Detail: Utagawa Hiroshige III (Ando Tokubei). 1876. "Ou gojunko matsushima tenran no zu.”

Read the Reflection, written 22 May 2021, below the following original Transmission.

Beginning in early March, 2020, conflicting advice about COVID-19 emanated from local, state, and federal leaders, as well as from public health spokespeople. While there was unanimous agreement that some level of social distancing was critical to reducing the daily incidence of new COVID-19 cases, there was wide disagreement as to what group size should be allowed. Through the media, one learned that group size should be limited to 200, or maybe to 50, or maybe to 20, or maybe to 10. On the same day, you could learn that you could attend a large lecture but not a sports event or political rally, or you could go to a bar or restaurant but not a concert hall, or you attend a small dinner party with friends but not a restaurant. Eventually many regions of the US settled on home confinement, which implies group sizes of at most a handful. 

So what is the effect of group size on the transmission rates of infectious disease? This question raises many secondary questions. How long does one stay within a group — perhaps two hours at a ball game, but all day in kids’ classrooms — and how does that interact with group size? How thoroughly within a group does transmission occur? Surely somebody in the bleachers cannot directly infect someone in a box seat above home plate. And what about whether the group is indoors or outdoors; what about wind and humidity?

Clearly, it’s complicated. But, to get at least a very simple insight, we can make some very simple assumptions and obtain a back-of-the-envelope result. Let us suppose that you are in a group of size n0, and that there are N0 such groups in a total population of size n0N0 = P0. Moreover, we assume that if an infected individual happens to be in a particular group, then everyone in that group becomes infected. Finally, assume there is no mixing among groups. Both of those last two assumptions are readily altered, but let’s look at this simplest case first.

We further assume that the group you are in, A, is comprised of your friends and/or family members. Hence, a simple but useful measure of your expected damage is the probability that an initially-infected individual happens to be in your group multiplied by the number of people in the group. 

Suppose that initially the population contains a single infected individual. That individual could be equally likely to be in each of the groups, and so the probability that it is in the group that you happen to be in will be proportional to 1/(number of groups). Multiplying by the number of individuals in the group, your expected damage is proportional to n0/N0

How does n0/N0 depend on group size, n0? Because N0 = P0/n0, the expected damage to you varies as n02. In other words, a doubling of allowable group size results in a four-fold increase in your expected damage. Group size matters a lot! 

Suppose, instead of assuming that everybody in a group gets infected if one of the initially infected people is in the group, we assume that the number of infected in a group increases as the square root of the group size (the bleachers are far from home plate). Then it is easy to show that the damage to you varies as n03/2

Suppose we allow inter-group mixing. Then, depending on the rate of mixing, the infection rate, the duration of infectiousness in a person, and other factors having to do with the spatial pattern of mixing such as distance over which one mixes, the expected damage can become much larger, but the dependence on group size probably does not ever become steeper than the quadratic dependence derived above. 

Clearly, more sophisticated modeling is needed here. It is imperative that, as we emerge from strict social distancing some months from now, we don’t go straight from group sizes of two or three or four to unlimited group gatherings, lest we trigger a resurgence in infection.

So what’s the magic number? There isn’t a single answer. However, because group size matters a lot, the precautionary principle urges us to err on the side of small group-size restrictions. If mathematics informs our decisions, then as we eventually ramp up our sociality and return to some approximation of normality, we can do so with more clarity than was available when we went into quarantine.

John Harte
University of California, Berkeley
Santa Fe Institute

T-002 (Harte) PDF

Read more posts in the Transmission series, dedicated to sharing SFI insights on the coronavirus pandemic.

Listen to SFI President David Krakauer discuss this Transmission on Episode 26 of our Complexity Podcast


May 22, 2021

Some COVID-19-Triggered Thoughts on Complexity

My March 2020 Transmission addressed a perplexing issue of scale, network structure, and risk. My essay was motivated by the bewildering variety of guidelines issued by public officials in the first months of the COVID-19 pandemic. The conflicting advice concerned the numbers of people that one could safely hang out with. There was well over an order of magnitude variation in safe group sizes advocated by experts, and often no distinction made between indoor and outdoor gatherings. I tried to shed a little light on the sources of ambiguity in epidemiological model predictions of the consequences of group size.

Similar ambiguities now arise in discussions of vaccination and herd immunity. A widely disseminated estimate is that the US could reach herd immunity when perhaps three-quarters of us are vaccinated. Using a standard epidemiological Susceptible-Infected-Recovered (SIR) model, such a value can be derived. But how the unvaccinated susceptibles are distributed over space and interact with the entire population matters. It is easy to imagine situations in which three-quarters of the population is immune and yet the network structure and spatial distribution of the susceptibles results in unacceptable death rates.

