Mathematical Models in Epidemiology

A 2019 book on epidemiology from three of the world’s top experts on the subject could not be more timely.  To the best of my knowledge, none of these experts have been contacted by public health authorities for advice.  Given that, it is worthwhile to start by quoting the first few sentences of the final chapter, titled “Challenges, Opportunities, and Theoretical Epidemiology.”

 

 

This 16-chapter book is an 8-fold expansion of the material on epidemiology that appears in the book Mathematical Models in Population Biology and Epidemiology by the first two authors.  It is a detailed treatment of deterministic epidemiology models.  The core of the book consists of a chapter that presents the basic SIS and SIR models, with and without demographics (births and deaths), followed by specific chapters on endemic models (with demographics), epidemic models (without demographics), models with heterogeneous mixing, and models for vector-transmitted diseases.  This core is followed by six chapters that present models for specific diseases of current interest, such as tuberculosis, HIV/AIDS, and Zika.

 

It would be impossible to review the book in its entirety, nor would a broad superficial review give the reader as much of a sense of the book as a more focused look at a small number of topics.

 

The standard Kermack-McKendrick epidemic SIR model is presented in Section 2.4.  This model combines a transmission process that moves individuals from susceptible to infected with a removal process that moves individuals from infected to removed.  The assumptions used to quantify these processes in the model are clearly wrong.  Using βSI as the transmission rate assumes that people in a community are like molecules in a gas, moving around randomly and having a normal distribution of contacts, while real communities have contacts based on interpersonal networks with an asymmetrical distribution of contact rates among individuals.  Using αI as the removal rate assumes that the time spent in the infected class is exponentially distributed, which means that more individuals are infected for exactly one day than for exactly five days, even if the mean infectious period is five days.  We often use this standard model without consideration of the amount of error caused by these flawed assumptions, but this book addresses them in some detail.  Chapter 4 begins with a stochastic model that assumes contacts between individuals occur in a network of connections rather than through random mixing.  The development and analysis of the model ultimately leads to discovery of the specific assumptions needed for the model to reduce to the usual mass action transmission rate.  This development clarifies the conditions needed for the mass action assumption to be appropriate.  Later in the chapter, the authors replace the exponential distribution of infection duration with a general distribution and show that the relationship between the fraction of individuals who are never infected and the basic reproductive number is the same for any distribution.  The exponential distribution may yield systematic error in the early stages, but it yields the same ultimate outcome.

 

Among the models presented for specific diseases are two models for dengue fever.  The first of these modifies the basic model from the core chapter on vector-transmitted diseases by incorporating vertical transmission of the pathogen in the mosquito population.  The second model incorporates an additional class of asymptomatic infectives that has a reduced transmission rate and an increased recovery rate.  The addition of asymptomatics to the dengue model is timely, as a similar change is needed to adapt the standard SEIR model to a model for COVID-19.  The study of both dengue fever models focuses on calculating the basic reproductive number using the direct method of working with the average numbers of transmissions from one human, from one human-infected mosquito, and from one mosquito infected at birth.  The computation of the basic reproduction number using the next generation matrix method is left for the exercises.

 

No book can be both a gentle introduction for beginners and a serious reference book for specialists; this one is the latter and not the former.  It can be used as a textbook for courses, but anyone who considers it for that purpose should make sure it is a good fit for their students.  I would not hesitate to use it for a graduate-level course, but I would be cautious about using it with undergraduates.  The authors assume a significant level of general mathematical modeling background, which would make the book a difficult read for a typical advanced undergraduate student whose modeling experience is limited to what they have seen in mathematics courses.  A model with multiple parameters, whose behavior is to be studied without specific parameter values, is a quantum leap beyond the typical “application” that contains only one parameter whose value can be fixed by one piece of data.  Sections 2.2 through 2.4, on the SIS model and on SIR models with and without demographics, focus on analytical results (both quantitative and qualitative), while presenting only one figure that shows plots of class populations versus time and only one phase portrait.  On the plus side, the book is well endowed with exercises of varying levels of difficulty, along with a number of sections that outline possible student projects.

 

As with all first edition books, one should be wary of proofreading errors.  One particularly unfortunate error is at the end of Section 2.3, where the authors derive an equation for the total population N in an SIR model with constant birth rate and disease-related death, \( N'=\Lambda - \mu N - \alpha I \), and then discuss the effect of the pathogenicity parameter on the equilibrium population.  Unfortunately, this equation is derived from one a few pages back that uses \( d \) as the pathogenicity parameter and \( \alpha \) as the mean recovery rate of infectives.  Presumably, this will be corrected in a subsequent printing.

 

Whether one chooses this book for use in a class or not, every practicing mathematical epidemiologist will want this book to occupy a prominent place in their library.

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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.