# How large a gathering is too large during the coronavirus pandemic?

The math of social networks can create a roadmap to a group size that still curbs spread

A question has perplexed public officials trying to curb the COVID-19 pandemic: How large of a group of people is too large?

As the spread of the coronavirus has gathered speed, U.S. officials urged limits on large gatherings, constantly scrambling to reduce the definition of “large.” First, meetings of more than 1,000 were discouraged, then 250, 100, 50 and 10. As many states institute orders to stay at home, all nonessential gatherings are being banned.

But no scientific rationale has been cited for any particular number. Getting the right answer is crucial. Too large and you don’t control the epidemic. Too small, and people’s lives and livelihoods may be upended, for insufficient social benefit.

“I am not aware of any quantitative modeling informing those decisions,” says Lydia Bourouiba, a physicist and epidemiologist at MIT. “They weren’t based on events.”

Now, a new study is providing one roadmap for coming up with an answer. There is no gathering size that can eliminate all risk. But there is a threshold between curbing the epidemic and having it spread like wildfire, and that number is most likely not zero, the researchers conclude. The finding could have implications not only for slowing the pandemic, but also for figuring out how to eventually return to normal life without causing a new surge in cases (SN: 3/24/20).

In the study, posted online March 12 at arXiv.org, five epidemic modelers showed mathematically how an epidemic can be controlled without banning all get-togethers. Their model includes a version of the “friendship paradox,” which says that your friends in a social network on average have more friends than you. When an epidemic strikes such a network, large gatherings are especially bad because they attract people who have more contacts than average — and hence are more likely to already be infected.

It’s possible to determine the dividing line between an effective and an ineffective intervention, the team found. In one hypothetical epidemic, if you banned gatherings larger than 30, the epidemic would rage on. But if you banned groups larger than 20, it would eventually die out. The threshold of effectiveness, for this particular social network model (one in which the friendship paradox was fairly strong), was 23.

“I’m confident that there is a threshold,” says Laurent Hébert-Dufresne, a computer scientist at the University of Vermont in Burlington who developed the model. “I don’t have confidence in the exact number 23.” The threshold for COVID-19 is still unknown, and, he adds, “the cutoff could be very population specific.”

What’s significant, Bourouiba says, is the idea of computing the size of the safe group, not the actual number in this hypothetical case. A maximum gathering size of “23 leading to a collapse of the epidemic has to be taken with a grain of salt,” she says. “But the concept is important, because sheltering at home is not going to be sustainable forever.”

So far, public officials have been reducing maximum allowed group size without any precise formula. “The declining number of recommended people is a way of signaling that we are getting more and more serious about the need to be socially distanced,” says Marc Lipsitch, an epidemiologist at the Harvard T.H. Chan School of Public Health in Boston. “I’m not sure that there is a particular number that is magical.”

In part, the recommendations are based on the idea that the risk of a large gathering increases as the square of the gathering size. That is, a gathering that is 10 times larger will offer 100 times more “transmission opportunities,” says Lipsitch.

But according to Hébert-Dufresne, this rough calculation actually underestimates the danger of large meetings, because of the friendship paradox. It also doesn’t take into account the dynamics of the epidemic, which is precisely what creates the threshold between large and small gatherings.

The model in the new study, which hasn’t yet been peer-reviewed, represents gatherings as highly connected cliques, in which all people present are exposed to all the others. Hébert-Dufresne, who worked with colleagues from Université Laval in Quebec, compares an epidemic in such a network to a bonfire. You need two things to build a fire: kindling, which gets the first flame started, and larger branches, which transmit the fire from place to place. In Hébert-Dufresne’s model, small gatherings form the kindling, and large gatherings are the branches. To keep the fire from spreading, you don’t need to remove the kindling — only the branches.

Telling the difference between kindling and branches is where the mathematical model comes in. The dividing line between small groups and large groups depends on three factors: the disease transmission rate, the distribution of clique sizes, and the distribution of clique membership (how many cliques do highly social people belong to?).

Right now, the last two numbers are completely unknown, Hébert-Dufresne says. But with enough data on social networks, it might be possible to figure them out.

“The people with vast network knowledge are Google, Amazon, Apple, Twitter,” says Simon DeDeo, a professor of decision science at Carnegie-Mellon University in Pittsburgh. “If I were the government right now, I would fly out to Silicon Valley and get this data.”

Lauren Ancel Meyers, an epidemiologist at the University of Texas at Austin, agrees: “I’ve written a plea for sharing of geolocation and social media data,” she says. “We really need a better understanding of how people move and come into contact with each other in schools, workplaces and their everyday lives.”

Hébert-Dufresne’s network is far from being the last word. It ignores many other kinds of heterogeneity, such as the age structure of the population (which is especially important for COVID-19, as the elderly are the most vulnerable (SN: 3/4/20)) and differences between cliques. “A school is different from a factory,” says Bourouiba.

Many other network models do take into account these variables. Lipsitch, Meyers and others all work with models that include a great deal more detail, going down to the level of contacts between individuals. “You can incorporate an incredible amount of detail,” Meyers says, “but then it takes many simulations to extract general results.” And that can take a lot of time.

The one developed by Hébert-Dufresne and his colleagues is comparatively simple, but unique in treating gathering size itself as a source of diversity. “Some people are doing more complex models, but just in terms of getting at the idea of a cutoff, it’s a powerful idea,” says Hébert-Dufresne.