Introduction to Biomathematics
After reading this section of notes, you should
be aware of the general structure of the course Topics in Biomathematics, and
have a general sense of what the field of biomathematics is about and how we will approach learning biomathematics in the course Topics in Biomathematics.
Welcome to Topics in Biomathematics! The set of notes to which this document belongs have been developed from a course, MATH 463 Topics in Biomathematics, taught at the University of Scranton since 2013. The purpose of this course is to provide students having a background in linear algebra and differential equations with a grounding in mathematical modeling for biological phenomenon. We hope that these notes may also be useful for a somewhat wider audience of individuals that are interested in mathematical or theoretical biology, or areas of applied mathematics that consider scientific problems from biology or medicine.
There are several reasons for the existence of these notes. The principal ones are
to provide a reference that strikes a desired balance between the subjects of mathematics and science,
to provide a reference that includes discussion of some important topics from mathematical modeling that are relevant in contexts more general than just mathematical biology,
to provide a reference that helps to bridge the gap that is necessary to close in order to begin reading the research literature, and
to provide a reference that encourages good work flow practices such as integrating analysis, computing, and the written presentation of results.
When teaching MATH 463 Topics in Biomathematics, an important component of the course is student projects. There are two different types of projects that students in Topics in Biomathematics are asked to complete: model reports, and a final independent project. The model reports are meant for students to get early exposure to reading research literature that involves the development or application of mathematical models in biology and medicine. The idea is for students to get experience with the “translation” process of going from a real world biological problem to a mathematical abstraction and back again. Furthermore, students are asked to think and write critically about a mathematical model published in a recent research article. One might reasonably view the model report assignment as asking students to write a peer-review referee report for a manuscript that has already been reviewed and published.
The final independent project is meant as a capstone to the course. It is not the expectation that every student will do an entirely original research project that involves developing a novel mathematical model from scratch, although in practice this does occur fairly often. However, the final project requires a more substantial synthesis of scientific and mathematical knowledge and understanding than the model report or other assignments such as homework or written exams.
This is all meant to highlight the fact that throughout these set of lecture notes, we have tried to provide pointers to current or at least fairly recent research that is reasonably accessible to the average student in MATH 463 in order to facilitate the model report and final independent projects. Every time we discuss a topic in the notes, we follow the discussion up with specific references where students may go to in order to learn more, acquire project ideas, and see what else is out there regarding mathematical and theoretical biology.
Throughout these notes and the Topics in Biomathematics course we make regular use of the R software and ecosystem for computing. It is impossible to do very much interesting work in science and applied mathematics, including mathematical and theoretical biology, without computation. For the purposes of this course, we will need to employ numerical methods for solving equations, optimization, and linear algebra. In addition, we will find it very helpful to be able to create visualizations that are simple yet professional in appearance. The R environment and language possesses several features that make it highly appropriate for all of these purposes. Among it’s many advantages, R is also freely available.
In addition to R, we also use RStudio, an integrated development environment that enhances the use of R. Using RStudio makes it possible to access the Topics in Biomathematics lecture notes in an interactive fashion because all of the notes are written as R Markdown notebooks. This allows us to integrate text, equations, figures, R code, plots, etc. and to compile the result as both html and pdf files. On top of that, students will have a template they can follow in order to write and present their work for the model report and independent final projects.
Biomathematics is concerned with the use of mathematical methods (e.g., linear algebra, differential equations, dynamical systems, and probability theory) to understand phenomenon in the life sciences, it is part of the larger field of mathematical and theoretical biology. Mathematical and theoretical biology provide a solid foundation for computational and quantitative approaches to investigations in biology and medicine. The general work flow in biomathematics proceeds as follows:
It may be necessary to repeat steps 2-5 multiple times with refinements at each iteration in order for your efforts to yield something that is truly useful for addressing a scientific question.
