Dynamic Models in Biology. Stephen P. Ellner and John Guckenheimer. Princeton University Press, Princeton, NJ, 2006. 330 pp. $99.50 (ISBN 0691118434 paper).
Why should biologists be interested in mathematical modeling? Never at a loss for an anecdote, the late eminent biomathematician Lee Segel loved to quote Picasso: “Art is the lie that helps us see the truth,” and, Lee quipped, “the same can be said for mathematical modelling.” His assessment certainly proved true. Over the last 10 years we have witnessed dramatic changes in biological research in terms of its dependence on the quantitative sciences. In some university corridors, it is even possible to hear whispers of “the New Biology,” which, according to one informed view, is made up of approximately one-third statistics, mathematics, and computer science; one-third physics, chemistry, and engineering; and one-third traditional biological sciences. The impact of the New Biology may be gauged by the recent estimate that 30 percent of all high-performance computing worldwide is now dedicated to biological analyses. The new experimental setups, huge data sets, statistical analyses, and modeling approaches are having their effect. Many of the brightest young scientists are following the excitement and moving into the areas of biophysics, nanotechnology, bioinformatics, and biomathematics.
Unfortunately, these changes have hardly filtered into the undergraduate curriculum at most universities, yet there is an obvious need to train and prepare the next generation's young scientists. To help fill this growing gap, Stephen P. Ellner (Department of Ecology) and John Guckenheimer (Department of Mathematics), both highly respected mathematical biologists at Cornell University, have published their new book Dynamic Models in Biology.
The book is based on an interdisciplinary course given by the authors to a heterogeneous group of undergraduate students majoring in the biological and exact sciences, including medicine, computer sciences, mathematics, and engineering. Anyone wanting to bring together students from such wide-ranging backgrounds will find this no easy feat. It is in fact a major teaching challenge. The authors' strategy is to maintain a careful balance between the mathematics and biology presented, while following a “business school” model that gives an in-depth treatment of a selective set of case studies. Working through particular case studies allows students to come to grips with nitty-gritty aspects of biology that might push math students to their limits, while exposing biology students to areas of mathematics they may not otherwise have encountered, and possibly inspiring them to learn more in the process. The case studies chosen span major areas in biology, with chapter topics that include structured population models, membrane channels and action potentials, cellular dynamics and simple gene networks, infectious diseases, spatial patterns in biology, and agent-based models of digital evolution. In addition, there are several technical chapters conveying important background material on dynamical systems and the art of building biological models.
As part of their strategy, the authors pair up very different biological case studies that are amenable to analysis with similar mathematical frameworks. For example, chapter 2 introduces matrix models of structured populations, a good home base for ecologists or biologists who are likely to have encountered parts of this material previously. The chapter walks the reader through essential mathematical concepts that include projection matrices, eigenvalues, left and right eigenvectors, the Perron–Frobenius theorem, stable age distributions, and eigenvalue sensitivity (elasticity). These same concepts are heavily drawn on in the paired chapter 3, which moves on to the study of gating in membrane channels via Markov chain matrix models. Now, however, the probabilistic transition matrix replaces the projection matrix, and the right eigenvector and Perron–Frobenius theorem are used to calculate residence times rather than stable age population distributions. The pairing of chapters can be exploited by the course instructor to reinforce learning, or, alternatively, the instructor has the freedom to skip one of the pairs without loss of material required later in the book.
It becomes clear that the chapters are written by true specialists who have a deep knowledge of the subject matter and an extensive and up-to-date awareness of the literature. A good example is the chapter dealing with modeling infectious diseases, which gives a wonderful overview of the field. The first 17 pages deal with the basic textbook theory covering the classical SIR (susceptible, infectious, recovered) epidemic model (with and without the birth/death process), model scaling and dimensionless variables, the reproductive rate R0, force of infection, and the model's natural oscillations, plus a little bit of history. These topics are dealt with in a linear and very readable fashion, and followed by a set of challenging computer exercises.
The remaining 15 pages move into some lesser-known terrain, where simple models lay bare (a) the role of core groups in sexually transmitted diseases (STDs) and methods for controlling disease spread; (b) the dynamics of drug resistance of infectious diseases such as tuberculosis and HIV; and (c) within-host dynamics of HIV and its T-cell targets. Each section is conveyed cleverly and compellingly. The section on STDs, for example, brings to light a paradox introduced by Hethcote and Yorke, namely, why does gonorrhea demonstrate long-term persistence when it shows all the signs of being on the brink of extinction? The reader learns that this can be resolved by introducing the concept of a core group, and it takes the authors a matter of seconds to set up a simple, elegant model that manages to illuminate exactly how this is done. The deep insights gained make the power of dynamic modeling directly evident.
The book's business school approach to modeling comes at a price, in that a large amount of course material accumulates as unrelated and weighty case studies are introduced and dealt with in succession. As a result, the material outlined in the book is more than can be covered in a one-semester course. The authors suggest guidelines for different course variants based on subsets of chapter combinations that should be realistic over a semester. Because of the amount of material covered in the book, the pace is sometimes uneven. Certain concepts are dwelt on at length, while others are necessarily covered too briefly and will need more careful preparation by any instructor planning to lecture on the material in class. However, overall, the book is well organized and well written, and the authors have a captivating style that keeps the reader interested and tuned in. This is facilitated by the witticisms scattered through the book, with the authors admonishing the reader with such warnings as “Thou Shalt Not Extrapolate Beyond the Range of Thy Data” or invoking fear of “the Curse of Dimensionality.”
Speaking from my own experience of teaching a similar course, students will especially enjoy the hands-on computer laboratories and exercises that have been prepared for the book. The authors provide a well-documented laboratory manual that comes in two versions, for Matlab or for R (freeware). The manuals have been carefully thought out and make it possible to learn to program with these powerful software packages even if beginning from scratch. The manuals ensure that students will be able to build and test their own dynamic models in minimal time.
Dynamic Models in Biology stands apart from existing textbooks in mathematical biology largely because of its interdisciplinary approach and its hands-on, project-oriented case studies and computer laboratories. In an effort to explore biology in more detail, the authors bravely choose a style that differs from the classical biomath texts of, say, Murray and Edelstein-Keshet, whose focus is more on formal mathematics. The success of a course built around Ellner and Guckenheimer's textbook will depend on the instructor's skill in assessing the diversity of the students' backgrounds and catering to their different needs, but the task will be far easier and more enjoyable with this well-crafted book as a guide.
