An Introduction to Mathematical Epidemiology

Epidemiology studies the patterns of health and illness for whole populations. Once — before the twentieth century — epidemiology was primarily concerned with infectious diseases, but now its study incorporates a variety of other diseases such as coronary artery disease and stroke. This book, however, is largely concerned with the epidemiology of infectious disease.

While epidemiology has a long history reaching back to Hippocrates, the mathematical study of the spread of disease is only about 350 years old. It appears to have begun with a statistical study of disease and public health data by John Graunt about 1663. About a century later Daniel Bernoulli applied mathematical methods to analyze mortality rates from smallpox. He published what is probably the first epidemiological model.

Mathematical modeling of infectious disease made significant progress in the twentieth century. William Hamer, who was trying to understand the recurrence of measles, was apparently the first to apply the law of mass action (that the rate of infection is proportional to the number of contacts between those infected and those susceptible). Sir Ronald Ross — considered the father of mathematical epidemiology — did pioneering work on the transmission of malaria and won the Nobel Prize for Medicine.

The current book is an introductory text that starts at the level of the neophyte and gradually brings the student to the level of current research. There can’t be many areas within the mathematical sciences where this is still possible, but the author carries it off quite plausibly. The target readers include advanced undergraduate and graduate students in mathematics as well as graduate students in other fields.

This is a field that depends heavily on differential equation models. Ordinary differential equations are the primary tools for basic modeling of the distribution of diseases in a population. More complex models (age-structured epidemic models and spatial diffusion models, for example) are based on partial differential equations. The final chapter addresses discrete epidemic models using difference equations.

The treatment begins with the Kermack-McKendrick epidemic model (the SIR model) that describes three distinct groups within a population: the susceptibles, who are healthy but vulnerable to the disease, the infected, and the recovered. A good portion of the book is devoted to variations and extensions of this basic model: to epidemics where the recovered are again immediately susceptible, to populations with realistic birth and death models, to multi-strain diseases, and to age-dependent models. In many such models, the reproduction number — the number of secondary cases one infected individual will produce — is a most significant parameter, and one that has to be estimated from data.

The author devotes a chapter to fitting models to data and does a pretty good job of identifying the major issues. One might wish that more attention had been given to this question. For practical applications, estimation of the model parameters is critical for understanding the character of the epidemic and predicting its future. Of course, model validation is also very important.

Several other epidemiological topics are taken up in following chapters. An especially important one focuses on control strategies and vaccination. Another explores models of vector-borne diseases where the infection is carried from an infected person to another via a disease-carrying agent such as an insect.

The author provides a limited number of exercises and an extensive bibliography. This is an appealing book, well-written and thoughtfully organized. The main prerequisite would be a good undergraduate-level course in differential equations. More advanced topics such as stability analysis and the Hopf bifurcation are introduced in the text as necessary.

Bill Satzer (wjsatzer@mmm.com) 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.