Biomathematics, which for the purposes of this course we consider synonymous with mathematical biology, emerges as a distinct field due to the fact that carrying out the previously described steps has resulted in the (continuing) development of some common methods and terminology. Furthermore, we carefully distinguish biomathematics from the field of biostatistics. While biomathematics certainly involves the use of probability and statistics, these areas play a different role within biomathematical investigations than they do in biostatistical investigations. Additionally, the relationship between theory and experiment is different in biomathematics than in biostatistics.
In the next lecture, we will spend some more time fleshing out the steps of the basic biomathematical workflow. From there, we will begin to examine some general principles related to the construction of mathematical models. That is, we will study those ideas from mathematical modeling that are essential for biomathematics. Our goal is to get to a point where developing a mathematical model and using it to answer some biological question does not seem like an impossible task.
Throughout this course, you are encouraged to look at additional references for the purposes of seeing the perspectives of others and for learning about topics and methods that are not covered explicitly in lectures or assigned reading. The following sections provide an overview of some additional texts you might want to examine at some point.
Several established textbooks in biomathematics are widely available. See for example, (Allen 2007; Britton 2003; de Vries et al. 2006; Edelstein-Keshet 2005; Friedman and Kao 2014; Murray 2002, 2003). The two-volume text by Murray, (Murray 2002, 2003), is considered something of a standard, the books cover a variety of different topics well beyond what we can, but assume a greater mathematical background than we do in this course. There are also several nice sets of lecture notes that are available online for free, such as those by Byrne, Chasnov, Goldstein and Pedley, Ingalls, and Sontag.
The earliest motivations for the development of biomathematics comes
from theoretical
ecology and population
dynamics. A sub-area of these fields is mathematical
epidemiology. Excellent references for learning more about these
fields are provided by
(Brauer and Castillo-Chavez 2012; Kot
2001).
Mathematical physiology is another large sub-discipline within biomathematics. A standard reference is provided by (Keener and Sneyd 2009b, 2009a). Note that mathematical physiology intersects with biophysics and biochemistry.
An important and well-developed topic in mathematical physiology and biophysics is mathematical and computational neuroscience. Excellent references for this field are (Börgers 2017; Ermentrout and Terman 2010). You are also encouraged to watch the videos from two conference talks on Brain Control and Modeling Large Scale Brain Activity.
In this course, we will primarily use techniques from linear algebra and ordinary differential equations. In much of our study of mathematical models arising in biomathematics, we will be equally interested in qualitative and quantitatve results. The area of applied mathematics known as dynamical systems theory provides the mathematical techniques for a qualitative analysis of mathematical models. During the course, we will need to develop and employ some techniques for differential equations that may not be covered in a first course in ODEs. The following two references may be very helpful in learning more about applied dynamical systems and differential equations as they relate to biomathematics (Jones, Plank, and Sleeman 2010; Strogatz 2015).
When it comes to both the quantitative and qualitative analysis of differential equation models, it is very helpful to do so with the aid of a computer. Both the R language for statistical computing and the Julia language for scientific computing are excellent for our purposes. Each of these has an associated ecosystem with packages that help make mathematical modeling and analysis easier. In this course, we primarily use R because of the phaseR package for the phase-line and phase-plane analysis of one- and two-dimensional systems of differential equations. Furthermore, the notes for this course are written using the RStudio integrated development environment associated with R. Thus, using R and RStudio allows the reader to go through these notes interactively to enhance the learning experience.
There are many different classes of mathematical models and a variety of different approaches to biomathematics. Notably absent from the main discussion in this course is stochastic models (an excellent reference with applications in biology is provided by (Allen 2011)) and game theory.
Biomathematics and related fields are active areas of research in which there are new peer-reviewed articles appearing on a daily basis. The Society for Mathematical Biology is a great source for information on what is happening in the field of biomathematcs. If you are interested to see what is going on at the cutting edge, you might also want to look at some of the following links:
Text and figures are licensed under Creative Commons Attribution CC BY-NC 4.0. The figures that have been reused from other sources don't fall under this license and can be recognized by a note in their caption: "Figure from ...".