Mathematical Models in Biology: An Introduction is an introductory textbook in discrete mathematical modeling covering a wide variety of biological topics: dynamic models of population growth, models of molecular evolution, the construction of phylogenetic trees, genetics, and infectious disease modeling. The authors, Elizabeth S. Allman and John A. Rhodes, describe their target audience as undergraduates "having a strong interest in biological science and a mathematical background sufficient to study calculus." While they do not assume any training in calculus or beyond, they do include (well-marked) supplemental exercises that give students who have had calculus an opportunity to pursue the connection between the discrete and the continuous. The authors have made some MATLAB programs available to accompany the text "to give readers hands-on experience with the mathematical models developed." However, technology does not play a leading role in the text. Depending on the availability of software packages and the chapters to be covered, I can imagine a situation where a professor might choose to use a computer algebra system like Maple or Mathematica instead of MATLAB. However, this would require some additional effort and would probably not be a reasonable option if Chapters 4 and 5 are covered, since the MATLAB files supporting these chapters rely more heavily on MATLAB's numerical capabilities.
The biggest strength of the text is that it touches on a variety of topics from a wide range of mathematical subdisciplines (e.g., mathematical modeling, discrete dynamical systems, linear algebra, and probability,) while following a coherent and logical pathway through an interesting set of biological topics. Hence an undergraduate studying with this text gains a broader view of mathematics as well as a feel for how mathematical models are constructed and used in biology.
Chapter 1 (Dynamical Modeling with Difference Equations) covers discrete models of single populations, starting with the exponential growth model and proceeding on to the logistic growth model, before discussing the different types of long-term behavior (equilibrium points, n-cycles, and chaos.) The final sections of the chapter cover variations of the logistic growth model and include a few closing comments about discrete models versus continuous models. There are plenty of exercises supporting each section, but many of the exercises involve the analysis of a model that is already set up for the student. I would prefer more exercises requiring the students to construct population models for themselves — given certain assumptions and conditions. I was also surprised by the scarcity of graphs in this chapter as I find that first- and second-year students gain a great deal from visualization.
Chapter 2 (Linear Models of Structured Populations) focuses primarily on discrete models of a single population that is partitioned into subpopulations. For example, the first example in the chapter divides a population of insects into three groups: eggs, larva, and adult. Given that death rates differ across these different stages in insect development, it makes sense to break the population up in this way. The authors extend the models from Chapter 1 by introducing some matrix algebra. They introduce the Leslie population model, mentioning it is a Markov model, and then finish up the chapter with a section on eigenvalues and eigenvectors. Because of the level of mathematical maturity required to tackle the material, I do not think that this last section is accessible to the typical college student who has not taken calculus. A more realistic audience would be a math student at the sophomore level. Nonetheless, I liked the progression of topics. The material builds nicely upon Chapter 1.
Chapter 3 (Nonlinear Models of Interaction) introduces the dynamics of interacting populations, starting with a simple predator-prey model. There is a nice coverage of phase plane plots, equilibria and stability before the authors introduce other examples of interacting population models. The final section considers three different situations: Competition (where two species are competing for resources), Immune system vs. infective agent (where a disease-causing agent infects an organism resulting in an interaction that is detrimental for both), and Mutualism (where the interaction is positive to both parties.) The authors provide numerous exercises throughout the chapter. Once again, however, most of the exercises require an analysis on models that are already given to the student.
Chapter 4 (Modeling Molecular Evolution) presents mathematical models that describe the process of DNA mutation. The first section provides a clear and informative presentation of the necessary background information on DNA. Because mutations are random events, the models of molecular evolution require a basic knowledge of probability theory. To this end, the authors include sections on introductory probability and conditional probabilities. The final two sections develop and analyze the mathematical models, all of which are Markov models. The authors start with the Jukes-Cantor one-parameter model and then present the Kimura 2-parameter model. The chapter closes with a discussion of phylogenetic distances, which quantify the amount of mutation that has occurred, allowing scientists to make meaningful comparisons. I found this chapter to be delightful! The authors are successful in explaining the preliminary information about DNA without getting too caught up in jargon, and the models seem to be quite accessible. The MATLAB files that are available are also helpful here in that students can experiment with the models and observe the results. Biology students (and others) are certain to find this material interesting.
Chapter 5 (Constructing Phylogenetic Trees) considers methods of constructing phylogenetic trees that best reflect the ancestral history of an organism or a group of organisms. The first section provides preliminary information about trees, defining topological trees (in which only the branching structure matters) and metric trees (in which the branch length represents the amount of mutation that occurred between splittings of the lineage.) In section 2, the authors introduce two distance methods for constructing phylogenetic trees: the unweighted pair-group method with arithmetic mean (UPGMA) and the Fitch-Margoliash algorithm. While neither of these distance methods is typically used in practice, the authors present them to motivate the Neighbor Joining algorithm, which "has become the distance method of choice for tree construction." Section 4 covers a tree construction method known as the method of Maximum Parsimony, and Section 5 provides a brief discussion about other methods (Maximum Likelihood and Bootstrapping.) The chapter closes with a list of applications and some suggestions for further reading. At the end of the chapter, the authors also provide a nice set of exercises that make good use of technology, as well as a project titled "Dental transmission of HIV" that requires students to construct phylogenetic trees to determine clustering patterns that will allow them to identify patients that were infected with the HIV virus by their dentist. The project uses DNA sequences mined from a paper that appeared in Science in 1992. The project looks fascinating! As with Chapter 4, I found this chapter to be a delight.
