## Mathematical Models in Population Biology and EpidemiologyThis 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 |

387 | |

409 | |

### Other editions - View all

Mathematical Models in Population Biology and Epidemiology Fred Brauer,Carlos Castillo-Chavez Limited preview - 2013 |

Mathematical Models in Population Biology and Epidemiology Fred Brauer,Carlos Castillo-Chavez Limited preview - 2001 |

Mathematical Models in Population Biology and Epidemiology Fred Brauer,Carlos Castillo-Chavez Limited preview - 2011 |

### Common terms and phrases

approach assume assumption asymptotically stable asymptotically stable equilibrium basic reproductive number behavior of solutions biological birth rate budworm capita growth rate carrying capacity Castillo-Chavez characteristic equation chemostat coexistence community matrix competition computer algebra system condition consider constant critical depensation curve death rate decreases delay denotes depends describe determine difference equation differential equation differential-difference equation disease disease-free equilibrium dynamics eigenvalues endemic equilibrium epidemic epidemiological equilibrium is asymptotically eutrophic example Exercises exponential extinction Figure function gives immunity increases infective period initial value isocline limit cycle linear logistic equation logistic model Lotka-Volterra mathematical maximum maximum sustainable yield nonlinear obtain oligotrophic orbit tends oscillations periodic orbit phase plane population model population sizes positive possible predator predator-prey system prey isocline qualitative recruitment roots saddle point Section separatrices Show SIS model species survival susceptible tend to zero Theorem tion total population unstable variables xn+i

### Popular passages

Page xiv - We have received much support and encouragement in the preparation of this book. In particular we would like to thank...