Understanding "k", "overdispersion" and "R".


This is a good discussion of K estimates from two outbreaks. K is estimated at 0.06 to 0.2. (K of 1 means about 40% do not pass it on at all, at 0.1 it's about 70% that don't pass it on.)

https://www.medrxiv.org/content/medrxiv/early/2020/07/24/2020.06.28.20142133.full.pdf

We need to understand "k", "R", "Overdispersion" and what they mean. And frankly, a lot of reports are getting it wrong in very important ways. And they're describing "superspreaders" in a very misleading way that is likely to influence public behavior to our detriment if it doesn't get corrected.

Understanding "k" will add to your understanding of how it spreads, but it DOES NOT invalidate forward contact tracing, quarantine, and social distancing. This is a LONG post, but this is important stuff that is now trending in news coverage.

Knowing that COVID is overdispersed (which we've known since January at least) means that the top targets to avoid are large gatherings of densely packed people near each other for extended periods, doubly so indoors. Huge indoor stadiums and megachurches are probably the worst, packed outdoor stadiums, political rallies, political protests if people are standing shoulder to shoulder, bars, indoor restaurants, classrooms, indoor house parties, should be avoided if possible. (Of these, closing schools has by far the highest social cost, and is probably the last one that should be shut down.)

This article : https://www.theatlantic.com/health/archive/2020/09/k-overlooked-variable-driving-pandemic/616548
gets a lot of it right, and comes really close to getting other parts right, but states them in a way that makes them absolutely wrong.

(A side note - the author also gets rapid antigen testing VERY wrong. They are NOT extremely good at identifying negatives, they have MUCH higher false negative rates than PCR. They do have very low false positives - if they say you have it, you do - but they can give false negatives at disturbingly high levels because they're not sensitive enough to pick up early cases. This was part of the problem with the Rose Garden event.)

"k" and "overdispersion" are concepts you already understand if you've understood the "80/20 rule". 20% of the people will account for 80% of...well, practically anything you measure. Wealth, phone calls made, internet usage, guns owned, social media posts, you name it. Most people will do a smallish number, but a few at the high end do crazy numbers.

"k" just acknowledges that it isn't always 80/20 - sometimes it's 60/40, or 90/10, or even 99/1. This is why, with Rt = 1.0 (average one new case per infected person) it's also true that most infected people don't pass it on to anyone. With 10 infected people, you may have one that infects 5 more, one that infects 3, two that infect one each, and six that infect nobody. 10 people still infect 10 others, but most infect nobody.

This Atlantic article reads like an intelligent author interviewed a real expert, understood much of what was said...and then wrote it up in a way that makes it misleading at best and outright wrong at worst.

Consider this: "We’re often told that if Rt, the real-time measure of the average spread, is above one, the pandemic is growing, and that below one, it’s dying out. That may be true for an epidemic that is not overdispersed, and while an Rt below one is certainly good, it’s misleading to take too much comfort from a low Rt when just a few events can reignite massive numbers. No country should forget South Korea’s Patient 31."

First, if Rt is less than one, the pandemic IS dying out. That is what Rt means! It doesn't mean that a big outbreak won't change it, or that it's being spread equally by all, but Rt is a measure across the population of whether the number of infected people is rising or falling.

Second, Korea's Patient 31 ended up causing a cascade of about 5000 infections BECAUSE Korea's Rt was way above 1 at that time. If Korea's Rt was 1 or less, that church couldn't have been open and the cases wouldn't have exploded.

The article goes on to explain backward contact tracing, which is useful and important, but the article makes it sound like this is a new revelation or that it is better than forward tracing...neither of which are true. It's a powerful complement to forward tracing that extends its reach.

The article describes Korea's "Patient 31" as "generating more than 5,000 cases in a megachurch cluster". That's technically sort of true, but paints a wildly inaccurate picture. It is not the case that she walked into the church and infected 5,000 people there.

This is a brilliant infographic on the case.

https://graphics.reuters.com/CHINA-HEALTH-SOUTHKOREA-CLUSTERS/0100B5G33SB/index.html
I've reproduced a key piece of the graphic below. Note that she attended the megachurch on March 9. It was two hours long, and the attendees sit on the floor, literally shoulder to shoulder, in a gymnasium-like setting. She likely infected somewhere between 10 and 50 people, although she was asymptomatic.

She went back on March 16, this time with symptoms. But here's the key - the 10 or 50 people she infected the first time also came back, nearly all of them highly contagious. They spent two hours packed shoulder to shoulder in a congregation of thousands. Now hundreds were infected, and the first wave was already infecting their households and coworkers.

