Mathematical Models in Population Biology and Epidemiology

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Format: Hardcover
Pub. Date: 2001-03-01
Publisher(s): Springer Verlag
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Summary

This book is an introduction to the principles and practice of mathematical modeling in the biological sciences, concentrating on applications in population biology, epidemiology, and resource management. The core of the book covers models in these areas and the mathematics useful in analyzing them, including case studies representing real-life situations. The emphasis throughout is on describing the mathematical results and showing students how to apply them to biological problems while highlighting some modeling strategies. A large number and variety of examples, exercises, and projects are included. Additional ideas and information may be found on a web site associated with the book.Senior undergraduates and graduate students as well as scientists in the biological and mathematical sciences will find this book useful.Carlos Castillo-Chavez is professor of biomathematics in the departments of biometrics, statistics, and theoretical and applied mechanics at Cornell University and a member of the graduate fields of applied mathematics, ecology and evolutionary biology, and epidemiology. H is the recepient of numerous awards including two White House Awards (1992 and 1997) and QEM Giant in Space Mentoring Award (2000).Fred Brauer is a Professor Emeritus of Mathematics at the University id Wisconsin, where he taught from 1960 to 1999, and has also been an Honorary Professor of Mathematics at the University of British Columbia since 1997.

Author Biography

Carlos Castillo-Chavez is professor of biomathematics in the departments of biometrics, statistics, and theoretical and applied mechanics at Cornell University.

Table of Contents

Series Preface v
Preface vii
Acknowledgements xi
Prologue xvii
I Simple Single Species Models 1(124)
Continuous Population Models
3(48)
Exponential Growth
3(5)
The Logistic Population Model
8(5)
The Logistic Equation in Epidemiology
13(4)
Qualitative Analysis
17(11)
Harvesting in Population Models
28(4)
Constant Yield Harvesting
28(1)
Constant Effort Harvesting
29(3)
Eutrophication of a Lake: A Case Study
32(8)
Appendix: Parameters in Biological Systems
40(5)
The Spruce Budworm
45(3)
Estimating the Population of the U.S.A
48(3)
Discrete Population Models
51(44)
Introduction: Linear Models
51(4)
Graphical Solution of Difference Equations
55(3)
Equilibrium Analysis
58(6)
Period-Doubling and Chaotic Behavior
64(7)
Discrete Time Metered Models
71(3)
A Two-Age Group Model and Delayed Recruitment
74(6)
Systems of Two Difference Equations
80(3)
Oscillation in Flour Beetle Populations: A Case Study
83(7)
A Discrete SIS Epidemic Model
90(2)
A Discrete Time Two-Sex Pair Formation Model
92(3)
Continuous Single-Species Population Models with Delays
95(30)
Introduction
95(3)
Models with Delay in Per Capita Growth Rates
98(4)
Delayed Recruitment Models
102(7)
Models with Distributed Delay
109(4)
Harvesting in Delayed Recruitment Models
113(4)
Constant Effort Harvesting
113(1)
Constant Yield Harvesting
114(3)
Nicholson's Blowflies: A Case Study
117(4)
A Model for Blood Cell Populations
121(4)
II Models for Interacting Species 125(148)
Introduction and Mathematical Preliminaries
127(44)
The Lotka-Volterra Equations
127(4)
The Chemostat
131(1)
Equilibria and Linearization
132(9)
Qualitative Behavior of Solutions of Linear Systems
141(13)
Periodic Solutions and Limit Cycles
154(9)
Appendix: Canonical Forms of 2 x 2 Matrices
163(2)
A Model for Giving up Smoking
165(1)
A Model for Retraining of Workers by their Peers
166(1)
A Continuous Two-sex Population Model
167(4)
Continuous Models for Two Interacting Populations
171(60)
Species in Competition
171(9)
Predator-prey Systems
180(12)
Laboratory Populations: Two Case Studies
192(4)
Kolmogorov Models
196(3)
Mutualism
199(7)
The Spruce Budworm: A Case Study
206(7)
The Community Matrix
213(4)
The Nature of Interactions Between Species
217(3)
Invading Species and Coexistence
220(2)
Example: A Predator and Two Competing Prey
222(4)
Example: Two Predators Competing for Prey
226(1)
A Simple Neuron Model
227(4)
Harvesting in two-species models
231(42)
Harvesting of species in competition
231(6)
Harvesting of Predator-Prey Systems
237(9)
Intermittent Harvesting of Predator-Prey Systems
246(4)
Some Economic Aspects of Harvesting
250(6)
Optimization of Harvesting Returns
256(4)
Justification of the Optimization Result
260(3)
A Nonlinear Optimization Problem
263(6)
Economic Interpretation of the Maximum Principle
269(4)
III Structured Populations Models 273(98)
Basic Ideas of Mathematical Epidemiology
275(64)
Introduction
275(6)
A Simple Epidemic Model
281(7)
A Model for Diseases with No Immunity
288(4)
Models with Demographic Effects
292(10)
Disease as Population Control
302(7)
Infective Periods of Fixed Length
309(6)
A Model with a Fixed Period of Temporary Immunity
315(3)
Arbitrarily Distributed Infective Periods
318(3)
Directions for Generalization
321(5)
Pulse Vaccination
326(2)
A Model with Competing Disease Strains
328(3)
An Epidemic Model in Two Patches
331(1)
Population Growth and Epidemics
332(7)
Models for Populations with Age Structure
339(32)
Linear Discrete Models
339(7)
Linear Continuous Models
346(8)
Nonlinear Continuous Models
354(7)
Numerical Methods for the McKendrick-Von Foerster Model
361(10)
A Numerical Scheme for the McKendrick-Von Foerster Model
363(8)
Epilogue 371(2)
IV Appendix 373(14)
A Answered to Selected Exercises
375(12)
References 387(22)
Index 409

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