The immune system provides our primary defense against the many pathogens that surround us. It is tremendously complex and much of it is still not well understood. The current book offers an introduction to the mathematical study of immunology. It is based on a course the author gave at the Courant Institute. This is by no means a comprehensive study of the mathematics of immunology. It is more like dipping one foot into a wild river. For readers without some additional background in medically-oriented biochemistry it will be a difficult read, although in principle the book is self-contained.

How can mathematics contribute to the study of immunology? In his memoir *This Mad Pursuit, *Francis Crick wrote:

To produce a really good biological theory one must try to see through the clutter produced by evolution to the basic mechanisms lying beneath them, realizing that they are likely to be overlaid by other, secondary mechanisms. What seems to physicists to be a hopelessly complicated process may have been what nature found simplest, because nature could only build on what was already there.

So, if mathematics has something to offer here, it must be as a tool to help penetrate the clutter. Elsewhere, Crick wrote, “In physics, they have laws; in biology we have gadgets.” More than anything else, the immune system looks like a very complicated gadget with lots of apparently ad hoc features. The author of the current book approaches the inherent complexity of the subject by focusing on a narrow area and using it as a window into the wider field.

That narrow area is the immunological battle between the human immunodeficiency virus (HIV) and the adaptive immune system. The first chapter introduces the topic in various simplified forms as a conflict between HIV and helper T-cells. Mathematically, this leads to something that looks like population dynamics in a well-mixed population, which in turn looks like much-studied questions in chemical kinetics.

The author postpones to the second chapter a summary of the basic facts of immunology. This is very concise and just begins to give the novice reader a sense of the situation and a basic vocabulary. Then the following chapter discusses techniques that have been developed to quantify the immune response. Among others this includes the commonly used RIA (radio immune assay) and ELIZA (enzyme-linked immune-absorbent assay) methods. The author uses this chapter to show what can be determined empirically as well as how these assay processes can be modeled.

The longest chapter in the book addresses humoral immune responses, and in particular the role of B-lymphocytes in acquired immune response. The humoral immune response depends on antibodies secreted in blood and other bodily fluids, whereas the cell-mediated immune response discussed in the following chapter depends on T-lymphocytes. The final two chapters then describe control of the immune response (recognition of self and mechanisms that prevent autoimmune responses), and immunology from the virus’ perspective.

This book is aimed at readers with a background at least at the advanced undergraduate mathematical level. The major mathematical tool is differential equations, with just a little bit of probability mixed in. Although intended as an introduction for non-specialists, this is a challenging book.

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.