Back of the Envelope Modelling of Infectious Disease Transmission Dynamics for Veterinary Students

The title of this book suggests an unsophisticated treatment of use only to veterinary students. Indeed, the preface indicates that only minimal mathematics background is required—certainly less than a calculus course—and the author owns up to the charge that the pace is “plodding.” I, therefore, expected a very elementary treatment of epidemic modeling. This turned out to be wrong, as shown by a detailed look at the first three chapters. 

The first chapter introduces compartment models via an algorithm for producing the boxes and arrows, labeling the arrows, and writing a differential equation for each box. An example of the level of detail is “Begin with the flow chart in front of you. I like to draw the flow chart as far to the left of my pad of paper as I can.” This sounds like a cookbook written for 10-year-olds. After eight (!) pages like this, we finally have the equations

\( \frac{dS}{dt} = -\lambda S \)

\( \frac{dI}{dt} = \lambda S - \delta I \)

\( \frac{dR}{dt}=\delta I. \) 

The verbiage in these eight pages is not merely a description of instructions, however. Some of it focuses on what the model elements mean: the derivative symbols are math-speak for rates of change, the rates of change are bigger when the group population is bigger, the sign in the equation indicates whether the change increases or decreases the group size, the rate \( \delta I \) represents a transition while the rate \( \lambda S \) represents a transmission. 

A better feel for the book comes from chapters 2 (on transitions) and 3 (on transmissions). Here, we see the modellng to be as sophisticated as is found in books written for far better-prepared readers. The reader learns the dimensions of transition rate constants and their interpretation as the reciprocal of the mean time in class. There is a detailed discussion of the difference between rate constants in discrete and continuous models; for example, how does one calculate the continuous-time rate constant corresponding to a 60% decrease in one month? The transmission chapter carefully identifies the factors that are required for transmission: the susceptible host must have contact with a member of the population, that member has to be infectious, and then there is only a probability of transmission. Formulas are derived for frequency-dependent and density-dependent contact rates and the more fundamental parameters combined to obtain a form of \( \beta S I \) or \( \beta S I / N \). We learn how the constants called \( β \) in these two formulas are actually very different, with different values for the same scenario and different dimensions. This is a point that my upper-division mathematical modeling students struggle to grasp. Perhaps they would understand it better if I introduced epidemic models using this book instead of a more advanced one. 

The remainder of the book builds on the first three chapters. We learn to modify our epidemic models to endemic models by adding demographic features. We learn the meaning of the basic reproductive number, how to calculate it from first principles, and how to calculate it using an algorithmic implementation of the next generation method. There is no discussion of linearized stability analysis, but stability in an epidemiology model is easily determined by interpreting the basic reproductive number. Graphs from numerical simulations illustrate patterns, such as oscillations toward the equilibrium. There are two chapters of case studies, chosen to teach modeling principles as well as the details of the specific diseases. This goes far beyond the “back of the envelope modeling” indicated in the title. The case studies compare model predictions with real data and discuss reasons why they might not agree and how the model might be improved. There are chapters on how to estimate parameters, different methods for trying to control epidemics, sexual transmission, and vertical transmission. 

I don’t teach veterinary students, but I could still see myself using this book in a class. It would be an excellent choice for a course where students would use epidemiological modeling to achieve a general education quantitative requirement, rather than having to retake a precalculus course they hated in high school. It would be a useful supplement for a more advanced epidemiological modeling course offered to students with a background of differential equations.

Glenn Ledder has done research work in combustion theory, groundwater flow, population dynamics, plant life history theory, and plant physiology.  In addition to his research, he has worked in mathematical pedagogy, editing an MAA notes volume and writing textbooks in differential equations and mathematics for the life sciences.