Book review

# Mathematical Tools for Understanding Infectious Disease Dynamics. O. Diekmann, H. Heesterbeek, and T. Britton (2012). Princeton: Princeton University Press. 502 pages, ISBN: 978-0-691-1-5539-5.

Article first published online: 26 AUG 2013

DOI: 10.1002/bimj.201300132

© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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#### How to Cite

Kretzschmar, M. (2014), Mathematical Tools for Understanding Infectious Disease Dynamics. O. Diekmann, H. Heesterbeek, and T. Britton (2012). Princeton: Princeton University Press. 502 pages, ISBN: 978-0-691-1-5539-5. Biom. J., 56: 176–177. doi: 10.1002/bimj.201300132

#### Publication History

- Issue published online: 2 JAN 2014
- Article first published online: 26 AUG 2013

- Abstract
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The preface of the book quotes Picasso with the statement that “art is the lie that helps us discover the truth”. This quote—applied to mathematical modeling—characterizes the philosophy of the book and the view the authors take on mathematical modeling as a tool in infectious disease epidemiology. In contrast to a large proportion of the ever-increasing modeling literature, Diekmann, Heesterbeek, and Britton approach mathematical modeling of infectious diseases not from the data-driven point of view; their aim is rather to gain an understanding of the mechanisms underlying the dynamics of spread of infections in populations. Therefore, their starting point is a clear formulation of the basic assumptions about biological processes in mathematical terminology. Then, reasoning and analytic mathematical tools are used to obtain insight and understanding of how biological mechanisms at the individual level lead to observed phenomena on the population level. Reality is dissected into its building blocks and stripped from all complexities that cloud our vision of how it works. In a section entitled “The challenge of reality”, the authors concede that dealing with reality is not always easy and making the link between insight gained from modeling and observed phenomena may not be straightforward.

This book is a reworked and extended edition of the earlier volume “Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation” by Diekmann and Heesterbeek (2000). The main change, with respect to the earlier book, is the addition of the stochastic perspective contributed by the third author, Tom Britton. While the first book focused almost exclusively on deterministic theory, in this new edition, there are several chapters that treat epidemics from the stochastic viewpoint. Not only the basic theory of stochastic epidemics is explained, but also methods are introduced for estimation of parameters from data. These methods are then illustrated with some worked examples; in particular, the authors show ways to estimate the basic reproduction number from different types of data. However, the focus is not so much on application to particular data sets but on understanding the concepts used to extract information from available data.

The core of the book is the development of a theory for describing transmission dynamics in heterogeneous populations. Heterogeneity means that individuals within a population are not all the same, but they differ from each other in some aspects that impact on the way an infectious disease is transmitted and spreads through the population. The most obvious example is age of individuals, but it can also be contact patterns, immune status, behavior, geographical location, or anything that is usually measured also by epidemiologists. To deal with heterogeneity in terms of modeling, the authors distinguish between a description of possible states on the individual level (the so-called i-state) and a description on the population level (the p-state). The actual process of modeling takes place in describing the i-state, because here one has to decide about the biological mechanisms and processes that determine and change an individual's state during the course of his (infectious) lifetime. If the processes on the individual level are specified consistently, description of the p-state is then a question of correct bookkeeping, that is, of keeping track of how many individuals are in a specific state at a given time.

The authors use this way of thinking to build a theoretical framework for understanding transmission dynamics, where the basic reproduction number *R*_{0} occupies a central position. Defining *R*_{0} as the dominant eigenvalue of a next-generation matrix, or more generally, a next-generation operator, encapsulates many of the central ideas of how infectious diseases spread in heterogeneous populations. The next-generation operator specifies how one generation of infected individuals is mapped onto a subsequent generation taking into account who infects whom in a heterogeneous population. Its view of transmission in terms of generations of infected individuals links to infectious disease epidemiology, where outbreaks are investigated that start with one index case who causes a first generation of secondary cases; these new cases then infect a subsequent generation of cases and so on. The next-generation framework is then detailed for describing age-structured populations, populations stratified by contact rates, and other more intricate forms of heterogeneity.

An important ingredient of the book is the way practical exercises are used to present substantial parts of the content. The practical exercises are not simply problems or questions to practice the theory laid out before, but they contain steps in derivation of theory or applications of general theory to specific subproblems, chopped up into pieces that the diligent reader is expected to master on his own. Ideally, reading this book is therefore not done in a passive mode of consuming, but in an active mode of participation in deriving and understanding the basic insights. Many exercises are based on articles from published literature, in which one or the other specific problem was worked out. So the message to potential readers is that when reading this book there is no time to lean back, but to get immersed into the fascinating world of infection dynamics and the abstract framework built to understand it. Luckily, however, for those of you who are pressed for time, the solutions of the exercises are all given at the end of the book.

There are, by now, a number of textbooks on mathematical models of infectious diseases, notably the classics by Bailey (1975) and Anderson and May (1991), and the more recent books by Keeling and Rohani (2008), and Vynnycky and White (2010). All of them have their merits, but none of them contains such a lucid and well-composed mathematical exposition of what the “bare bones” of infectious disease transmission actually are. Everybody whose aim is to understand the driving forces of infectious disease epidemiology—in other words “daß ich erkenne, was die Welt im Innersten zusammenhält”, to speak with Goethe's Faust—should study this book. This book will soon be a classic in the theoretical epidemiology and modeling literature.

### References

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- 2000). Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. John Wiley & Sons, Chichester. and (
- 2008). Modeling Infectious Diseases in Humans and Animals. Princeton University Press, Princeton. and (
- 2010). An Introduction to Infectious Disease Modeling. Oxford University Press, Oxford. and (