Many textbooks in mathematical biology are “large M” “small b” books that tend to treat the biology as an interesting and valuable application of mathematics. This is not necessarily a bad thing — particularly when it’s used to show mathematics students part of the wide universe of applications. The approach is less well-suited for an audience primarily interested in the biology; there it looks too much like a fixation on the tools instead of what you do with them.
Modeling Infectious Disease in Humans and Animals presents more of a balance. It is an introduction to real-time and predictive modeling of infectious disease intended primarily for health care professionals, epidemiologists and evolutionary biologists. It focuses on directly transmitted infection by microparasites (viruses, bacteria, protozoa and prions) where there is extensive long-term data and a good understanding of the transmission dynamics. This is only a small portion of the whole field of epidemiological modeling and analysis, but it is an area where substantial progress has been made over the last two decades.
The book begins with a very nice introductory discussion of the types and characteristics of diseases. Here it becomes clear that modeling infectious disease is not a “one size fits all” operation. This also establishes the context and scope of the authors’ presentation. The second chapter introduces simple epidemic models, and it does so in a way that is considerably more nuanced than comparable treatments. Most often authors will discuss the basic susceptible-infected-recovered (SIR) model, perhaps examine a few examples, and then move on. In such treatments, it’s difficult to see what implicit assumptions have been made, so it’s unclear when the model is indeed applicable. In this book, the authors look carefully at the model with several different sets of assumptions. These include inclusion or exclusion of births, deaths and migrations, density- and frequency-dependent transmission of infection, as well as models without immunity, with waning immunity, or with a latent period.
Succeeding chapters treat more complex environments. A chapter on host heterogeneities describes models appropriate to populations where distinct groups have different susceptibilities to catch and transmit an infection. A chapter on multi-pathogen, multi-host models treats diseases that can be caught and transmitted by numerous host species. Such models may be critical for the prediction of worldwide influenza strains and the possibility of pandemics.
Other topics treated here are temporally forced models (when diseases are subject to periodic forcing by some environmental variable), stochastic dynamics and spatial models (dealing with spatial variations of infected populations and the propagation of infections across geographic regions). A final chapter on controlling infectious diseases describes how the models from the previous chapters can be used to optimize control measures in order to minimize the spread of infection.
Somewhat surprisingly, the authors do not address data analysis or any related statistical issues. This is disappointing because – especially in this area of epidemiology that is rich in good data — the study of modeling is much enhanced when challenged with real data.
Many of the epidemiological problems addressed in this book are analytically intractable. The authors have provided a website with software (in Java, C, Fortran and Matlab) to allow students to explore at length the models presented in the text or to develop new models. This book includes no exercises, and, if used in as a course text, would need supplementing with problem sets and project suggestions. Probably its best use in a mathematics class would be as background or supplementary reading.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.