Applications of Calculus to Biology and Medicine: Case Studies from Lake Victoria

This book offers a new and rather unconventional approach to a first level undergraduate course in applications of mathematics to biology and medicine. The motivation is explained clearly in the authors’ preface. They note that they have taught courses based on the material in it, both to a mixed class of biology students with little mathematics background and to mathematics majors who have completed courses in linear algebra and sometimes differential equations. Their unfortunate conclusion is that both groups came to their course with “remarkably little recollection of what they learned in either their biology or math classes.” This book is an attempt to engage students more fully with both subjects and encourage development of critical and creative thinking.

The authors suggest a different pedagogical approach: turn the students into active independent researchers by asking them to pose and answer their own research problems. The concept is something like adopting a “research experiences for undergraduates” framework and applying it to a course in the regular curriculum.

The Lake Victoria region in Africa provides the setting for most of the applications in the book. That region and its people face a number of challenges in the areas of human health, water quality, the struggling fisheries industry, and more. Using real data collected by scientists working in the region, the authors discuss mathematical modeling for many of the challenges that arise there. They consider growth issues (algal, water hyacinth, and fish), predator-prey interactions among insects or insects and plants, lead contamination in human blood, and the spread of HIV.

It is a very ambitious program and the authors assume a fairly minimal background for their students. They begin with a review of basic calculus concepts motivated by an example of tumor growth using a Gompertz model. This provides the opportunity to revisit the derivative, antiderivative, and a simple separable differential equation.

The next four parts of the book address four general areas in biology where mathematicians and biologists have collaborated fruitfully. Some chapters in each part explain the biology and mathematics needed to understand problems that arise in a particular area, and some describe computational techniques that are needed to solve the problem.

Population modeling is considered first. It is focused initially on the population of algae in Lake Victoria. A collection of chapters introduces more material on differential equations, logistic growth and Euler’s method for numerical solution. At the end of this part the authors provide a short introduction to the Sage software system, one that they use and expect students to use through the remainder of the text. Sage is well suited for solving systems of differential equations, and that is what is needed in the rest of the book.

Throughout the book the exercises start with basic questions interspersed within the text to insure that students understand what they have read. “Problems for Exploration” include practice exercising what has been presented in the context of research. More extensive project work starts after the completion of the discussion of population modeling.

Several parts of the book provide useful “interludes”. These include modeling interludes (describing the modeling process, sensitivity analysis and potential problems with models) as well as research interludes (reading and writing a research paper, and making figures). The effect of these is to help the students understand the context of mathematical modeling and how it is integrated into a larger research process.

Each major section of the book is accompanied by a bibliography that is intended to provide useful references for the students in their project work. A final chapter provides lists of classroom-tested projects in each of the areas discussed in the text.

This is likely to be an exciting and challenging book for motivated students, and would be fun to use in the classroom. For less motivated students or for those who prefer a more passive approach, it might be a challenge of quite a different sort.

Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.