Fortunately, South Korea had massive testing capacity at this time (far more than the US) and a huge and very competent tracing force. When case 31 finally got tested after the second visit, they used FORWARD TRACING to start testing 1300 church members, and found many cases. (Backward tracing - where did patient 31 pick it up?- was much less fruitful.) They used forward testing and quarantines to limit and track the spread, and over the weeks that followed, that cluster ended up being responsible for over 5,000 infections, a large fraction of the total in the country.

We don't know if patient 31 was unusually contagious or not. We don't know if she directly infected 10 people or 100. That actually doesn't matter. What does matter is that she was in a tinderbox with so many people, so close, indoors, for two hours. And that everyone she infected was able to come back for a second round a week later, before anyone was identified as a carrier. And that's the key to understanding superspreading events.

Stop thinking of "superspreaders" as amazingly infectious people. There certainly is variability in viral loads. Most people are very mildly contagious, if at all. Many are contagious, but perhaps only for three days or a week. Given equal viral loads, some people produce more aerosols than others. But think of a superspreader event as requiring someone who's highly contagious (but maybe not at all unusual) in an environment with close contact with many others for a long period of time.

Dr. Zubin Damania (ZDoggMD) is a brilliant populizer of medicine, but apparently not all that intuitive with numbers and statistics. His recent shows have mentioned superspreaders and overdispersion, but he keeps describing it as if it's a biological characteristic of the spreader, and downplays or overlooks the importance of the event setting.

The danger here is that people start thinking "almost all of this is caused by a few superspreaders, why should we all wear masks?"

It's exactly the logic that leads to "I'm feeling fine, I don't need a mask". You don't know who is a superspreader, just like you can spread COVID-19 in your presymptomatic phase. Also note that superspreaders - say, people who spread to more than 10 others - are still likely to be a fairly small fraction of the total spread, as those countless cases spreading to one, two, or five others add up.

I'll wrap up with some comments on isolation, quarantine, and forward vs. backward contact tracing, using the recent white house debacle as an example of how it SHOULDN'T be done.

When you test positive, you need to ISOLATE immediately. You are infected, you are likely contagious, you need to stay away from others for at least a week, ending with a negative PCR test. (By 2 weeks you're probably good even if still PCR positive because it's so sensitive.) Trump may have done public events without a mask after testing positive, which is utterly unforgivable, but that was far from the only outrage.

When you're infected, it's important that you or a contact tracer trace FORWARD to all the people you may have infected. These people, once notified, need to QUARANTINE immediately for two weeks from date of exposure, possibly less with negative PCR after day 7 or so. This is because they may be carrying it, and we want to stop the spread. This is the main tool that the Koreans used to trace down the ever-expanding outbreak from the megachurch. Most of the infected were not in the church building, they got it from people who got it from members.

It's helpful to do backward tracing. You'll recall that when POTUS, FLOTUS and Hope Hicks all had symptoms within a day or two, I predicted they would find an event within a couple days of 9/25 where they were all infected by somebody else. Turns out it was 9/26 at the Barrett Rose Garden event.

We still don't know who was the index case, but BACKWARD tracing should be used to have everyone who attended quarantine themselves and get tested. Not because Trump may have infected them, but because whoever infected Trump probably infected some of them. And anyone who tests positive in that group needs to do forward tracing to quarantine and test their contacts. By the time you start getting symptomatic cases, backward tracing has limited value because the infected attendees are already infecting others and will soon be out of their infectious phase. It's the forward tracing from the next wave that really stops the spread, backward tracing helps us understand the process better. For backward tracing to work, the contact tracing has to be lighting fast, and it's best if the first case is detected before being symptomatic.

It is abundantly clear that this didn't happen, especially the nearly complete disregard for quarantine. Senators and administration officials continue to perform duties, take airline flights, etc., exposing hundreds or thousands more.

In conclusion, some take-home messages:

"k" and "overdispersion" mean we should prioritize avoiding superspreader events.

Masks help. Distance helps. Outdoor helps. Small group size helps. Short duration helps. Rapid test screening helps. None of these are a guarantee. (The rose garden event scores 2/6 here.)

Forward tracing is essential and extremely effective.

Backward tracing improves understanding and can help stop spread if it's done very quickly with aggressive, immediate testing of contacts.

Rapid antigen tests are less accurate, but very, very useful in the right circumstances. Screening for a mask-free event is NOT one of those circumstances.


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