Tipping Points: Modelling Social Problems and Health

This book is not really about tipping points: the book’s subtitle, “Modelling Social Problems in Health”, should have been the title. Each chapter is based on a different paper originally presented at a conference which was funded as part of the so-called “Tipping Points Project” — so that the authors of each paper apparently had to find some way to jump on the “tipping point” bandwagon. Although tipping points do appear in various ways in the different chapters, they are in general not the major focus. Furthermore, the term “tipping point” is interpreted in vastly different ways by the different authors. For some, it takes the conventional mathematical meaning of a parameter value (or set of values) which marks a discontinuous changeover in the stable solutions of a dynamical system. For others, it denotes a threshold in a medical condition which, when attained, indicates that active intervention is required. For still others, it is used as a synonym for a rare, critical event, in general one that should be prevented.

Despite the inappropriate title and inconsistent terminology, I enjoyed reading the book. The chapters are in general well-written (especially the more mathematical ones) and thought-provoking. The mathematical content could be characterized as specific applications of conventional modelling techniques. Multicompartment models (systems of ODEs) feature prominently. The epidemiological smoking model (which figured in three of the twelve chapters) was clearly presented, and the model behavior was well-documented. It was also nice to see papers by researchers in non-mathematical disciplines treating the same topics from different perspectives. The conference must have been great fun for the participants.

Chapter 1 gives an introduction to an epidemiological smoking model which also appears in two subsequent chapters. The model is a conventional 3-compartment model with bilinear terms, and a straightforward stability analysis is given. A generalized version has multiple nontrivial equilibria, whose stability properties are characterized in terms of reproduction numbers.

Chapter 2 discusses sensitivity analysis of a stochastic version of the model presented in Chapter 1. Although the chapter doesn’t have much to do with tipping points, it does give a very easy-to-follow example of the use of Sobol indices to characterize the model’s sensitivity to parameters. Some minor quibbles: the values of B_ij are not specified in Figure 2.2, and Figure 2.3(c) has a strange “bar code” pattern which is not discussed in the text, and is most likely a printing error.

Chapter 3 is non-mathematical, and explores possible reasons for the high incidence of smoking among women in northeast England. The variety and subtlety of factors involved should give pause to those who expect sociological modeling to develop into a kind of social physics, like the “psychohistory” imagined by Isaac Asimov in his “Foundation” science fiction series.

Part II (Chapters 4–6) has very little in common with Part I, either mathematically or in application. The applications are medical signal processing and blood flow, and the models are statistical fits and PDEs respectively.

I was frankly rather dismayed by Chapter 4, which talks about the assessment of cardiac surgeons’ performance based on patient survival statistics. Detailed statistical analyses are used to assess mean risk of death for various types of operations — but very simplistic and misleading methods are used to compare each physician’s performance against the means thus obtained. For each physician, (success minus failures) minus mean risk is plotted as a function of operation number — this will produce a mean-zero random walk for a physician whose performance matches the average. The authors do not acknowledge (and I suspect many administrators do not recognize) that random walks increasingly diverge from the mean over time, so that larger and larger deviations from the mean are no indication of significance. This would mean that many useless “interventions” would be conducted simply because of random variation. Furthermore, the interventions would be considered “successful” simply because of regression to the mean. One is reminded of W. Edward Deming’s “red bead” experiment.

Chapter 5 involves medical signal processing, and describes a Bayesian update-based algorithm for recognizing dangerous discrepancies in ECG patterns. Chapter 6 is essentially a short tutorial covering the derivation of various diffusion and fluid flow equations used in biomedical applications.

Part III (Chapters 7–10), like Part I, is concerned with modeling social phenomena. Indeed, there are many interrelations between these two sections; and it would have been helpful to place them together, rather than interposing Part II which has no relation to either.

Chapter 7 speculates on the design of a “system sociology approach”. It presents a modelling approach which may be productive in certain situations. But the stated ambition to develop a comprehensive mathematical theory of social systems (akin to the mathematical models of fundamental physics) is doomed to failure, for a number of reasons. A one-size-fits-all modeling methodology cannot possibly account for the enormous variety of social phenomena. Social systems do not always consist of fixed components interacting according to fixed rules, as physical systems do. Basic social structures may merge, fragment, or redistribute; rules of interaction may change dynamically. It follows that many social systems do not fit the authors’ paradigm of being divisible into “functional subsystems” consisting of “particles” (basic social units). Furthermore, even those systems which do fit this paradigm may not be Markov, or first order in time, or based on statistically independent particle interactions — all of which are presumed by the authors. Furthermore, the whole approach is based on the possibility of obtaining “a mathematical description of individual-based interactions according to a detailed interpretation of the dynamics at the microscopic scale”. But such “detailed interpretations” are rarely forthcoming. Unlike physical interactions, individual social interactions are hopelessly complicated. In general, it is more productive to treat microscopic interactions as a “black box”.

Although the article’s jumping-off point is Taleb’s concept of a “black swan” I suspect that Taleb himself, as a champion of heuristic methods, would strongly disapprove of the authors’ methodology. He would likely comment that they are following the same garden path as those economists who package economic behavior into neat analytical models, only to find out that events “out of the blue” knock their predictions cockeyed. It is surprising that the authors say nothing about agent-based models, which have shown some signs of being better equipped to account for “black swan” behavior than multicompartment models.

Chapter 8 (like Chapter 2) presents another sensitivity analysis via Sobol indices, this time for an agent-based model. This second application (to a very different type of model) illustrates the power and generality of this method. There is an error in the values of \(\mu\) in the caption of Figure 8.4, but the values are given correctly in the text. We also find another strained interpretation of “tipping point”, this time as an elbow in a system behavior versus parameter curve.

Chapter 9 discusses the interplay between genetic and cultural evolution in humans, referred to as “coevolution”. Many practical examples of coevolution are mentioned and some multicompartment models are presented, but without any further analysis.

Chapter 10 discusses a multicompartment opinion dynamics model which bears some resemblance to the smoking model discussed previously, except that bilinear terms are replaced by nonlinear functions with sigmoidal dependence to account for the effects of “conformity bias”. The introduced nonlinearity is shown to give rise to critical parameter values at which the system’s steady-state behavior undergoes radical change.

Section IV (Chapters 11–12) centers around the ‘critical event’ incarnation of tipping point. The two articles in this section are non-mathematical, and describe (in rather general terms) risk management issues in different contexts. Chapter 11 discusses the management in suicide risk in prisons, and appends a rather vague discussion of risk management in the London Port Authority. The main point with respect to suicide risk appears to be that instead of trying to identify risky individuals, it’s better to focus on improving the prison environment. This is probably a controversial point in the field, and it’s hard to assess the validity without hearing the other side of the story. The chapter refers to “tipping points” as conditions “beyond which we are unable to get back to a steady or improved environmental state” — but offers no evidence that such “tipping points” exist, in the context of prison environments. Chapter 12 gives interesting anecdotes concerning risk management in a psychiatric inpatient facility, and then applied lessons learned to risk management in general.

Chris Thron is a former systems engineer at Motorola/Freescale, and currently associate professor of Mathematics at Texas A&M University-Central Texas. His research interests include various applications of modeling and simulation including mathematical epidemiology (malaria and plant disease), agent-based modeling of social systems, and network design and performance optimization. He is the author, with Justin Hill, of Elementary Abstract Algebra: Examples and Applications.