Chapter 6 (Genetics) presents a brief introduction to genetics while providing an extension of the discussion on probability in Chapter 4. Section 1 focuses on Mendelian genetics, describing Mendel's groundbreaking work in understanding the genetics of pea plants. Section 2 discusses probability distributions in genetics (in particular, the binomial distribution and the Chi-squared distribution.) Numerous exercises follow this section. The authors then discuss linked genes and gene mapping, and the chapter ends with a section on gene frequencies in populations. In this final section, the authors discuss the Hardy-Weinberg equilibrium, fitness and selection, and genetic drift. Other than the probability theory introduced in Chapter 4, this chapter seems to stand alone.
Chapter 7 (Infectious Disease Modeling) revisits the population models covered in Chapter 3, extending the ideas further to model the dynamics of infectious disease. The chapter begins with the famous SIR-model, in which a population is divided into three categories: the susceptibles, the infected, and the removed. The authors present and analyze the model — considering threshold values and critical parameters — before presenting two variations of the model: the SI-model and the SIS-model. The former reflects a situation where a member infected by the disease never recovers (as in the case of untreated head lice.) In the latter model members can recover, but are then susceptible again (as in the case of syphilis and gonorrhea.) The chapter closes with an introduction to differentiated infectivity models. These models can account for the fact that certain subgroups of a population are more (or less) likely to become infected with a disease. The authors introduce the differentiated model using a "sexy" example that explores the spread of gonorrhea among a population.
Chapter 8 (Curve Fitting and Biological Modeling) appears to represent the authors' recognition that statistics plays an important role in biological modeling. Although their textbook focuses on mathematical models in biology, the authors want to note that statistical methods are necessary in both developing meaningful models from real data and in assessing the validity of models. To this end, the authors provide a quick overview of curve fitting. They introduce semilog and log-log graphs, the method of least squares, and finally polynomial curve fits. (They also include an appendix that touches on the basic analysis of numerical data.) It is worth noting that Chapter 8 does not build upon the models covered earlier in the text, but rather serves as a stand-alone overview of statistical approaches to biological modeling. As such, I don't see how I would use this chapter when teaching the course, and I imagine that many biologists would prefer that statistics play a more vital and central role in a textbook about biological modeling. That said, I understand why one would want to include this chapter, and I think that the authors' approach is reasonable for what they have set out to accomplish in this book.
Again, the most positive aspect of the text is the fact that it covers a diverse set of mathematical topics while pursuing an interesting and coherent study of biological models. Chapters 5 and 6 are particularly interesting, and they appear to be unique to this textbook. Given the importance of bioinformatics in scientific research today, these chapters are certain to raise enthusiasm among students and faculty alike.
Regarding the authors' claim that this textbook is suited for the student "having a strong interest in biological science and a mathematical background sufficient to study calculus," I have some doubts. While I agree that the content of the text does not rely on calculus, I do believe that parts of the text require a level of mathematical sophistication that isn't typically seen in students who have not taken calculus. For this reason, I suspect that a more realistic target audience would be students who have already had a year of calculus. Professors should be aware of this and plan accordingly by advertising the course to a sophomore-level audience.
I have already touched upon the negative comments I would make about the text. First, I would prefer to see more exercises that require the students to construct models for themselves. Second, I thought that the authors could have placed a greater emphasis on visualization throughout Chapters 1-3. Much of the material relating to population models can be reinforced with meaningful graphs, yet there were very few graphs in those sections. Finally, I felt that the role of technology could have been greater, but perhaps individual faculty can address this issue when teaching the course.
Overall, I found this to be a very nice textbook. It would be appropriate for an undergraduate course in mathematical modeling for math majors, or it could serve as a course designed specifically for students interested in pursuing mathematical biology (e.g., biology majors having a mathematics minor.) Given the growing interest in the field of mathematical biology, I imagine that such a course would be attractive to students.
Judy Holdener is currently an Associate Professor at Kenyon College in Gambier, Ohio. She can be reached at firstname.lastname@example.org.
1. Dynamic modeling with different equations; 2. Linear models of structured populations; 3. Non-linear models of interactions; 4. Modeling molecular evolution; 5. Constructing phylogenic trees; 6. Genetics; 7. Infectious disease modeling; 8. Curve fitting and biological modeling; Appendix A. Basic analysis of numerical data; Appendix B. For further reading.