**Mathematical Surveys and Monographs**

Volume: 102;
2003;
261 pp;
Hardcover

MSC: Primary 92;

Print ISBN: 978-0-8218-0499-5

Product Code: SURV/102

List Price: $80.00

Individual Member Price: $64.00

**Electronic ISBN: 978-1-4704-1329-3
Product Code: SURV/102.E**

List Price: $80.00

Individual Member Price: $64.00

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# Spatial Deterministic Epidemics

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*Linda Rass; John Radcliffe*

The study of epidemic models is one of the central topics of mathematical biology. This volume is the first to present in monograph form the rigorous mathematical theory developed to analyze the asymptotic behavior of certain types of epidemic models.

The main model discussed is the so-called spatial deterministic epidemic in which infected individuals are not allowed to again become susceptible, and infection is spread by means of contact distributions. Results concern the existence of traveling wave solutions, the asymptotic speed of propagation, and the spatial final size. A central result for radially symmetric contact distributions is that the speed of propagation is the minimum wave speed. Further results are obtained using a saddle point method, suggesting that this result also holds for more general situations.

Methodology, used to extend the analysis from one-type to multi-type models, is likely to prove useful when analyzing other multi-type systems in mathematical biology. This methodology is applied to two other areas in the monograph, namely epidemics with return to the susceptible state and contact branching processes.

This book presents an elegant theory that has been developed over the past quarter century. It will be useful to researchers and graduate students working in mathematical biology.

#### Table of Contents

# Table of Contents

## Spatial Deterministic Epidemics

- Table of Contents v6 free
- Preface vii8 free
- Chapter 1. Introduction 112 free
- Chapter 2. The non-spatial epidemic 920
- Chapter 3. Bounds on the spatial final size 2738
- Chapter 4. Wave solutions 5162
- 4.1 Specification and discussion 5162
- 4.2 The wave equations 5263
- 4.3 A discussion of the single population case 5768
- 4.4 Some preliminary results for the multi-type model 6879
- 4.5 The regions of convergence of certain transforms 7081
- 4.6 The characteristic equation 7687
- 4.7 Existence of waves at supercritical speeds 8192
- 4.8 Non-existence of waves at subcritical speeds 8596
- 4.9 Existence of waves at critical speed 8899
- 4.10 Uniqueness of waves modulo translation 92103

- Chapter 5. The asymptotic speed of propagation 99110
- 5.1 Some preliminaries 99110
- 5.2 The single population case 101112
- 5.3 An upper bound on the speed of propagation 108119
- 5.4 The critical speed as a lower bound 112123
- 5.5 The asymptotic speed of propagation 125136
- 5.6 Non-exponentially dominated contact distributions 125136
- 5.7 The pandemic theorem revisited 128139
- 5.8 The asymptotic shape 128139

- Chapter 6. An epidemic on sites 135146
- 6.1 A one-type finite site spatial model 135146
- 6.2 The multi-type finite site spatial model 136147
- 6.3 The infinite site spatial model 141152
- 6.4 The final size equations 142153
- 6.5 The pandemic theorem for the one-type case 145156
- 6.6 A matrix approach for the multi-type pandemic theorem 148159
- 6.7 The limit of the spatial final size 151162

- Chapter 7. The saddle point method 153164
- Chapter 8. Epidemics with return to the susceptible state 183194
- 8.1 Introduction . 183194
- 8.2 The non-spatial S → I → S model 184195
- 8.3 The open S → I → S model 184195
- 8.4 One and two type epidemics 185196
- 8.5 The equilibrium solutions of the multi-type epidemic 192203
- 8.6 The global asymptotic stability of the equilibria 194205
- 8.7 Epidemic models on a finite number of sites 199210
- 8.8 Saddle point results for spatial models 203214

- Chapter 9. Contact branching processes 207218
- 9.1 Introduction 207218
- 9.2 The simple contact birth process and the McKean connection 208219
- 9.3 The multi-type contact birth process 210221
- 9.4 Exact analytic results 212223
- 9.5 The multi-type contact birth-death process 221232
- 9.6 Saddle point results for birth-death processes with branching 222233

- Appendices 227238
- Bibliography 249260
- Index 255266

#### Readership

Graduate students and research mathematicians interested in mathematical biology.

#### Reviews

The material of the book is presented in full detail … a thorough account … with a nice and rather complete bibliography … a useful reference source for forthcoming research in the field.

-- Mathematical Reviews