At a Feb. 15 workshop for Zumba instructors in the South Korean city of Cheonan, one person infected with Covid-19 spread the disease to seven others, who then passed it on in the classes they taught, with the resulting outbreak infecting more than 100. In early March, one member of the Skagit Valley Chorale in Mount Vernon, Washington, seems to have infected as many as 52 others at choir practice. Then there’s the guy at the seafood processing plant near Accra, Ghana, who was reported this month to have infected 533 co-workers.
These “superspreading” events have become a trademark of the new coronavirus — at first impression quite a scary one. But most people who get the disease don’t pass it on to dozens of others, and many don’t pass it on to anyone at all. One new global study estimates that about 10% of those infected with Covid-19 cause 80% of the secondary transmissions; another study focused on Israel puts that share between 1% and 10%. This imbalance explains a lot about why Covid-19 has spread so unevenly and unpredictably around the world. It also, perhaps counterintuitively, appears to make the disease easier to control than it would be if superspreaders weren’t so important.
The crucial variable in standard epidemiological models is the basic reproduction number, or R0, which is the average number of people someone with the disease is likely to infect. Actual susceptible-infected-recovered models are a bit more complicated than this, but here’s what you get if you start at one infected person and multiply by 2.25 (most estimates of Covid-19’s R0 are between two and three) over the next 10 periods — which I have arbitrarily deemed to be weeks, not too far off from the time period over which new Covid-19 infections develop — with results rounded to the nearest integer.
For sexually transmitted diseases and those carried by the water supply or by “vectors” such as mosquitoes, it has long been understood that such models aren’t the most useful representation of how infections spread or can be controlled. Get that one really promiscuous person to stop being so promiscuous, or shut down the pump on that one contaminated well, and you can have a really big impact on the spread of AIDS or of cholera. For directly transmitted respiratory diseases such as influenza, though, this was seen as less of an issue or an option.
In a 2005 paper in the journal Nature that has been getting a lot of deserved attention lately, two epidemiologists and two mathematicians pointed out that while maybe it wasn’t an issue for the flu, there are other diseases that spread through casual personal contact where transmission does seem dominated by a minority of big events. “Using contact tracing data from eight directly transmitted diseases, we show that the distribution of individual infectiousness around R0 is often highly skewed,” they wrote. “Model predictions accounting for this variation differ sharply from average-based approaches, with disease extinction more likely and outbreaks rarer but more explosive.”
The disease that was the main focus of the paper was the coronavirus-caused Severe Acute Respiratory Syndrome that had emerged in China two years before and spread rapidly into several other Asian countries and Canada before being contained. The authors devised a new variable, “k,” to reflect the distribution of individual infectiousness, with a low k meaning a more skewed spread. They assigned SARS a k of 0.16. Estimates of the k of the pandemic influenza of 1918 hover around 1, journalist and molecular biologist Kai Kupferschmidt reported last week in an excellent Science magazine account of the superspreading phenomenon. In response to the article, the lead author of the 2005 Nature study, Jamie Lloyd-Smith of the University of California at Los Angeles, tweeted that his provisional estimate of Covid-19’s k is 0.17. (Epidemiology Twitter is where it’s at, people.)
What can we do with these estimates of k? In 2005, Lloyd-Smith and his co-authors ran computer simulations of thousands of hypothetical epidemics, and found that diseases with a k nearing 0.1 were much more likely to fizzle out on their own or be stopped by modest control measures than those with a k of 0.5 or higher.
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To better understand why that’s so, and in hopes of imparting some of that understanding to readers, I put together a much simpler model of a disease with a dispersion of infectiousness in which 9 out of 10 people have a reproduction number of 0.5 and 1 out of 10 a reproduction number of 18. This works out to 10% of cases causing 80% of secondary infections, as found in the study cited above, and an average R0 of 2.25, as in the chart above. To make easy work of this on a spreadsheet, I just assumed that in 10% of the weeks everybody with the disease gives it 18 others, and in the other 90% they give it to an average of 0.5 others, which is terrible epidemiological modeling but I think still gets at the basic dynamics at work.
The key to these dynamics is that the highly infectious weeks and the less-infectious ones are distributed randomly. Over long periods of time, 10% of the weeks will be highly infectious ones, but as with, say, betting on a number in roulette, there will be long droughts and occasional clusters. I generated a bunch of random series of numbers from 1 to 10, assigning 18 infections per person to the weeks that came up as 10 and 0.5 to the rest. Here’s how my first series worked out, with the numbers rounded to the nearest integer.
So that epidemic started strong, with 18 new infections in week one, then fizzled out as subsequent weeks kept cutting the numbers in half. The second had a little more momentum.
It wasn’t until the 11th run, though, that I generated an epidemic that surpassed the one I got when I just assumed that each person infected 2.25 others — and even it had fallen behind by week 10.
Note that I haven’t proved anything here other than I now know how to use the RANDBETWEEN function in Excel, but this does seem to be a clear example of how random chance coupled with significant skew can deliver huge variability in outcomes. Countless pixels and even more computer processing power have been devoted to sussing out which characteristics and policies have been responsible for deadly Covid-19 outbreaks in some places and mild ones in others, and some of the conclusions of these investigations surely have merit. But given the apparent large variability of infectiousness, it seems likely that there would be big differences in the speed and severity of Covid-19 outbreaks around the world even if everybody lived in similar circumstances and every government followed identical policies.
The most important lessons to be derived here may spring from the fact that the variations in infectiousness are not entirely random. In the future, a team of eight mostly U.S.-based researchers speculated in yet another new paper on the phenomenon, it may be possible to identify those likeliest to be superspreaders by demographics, viral load or other physical characteristics. In the present, it’s already pretty easy to identify specific behaviors and locations that lend themselves to large-scale Covid-19 transmission, with singing, yelling, talking loudly or otherwise engaging in behaviors likely to spread the virus in the crowded indoor spaces implicated in most of the major superspreading events.
The key role of such events may help explain why, as my Bloomberg Opinion colleague Elaine He demonstrated with a remarkable set of charts, the strictness of government lockdowns in different European countries did not seem to be correlated with success in slowing the spread of the disease, although their timing did. Once you’ve put a stop to large, indoor gatherings with lots of yelling or singing, there may be diminishing returns to other restrictions. This may also help explain why epidemic models that did not assume great variability in individual infectiousness so wildly overestimated how fast the disease would spread under relatively relaxed restrictions in Sweden.
Another implication of Covid-19’s superspreader skew, according to several recent papers, is that even in the absence of widespread testing for the disease, low-tech efforts to isolate those with symptoms and track down their contacts can be quite effective in slowing its spread. Preventing just one superspreading event in this way can have a big impact, whereas if transmission were more evenly distributed, isolation efforts would have to be quite exhaustive to succeed. I would also suggest that widespread wearing of even not-very-effective cloth masks should cut down on the likelihood of superspreading, but then I am always looking to justify the investments I have made in building a family mask stash.
Finally, as South Korea in particular has experienced again and again over the past few months, superspreader events can allow Covid-19 to make rapid comebacks after periods of decline. They’re a reason to remain extremely vigilant about the disease even when you think you have it on the run. But they are also a reason to hope that it can be contained in a way that, say, influenza probably cannot. – Bloomberg
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