseases. It includes model building, fitting to data, local and global analysis techniques. Various types of deterministic dynamical models are considered: ordinary differential equation models, delay-differential equation models, difference equation models, age-structured PDE models and diffusion models. It includes various techniques for the computation of the basic reproduction number as well as approaches to the epidemiological interpretation of the reproduction number. MATLAB code is included to facilitate the data fitting and the simulation with age-structured models.

Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases

Author: Piero Manfredi,Alberto D'Onofrio
Publisher: Springer Science & Business Media
ISBN: 1461454743
Category: Mathematics
Page: 329
View: 5717
This volume summarizes the state-of-the-art in the fast growing research area of modeling the influence of information-driven human behavior on the spread and control of infectious diseases. In particular, it features the two main and inter-related “core” topics: behavioral changes in response to global threats, for example, pandemic influenza, and the pseudo-rational opposition to vaccines. In order to make realistic predictions, modelers need to go beyond classical mathematical epidemiology to take these dynamic effects into account. With contributions from experts in this field, the book fills a void in the literature. It goes beyond classical texts, yet preserves the rationale of many of them by sticking to the underlying biology without compromising on scientific rigor. Epidemiologists, theoretical biologists, biophysicists, applied mathematicians, and PhD students will benefit from this book. However, it is also written for Public Health professionals interested in understanding models, and to advanced undergraduate students, since it only requires a working knowledge of mathematical epidemiology.