Mathematical Models in Population Biology and Epidemiology

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Springer Science & Business Media, Mar 30, 2001 - Science - 417 pages
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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.
 

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Contents

Prologue
xvii
Simple Single Species Models
1
Continuous Population Models
3
12 The Logistic Population Model
8
13 The Logistic Equation in Epidemiology
13
14 Qualitative Analysis
17
15 Harvesting in Population Models
28
152 Constant Effort Harvesting
29
54 Kolmogorov Models
196
55 Mutualism
199
A Case Study
206
57 The Community Matrix
213
58 The Nature of Interactions Between Species
217
59 Invading Species and Coexistence
220
A Predator and Two Competing Prey
222
Two Predators Competing for Prey
226

A Case Study
32
Parameters in Biological Systems
40
The Spruce Budworm
45
Estimating the Population of the USA
48
Discrete Population Models
51
22 Graphical Solution of Difference Equations
55
23 Equilibrium Analysis
58
24 PeriodDoubling and Chaotic Behavior
64
25 Discrete Time Metered Models
71
26 A TwoAge Group Model and Delayed Recruitment
74
27 Systems of Two Difference Equations
80
A Case Study
83
A Discrete SIS Epidemic Model
90
A Discrete Time TwoSex Pair Formation Model
92
Continuous SingleSpecies Population Models with Delays
95
32 Models with Delay in Per Capita Growth Rates
98
33 Delayed Recruitment Models
102
34 Models with Distributed Delay
109
35 Harvesting in Delayed Recruitment Models
113
352 Constant Yield Harvesting
114
A Case Study
117
A Model for Blood Cell Populations
121
Models for Interacting Species
125
Introduction and Mathematical Preliminaries
127
42 The Chemostat
131
43 Equilibria and Linearization
132
44 Qualitative Behavior of Solutions of Linear Systems
141
45 Periodic Solutions and Limit Cycles
154
Canonical Forms of 2 x 2 Matrices
163
A Model for Giving up Smoking
165
A Model for Retraining of Workers by their Peers
166
A Continuous Twosex Population Model
167
Continuous Models for Two Interacting Populations
171
52 Predatorprey Systems
180
Two Case Studies
192
A Simple Neuron Model
227
Harvesting in twospecies models
231
62 Harvesting of PredatorPrey Systems
237
63 Intermittent Harvesting of PredatorPrey Systems
246
64 Some Economic Aspects of Harvesting
250
65 Optimization of Harvesting Returns
256
66 Justification of the Optimization Result
260
67 A Nonlinear Optimization Problem
263
68 Economic Interpretation of the Maximum Principle
269
Structured Populations Models
273
Basic Ideas of Mathematical Epidemiology
275
72 A Simple Epidemic Model
281
73 A Model for Diseases with No Immunity
288
74 Models with Demographic Effects
292
75 Disease as Population Control
302
76 Infective Periods of Fixed Length
309
77 A Model with a Fixed Period of Temporary Immunity
315
78 Arbitrarily Distributed Infective Periods
318
79 Directions for Generalization
321
Pulse Vaccination
326
A Model with Competing Disease Strains
328
An Epidemic Model in Two Patches
331
Population Growth and Epidemics
332
Models for Populations with Age Structure
339
82 Linear Continuous Models
346
83 Nonlinear Continuous Models
354
84 Numerical Methods for the McKendrickVon Foerster Model
361
841 A Numerical Scheme for the McKendrick Von Foerster Model
363
Epilogue
371
Appendix
373
A Answers to Selected Exercises
375
References
387
Index
409
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About the author (2001)

Fred Brauer is a Professor Emeritus at the University of Wisconsin, Madison and an Honorary Professor at the University of British Columbia.

Carlos Castillo-Chavez is a Regents and a Joaquin Bustoz, Jr Professor at Arizona State University (ASU), a member of the Santa Fe Institute's external faculty and an adjunct professor at Cornell University.

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