UNIQUENESS & UNIVERSALITY
A BEYOND BORDERS column by David Krakauer, President of the Santa Fe Institute.
One might say that natural history captures the uniqueness of every organism and natural science, their generality. Natural science values parsimony and natural history, profusion. But when is it appropriate to pick one approach over the other — that is, when do the details really matter and when are general mechanisms most important? The honest answer is that we do not know.
In 1766 James Christie hosted his first auction on the Pall Mall in London. Included in the sale were a pair of sheets, two pillowcases, and two chamber pots. It is hard to imagine anything more particular than an auction lot. In the same year Daniel Bernoulli, a 66-year-old professor of botany, physiology, and physics (yes, all three) at the University of Basel published “Essai d’une nouvelle analyse de la mortalité causée par la petite vérole” — his mathematical treatment on increased mortality due to infection with smallpox — that blight of sheets and chamber pots.
Bernoulli’s paper is the ur-type, type-specimen, or structurally, the Bauplan for many mathematical models of epidemics decanted from the late 18th century up to the present. Bernoulli envisaged a population of susceptible hosts that die at a constant rate, and a population of immune individuals who have been fully vaccinated following a single infection. Infected susceptible populations who do not transition into the immune state experience an elevated death rate. Bernoulli sought in his work to account for the increased mortality due to infection.
The model assumes that all susceptible individuals experience the same rate of infection, a constant case fatality rate, and that all immune individuals die at the same rate. For the sake of generality, the model assumes extreme homogeneity.
Shortly after Bernoulli introduced his model — and prior to its first publication — his lifelong rival, Jean Le Rond d’Alembert, criticized his work for, ironically, its excessive complication, providing a more parsimonious model of greater generality for estimating mortality, albeit with reduced relevance to infectious disease.
In 1927 Kermack and McKendrick significantly extended Bernoulli’s model to include a dynamical transmission process (dropping the assumption of a constant rate of infection), allowing infected individuals to infect susceptibles, and making mortality and recovery rates a function of the duration of infection.
The Kermack and McKendrick paper introduced for the first time a threshold level of infection below which epidemics die out and above which the epidemic grows — the idea of the basic reproductive number or R0 which has haunted our imaginations and dreams for the last two years.
Over the course of a century, these models have been subjected to a large number of variations and augmentations — to such a degree that it would be fair to conclude that we now possess a natural history not of epidemics but of general models for epidemics. Which at some level must suggest a contradiction.
The pursuit of the most parsimonious model has been replaced by the pursuit of exceptions. Many publications are now based on pointing out one or more ways in which simple models fail. We now include our “chamber pots and sheets” in order to better fit contingent data sets. The most recent agent-based models for COVID-19 include age structure and population size, transmission networks for households, schools, workplaces, long-term care facilities, and communities, as well as detailed viral dynamics. The mathematician d’Alembert would have been stupefied by the complications.
The fact is we do not know how best to justify or choose between the complicated unique simulation and the simple universal analysis. For Bernoulli and Kermack and McKendrick the objective was to identify the most salient causal links in a chain of mortality. For modern agent-based models the objective is to provide a near real-time simulacrum for the pandemic to guide policy-making. And still none of these models include realistic psychological reactions, and very few embed the epidemic within a political economy.
Enrico Fermi recollected a conversation with John von Neumann who declared that “with four parameters I can fit an elephant, and with five I can make him wiggle his trunk.” All of us trained in the mathematical natural sciences are drilled with that quote and until recently have avoided any suggestion of models that wiggle. It is now clear that the intellectual austerity of d’Alembert and von Neumann has in certain domains become obsolete. It will be a significant challenge for complexity science to do a better job of justifying the unique versus the universal.
— David Krakauer
President, Santa Fe Institute