## Diff. Equa. & Dyna. Sys.

#### Modelling Disease Ecology with Mathematics

By Robert Smith?

ISBN-10: 1-60133-004-9

ISBN-13: 978-1-60133-004-8

The book is a self-contained introduction to the basics of mathematics for students and researchers in the areas of biology, epidemiology, medicine and public health. It provides an overview of basic modelling, data-fitting and the tangled issue that is the basic reproductive ratio. Diseases covered include malaria, yellow fever, measles and AIDS.

This monograph views mathematics through the lens of real-world problems in infectious disease:

1. Mathematical models representing current diseases are formulated and analysed in an easy-to-follow manner.

2. MATLAB exercises provide the reader with the ability to develop control strategies, test hypothetical interventions and explore disease-management options.

This monograph is especially suited to those without a background in mathematics, who are interested in learning about the way that mathematics can organise, analyse and enlighten when tackling biological problems in disease control and management.

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Reviews

Professor Fred Brauer

Mathematical epidemiology at UBC, Canada

University of Wisconsin, USA

Most books or portions of books dealing with mathematical epidemiology assume that readers have a mathematical background that includes a working knowledge of calculus, differential equations, and matrix algebra. However, many public health scientists and epidemiologists do not have this knowledge at their fingertips and have at best a past acquaintance with these topics. For them, a study of mathematical epidemiology from existing books would require a review of mathematics that might well discourage them from beginning.

Here is a new book assuming an acquaintance with a basic level of calculus but recognizes that it may require refreshing. It gives an introduction to the main ideas of mathematical epidemiology with review of mathematical topics as they arise but without going into esoteric mathematical refinements.

After a brief introduction to the idea of mathematical modelling (Chapter 1), disease transmission models with recovery both with and without immunity and with and without births and deaths are motivated and described, and a mainly qualitative analysis is given (Chapters 2 - 3). The basic reproduction number is introduced and various approaches are given to its calculation (Chapter 4). The next topic is a description of vector disease models such as yellow fever (Chapter 5). Chapter 6, on the spread of measles, introduces the idea of diffusion and partial differential equation models, and the more mathematical Chapter 7 gives some methods for solving partial differential equations.

The next two chapters are devoted to questions of fitting curves to data (choosing parameter values) and approximation. These important topics are normally omitted from more mathematical treatments of mathematical epidemiology.

Chapters 10 and 11 discuss discrete models and the possible complicated behaviour that they may display. Chapter 12 gives an application of the material developed previously to AIDS and end ?stage renal disease, including analysis and interpretation of experimental data, and the text concludes with a model for malaria with delay, giving an opportunity for the introduction of differential ?difference equation models.

Appendices include some mathematical topics that may be unfamiliar to readers, a solution of the end –stage renal disease model, and an introduction to Matlab. Matlab is a computer algebra system sophisticated enough to be useful for solutions of matrix algebra problems, differential equations and equations with delay, and is introduced for the numerical solution of exercise in each chapter. The sample programs given in the text together with the introduction in the appendix will allow readers to develop enough knowledge of Matlab to be able to write their own programs.

Throughout the book the emphasis is on examples with clear explanations rather than on completeness of coverage. The level is such that readers need not be frightened by the mathematics. The goal is that they will be convinced that mathematical modelling is useful for them, and the hope is that some may be sufficiently hooked on the subject to learn more, and perhaps even to want to learn more of the mathematics involved. No other book tries to do this, and this book should be a good stepping stone to more advanced books.

Linda Pelude MSc

Canadian Nosocomial Infection Surveillance Program

Public Health Agency of Canada/Agence de la sant?publique du Canada

1431-100 Eglantine Driveway, Ottawa, ON, K1A 0K9

613 946 3386

www.phac-aspc.gc.ca/nois-sinp/

Modelling Disease Ecology with Mathematics offers a gentle introduction into mathematical modelling to those non-mathematicians in the area of public health such as policy analysts, epidemiologists and clinicians

Overall, the author does a superb job of addressing a diverse audience. The writing style is straightforward. Topics ranging from simple epidemic models to solving partial differential equations (and more) are presented clearly while equation components and model descriptions are well defined. Each chapter builds on knowledge gained in the previous chapter. Practical examples of real world problems are given at the end of each chapter are worked through in their entirety followed by actual exercises which provide hands-on experience modelling outcomes using a computer program called Matlab.

The focus on the usefulness and practical applications of mathematical modelling as they relate to the spread of disease is what makes this book a must read for those non-mathematicians in the public health field. Not only do you come away with an increased understanding of mathematics, modelling and a computer program used by many mathematicians but you realize the importance of developing strong links between mathematical modellers and those in the public health field so that we can understand each other and potentially devise better interventions to manage the spread of disease in our communities.

Pauline van den Driessche

Department of Mathematics and Statistics, University of Victoria

Victoria, BC Canada

This volume admirably fulfils the author’s primary purpose of using infectious disease modeling as a framework in which to explore the usefulness of mathematical modeling. Biologists and public health researchers will benefit from the clear explanations of the modeling process and details of some of the mathematical topics used. The volume is laid out in an accessible format, serving as a link between biologists and mathematicians and an aid in communication between the two groups. Specifically, each of the Chapters 2-13 begins with a flow chart outlining the flow of ideas to be presented, and also states what the reader should know by the end of the chapter. After the main part of a chapter, some computer lab work illustrated with Matlab programs is given, followed by some exercises for the] reader. A valuable overview of Matlab is given in an appendix. The structure of the volume thus lends itself well to self-study or a workshop environment, as well as a more traditional course.

Chapter 2 sets the scene by constructing simple epidemic models with continuous time, yielding ordinary differential equations. Equilibrium points and their stability are examined, and this theme is continued in Chapter 3 where the important threshold parameter R0 is introduced. This parameter plays a pivotal role in designing disease control strategies. Chapters 4 and 5 deal with vector borne diseases, for example, yellow fever and malaria. The models developed build on those of previous chapters, with the calculation of R0 requiring some tools from matrix algebra, which are clearly described in some appendices. Spatial spread of disease is introduced in Chapters 6 and 7, and the reader is given an appreciation of the added complexity. It is good to see the inclusion of chapters on fitting curves to data (Chapter 8) illustrated with data from epidemics, and splines (Chapter 9). In Chapters 10 and 11, time is taken as a discrete variable, yielding difference equation models. Stability of equilibria and possible bifurcations are briefly discussed. The final two chapters consider applications, modeling AIDS and end-stage renal disease incorporating treatment with antiretroviral drugs, and extending the malaria model to include time delays for incubation in humans and mosquitoes. Some mathematical details used in the volume are expanded as appendices.

After working through this volume, a reader will have a broad understanding of deterministic modeling of infectious diseases, and should be able to read current articles and discuss disease control strategies. This volume is highly recommended for public health workers seeking to facilitate communication with mathematicians and for mathematics students interested in an effective introduction to disease modeling.

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