Book Review ?Modelling Disease Ecology with Mathematics by Robert Smith?
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.