To complicate matters, human behavior, as reflected in patterns of social network structure, will both affect the outcome of a vaccination campaign and be altered by awareness of the degree of success of that campaign, just as it was altered by the pandemic. Given this complexity, the notion of herd immunity may be too simplistic a target of modeling; a broader analysis of the consequences of a massive vaccination campaign could allow us to better anticipate locations in which various interventions, including masking and social distancing, are still needed.

This is indeed a complex system, with a self-referential structure, in which outcomes both influence, and will be influenced by, policy interventions. Arthur1 noted the importance of such structures in economics, particularly in non-steady-state economies. Is this fundamentally different from complex systems in physics, in which the human element is absent? In a provocative article, Goldenfeld and Woese2 suggested that, while physics makes a clean separation between the state of a physical system and the equations that govern the time-evolution of the system, successful biological theory will inevitably be self-referential in the sense that the equations that govern dynamics will evolve with the predicted changing state of the system. In their words:

In condensed matter physics, there is a clear separation between the rules that govern the time evolution of the system and the state of the system itself. . . . (T)he governing equation does not depend on the solution of the equation. In biology, however, . . . we encounter a situation where the . . . update rules change during the time evolution of the system, and the way in which they change is a function of the state and thus the history of the system. To a physicist, this sounds strange and mysterious.

This self-referential nature of theory described here is distinct from conventional feedback interactions that are found ubiquitously in both purely physical and also biological complex systems. Conventional feedback operates among the actual components of the system. In a sense Goldenfeld and Woese were also referring to feedback, but it is between the system operating rules (that is, the structure of the theory itself) and the state of the components of the system.

How does this play out in macroecology: the study of micro-level patterns in the distribution and abundance of species within ecosystems? Must ecological theory be self-referential or need it merely describe conventional feedback processes? To gain insight into this, it is useful to distinguish between relatively static, undisturbed ecosystems exhibiting patterns that at most fluctuate from year to year, versus systems in which natural or human-caused disturbances produce real trends, not just fluctuations, in the patterns. In analogy with pressure, volume, and temperature—the state variables of an ideal gas—we can consider the macro-level descriptors of the ecosystem, such as the number of species, the number of individuals, and the total metabolism, to be the state variables. We have shown3 that in ecosystems in which these state variables are relatively constant in time, the patterns alluded to above can be quite accurately predicted using the maximum-information-entropy (MaxEnt) principle derived from information theory. This powerful inference procedure finds the least biased distributions over the micro-level variables that satisfy the constraints imposed by the macro-level state variables.

Viewed from the perspective of the rapidly developing new subfield of “disturbance ecology,”4 however, the story above appears quite inadequate. In particular, ample data indicate that in ecosystems in which the state variables undergo secular change in response, for example, to anthropogenic disturbance, the patterns change and the purely information-theoretic predictions fail rather dramatically.

To remedy this, a theory of dynamic disturbed ecosystems can be constructed by hybridizing information-theory methods with explicit mechanisms causing disturbance. In this dynamic theory, as the state variables change over time, the form of the micro-level dynamics (e.g., birth, death, and ontogenic growth of individuals) is altered, and thus the probability distributions used to determine the consequences of the mechanisms that generate disturbance change over time. Thus, at each forward-in-time iteration of the theory, the distribution over which micro-level variables are averaged to update the macro-level constraints is itself updated in a way that depends on the constraints.5 Or, as Goldenfeld and Woese described, the update rules change during the time evolution of the system, and the way in which they change is a function of the state of the system.

Society desperately needs to deal more intelligently with the next pandemic than it did with COVID-19, and, more generally, it needs to prevent catastrophic climate change, preserve biodiversity, ensure sustainable food and water supply, reverse the trend toward increasing inequity in the distribution of wealth and opportunity, and prevent war waged with weapons of mass destruction. To do so, we need to improve our capacity to understand complex, dynamic, disturbed coupled systems of all sorts, including ecologic, economic, and social, which not only are replete with feedback, but which, in addition, require theory that flexibly evolves with evolving systems’ configurations. Arguably, such theory can help us avoid rigidly designed, inflexible interventions and point the way toward sustainable policies. Insights from ecological theory may help guide progress toward that goal.

Read more thoughts on the COVID-19 pandemic from complex-systems researchers in The Complex Alternative, published by SFI Press.

Reflection Footnotes

1 W.B. Arthur, 1999, “Complexity and the Economy,” Science 284: 107–109,

2 N. Goldenfeld and C. Woese, 2011, “Life is Physics: Evolution as a Collective Phenomenon Far from Equilibrium,” Annual Review of Condensed Matter Physics 2: 375–399, doi: 10.1146/annurev-conmatphys-062910-140509

3 J. Harte, K. Umemura, and M. Brush, 2021, “DynaMETE: A Hybrid MaxEnt-plus-Mechanism Theory of Dynamic Macroecology,” Ecology Letters 24(5): 935–949, doi:10.1111/ele.13714

4 E.A. Newman, 2019, “Disturbance Ecology in the Anthropocene,” Frontiers in Ecology and Evolution 7: 147,

5 Harte, Umemura, and Brush (2021).