Introducing Mathematical Biology

Reviewed by Chad Westphal, Professor of Mathematics and Computer Science, Wabash College on 9/11/25

Comprehensiveness rating: 4 see less

This books covers several biological/medical applications using a basic dynamical systems framework. While it is an introduction to "mathematical biology," it doesn't cover all areas of the discipline possible or all relevant tools used across the field. The focus is on modeling time-dependent phenomena with ordinary differential equations (ODE).

From the mathematical point of view, a basic understanding of ODEs is generally assumed. However, there is a review chapter dedicated to linear stability analysis and bifurcations that the book makes reference to in several places. The intro material indicates that a reader should be familiar with basic ODEs. In this case, a course based on this book wouldn't add much 'new' mathematics, but would allow a student to see a more cohesive set of applications than an ODE course could provide. Alternatively, it may be possible to teach an applied ODE course using this book as a reference for students who have taken single variable differential and integral calculus (i.e., Calculus I and II in most U.S. universities). In this case, it would be best to pair this text with a more complete treatment of ODE systems, stability analysis, and asymptotic analysis. The text assumes no formal biology background, but appeals to common knowledge in population interactions and epidemiological dynamics. In applications involving genetics and pharmacokinetics, the author provides descriptions appropriate for a typical mathematics undergraduate student with no specific knowledge. The book includes python code for some of the applications and a few descriptions of how to use the code to explore how parameters or initial conditions affect the outcome. There are a few "exercises" that ask the student to fill in a few details of a derivation or computation, but the book does not include any comprehensive homework exercises or projects. So, using this as the primary book for a course would require the instructor to supplement these areas (unless it's a survey course of some sort).

There is no index or glossary.

Content Accuracy rating: 5

The content of the book is accurate and there are only a few minor typos and inconsistencies in the text and formatting. These are probably pretty typical of a first edition book.

Relevance/Longevity rating: 5

The technical content of the material is appropriate as introductory coverage of the application areas. The only material that seems dated is the discussion of Covid-19, where the text is written for it as a near-current event. Nothing seems incorrect, but the other applications have a more timeless style of description.

Overall, the applications covered are all introductory and all use a lot of the same mathematical tools. None of this would really need to be updated, but additional material could be added fairly straightforwardly to extend the applications and/or add new features to the models presented.

Clarity rating: 5

The writing style of this book is not overly technical and is appropriate for the intended audience. The chapters end in a nice summary of "Key Takeaways," and the flow of the prose is somewhat conversational.

Consistency rating: 4

The book is generally consistent in its use of terms and mathematical conventions. There are a few minor details to note (e.g., it often uses the trace and determinant of the Jacobian matrix, and sometimes uses tr(J) and det(J) and other times just has tr and det). It should also be noted that the mathematics is employed in a very applied manner, so there are no theorems or proofs. The text sometimes switches between using "we" and "I" as the author. In mathematics it is conventional to use "we" even if there is a single author, so the few cases of "I" stand out.

It also doesn't label equations or figures, so it tends to read pretty linearly. Having equations and figures labeled would help in using this book for a course.

Modularity rating: 4

The book has a nice structure with relatively short chapters. The mathematical tools used are pretty consistent throughout, so it would be possible to skip some chapters and/or to rearrange the order (to some extent). As mentioned elsewhere, the book could be improved with the addition of projects and/or homework exercises. In places where python code is included it would be relatively easy to design small projects to model a specific application and answer a few questions or verify the analysis in the main text.

Organization/Structure/Flow rating: 5

Each topic covered in the book has an appropriate amount of background description, with references for additional reading. The author is from a mathematical background and the applications all involve biological and/or medical concepts. The organization and coverage is appropriate for a student with a background in differential equations, but not necessarily any formal biological training.

Interface rating: 5

There are a few very minor typographical/formatting issues, though nothing that is inherently confusing or distracting.

Grammatical Errors rating: 5

The text is generally well-written and error free.

Cultural Relevance rating: 5

There is nothing that seems like it could be culturally insensitive or offensive in this book. The material for this book does describe epidemiological and medical applications, and the coverage and descriptions are appropriate. The examples don't tend to have any context for race or ethnicity.

This would make a nice companion for a first course in differential equations, where the mathematical topics are covered in detail. It could also be used as a good reference for a student interested in applied mathematical modeling with ODEs. Several of the topics could easily lead to undergraduate research topics with some additional reading and literature searching.