Ignoring Country Cohorts
I assume that many of you have been watching the “total coronavirus cases outside of China” number. If you look at them on a log chart, they appear to be growing perfectly exponentially and doubling every 4-5 days.
This perfect straight line is obscuring a very, very important underlying fact. Within countries, the second derivative of cases has been falling rapidly (please reference the “first 60 days” tab).
This is a result of a few important factors:
1. Initially, in country growth is extremely high. Prior to reaching 100-1000 cases, countries do a poor job of testing. As a result, they accumulate a backlog of cases that are discovered quickly upon commencement of testing. This makes the growth rate appear to be very large for the first 1000 cases. The aggregate data we are seeing today is incredibly skewed as a result of this effect.
2. After 1000 cases, governments are actually jumping into action. They implement large scale testing (like we are currently seeing in South Korea), quarantines (like we are currently seeing in Italy), and other aggressive policies. Additionally, some measure of social distancing goes into effect as people start to worry about their own health. The effectiveness of action can be seen in the chart above.
My prediction:
By the week of March 16th, the slope of the line on the log chart will halve and subsequently continue to fall like we’ve seen in China. Total cases ex-China will approximately double between March 16th and March 24th (half the rate we’re seeing today), and will continue to fall.
Despite the sense you might be getting from the 100lbs of rice in my pantry, I think we’ve seen some very positive news over the past few days and I’m becoming much more optimistic that this will not be a generational event.
Note that the graphs on the “first 60 days” tab of the site you linked to, are on a log scale. The slope on that scale says something interesting, but it may not be what you think.
Decreasing slope on a log-scale plot means R0 has decreased. R0, the virus’s basic reproductive number, is the average number of people that a given infected person will go on to infect. If R0 is greater than 1, the epidemic grows; if it’s below 1, then the epidemic shrinks. Public health measures decrease R0. For example, if closing public transit would prevent half of all transmissions, then it would reduce R0 by half.
As more measures like social distancing and drive-through testing centers are added, they decrease R0 further. Since the cheapest measures are done first, each successive measure costs more; lowering R0 can’t be done without limit, because eventually we run out of mitigations we can afford. If the mitigations are enough to get R0 to <1, then the epidemic can be contained. If we can’t get R0 to <1, then the mitigations buy time and spread cases out, without significantly affecting the eventual number of cases.
If the epidemic grows to a large fraction of the population, then the fact that recovered patients are immune will itself lower R0, putting containment back within range of available mitigations. This is the point where the growth would stop.
So, the all-important question is: Can countries get R0<1, before they’ve emptied their toolkit? And, will they have to pay extreme costs (like China shutting down all industry) in order to do so?
This paper looked at Wuhan, and found that the extreme measures taken there lowered R0 from 3.86 to 0.32. This is good news, because it suggests that the measures taken in Wuhan were stronger than necessary; they prevented ~11/12ths of transmissions, when they only needed to prevent ~3/4ths. This is good news, because most countries can’t implement measures as strong as Wuhan’s; knowing that Wuhan’s measures were stronger than necessary means that containing outbreaks might be feasible.
If a country has R0<1, then the number of new cases per day will decrease, after a time delay about equal to the incubation period. Unfortunately, that’s not what we’re seeing in any of the places where SARS-CoV-2 has taken hold. What we see is a decreasing slope on a log scale—an indication that R0 has been reduced—but not an indication that it has been reduced to less than 1. However, it’s still quite early, test-kit and mask production has not yet caught up with demand, and strategies are still being refined, so there is hope.
Thanks for your comments! Yes, I’m aware this is a log chart (which is why I referenced the second derivative falling). A revised version of a paper actually came out today with some very good data backing up my analysis. Please reference the tab “reproduction numbers over time in the six countries with the most cases currently.” There is also more per country analysis down below on that page.
This is obviously in part because of the quarantine efforts, but my point is that we are significantly overestimating the doubling period currently because all are data on cases outside of China is from countries that only recently started testing at scale.
Last week, it looked like the doubling period was 4 days, by the end of next week we’ll be talking about an 8 day lower bound for ex-China cases.