# Princeton Election Consortium

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## Bending the Curve on Covid-19: Doubling Times as a Simple Metric

#### April 2nd, 2020, 8:27am by Sam Wang

How will we know if we’re making progress on containing the coronavirus epidemic? One of the best measures, relatively free of biases that can creep in, is how long it takes for total deaths to double.

When this “doubling time” is 3 days or less, that’s been an indication of runaway spread of coronavirus. If it gets longer, that indicates a slowdown – either because shelter-in-place is starting to work, or someday, that the virus is spreading less efficiently (through changes in weather, improved treatments, vaccination, and inoculation).

First the key graph:

And now, an explanation.

It has been the case empirically that without taking measures to contain the spread, the number of corona deaths has doubled approximately every 3 days. Here is world data to show you that point.
Over and over, the shortest time for deaths to double is 3 days. Why would that be?

The likeliest explanation is that the number of people infected doubles every 3 days. This would depend on how likely an infected person (with or without symptoms) is to infect others, on a per-day basis.

Think of it as being like “compound interest.” Professional investors know the “rule of 72”: if you have an annual rate of return of X percent, your money will take 72-divided-by-X years to double. It’s the same for pandemics – except we’re talking days, not years.

Imagine that you are infected. I know, you feel fine. But you can still be infectious. What if you had a 24% chance of infecting someone during the day? If everyone had that that probability on average, then the number of infected people would double in 72/24=3 days. And that’s what we’re seeing.

Social distancing reduces this percentage. If you can get your percentage down by half, then the probability of infecting someone would be 12% per day. And the doubling time would be 72/12 = 6 days. That’s what bends the curve.

In other words, changes in how you personally behave every day can prevent many deaths later. By now you know the drill:

• Stand at least 6 feet (2 meters) from people
• Wash your hands before touching things that other people touch.
• Don’t touch your face with an unwashed hand.

The goal of public directives is to get people to do all of this, on average. But people are different. Under normal conditions, some people stay at home and infect nobody. Extroverts and lovers of crowds infect lots of people. On average, it probably works out to about a one in four chance every day, about 25%.

What are the odds that you touch a door knob, someone else touches it, then they touch their face? Just once during the day? Surely at least one in four. Social distancing reduces the one-in-four chance to one-in-40. And that turns your awesome viral “investment” into the equivalent of a money market fund.

This buys time for the medical system to accommodate patients, and maybe even lets the epidemic peter out.

Now let’s look at doubling-time trends around the world and here in the United States.

Above is the doubling time of deaths in selected countries worldwide. The data come from the Johns Hopkins University covid-19 site. South Korea, which moved aggressively to contain the epidemic, has successfully gotten the doubling time to slow to 25 days. They reported their first coronavirus case on January 20, one day before the United States. Their epidemic is under control – while ours is still roaring along.

What’s the difference? Three reasons:

1. Social distancing. The United States was very slow to adopt this.
2. Testing. As of March 17, South Korea did 70 times as much testing per capita as the United States. Another estimate from around that time has the ratio as about 20. That ratio is likely even higher now that the infected population has exploded. In short, testing in the United States is inadequate. (Here in New Jersey, I currently know eight people with probable coronavirus, but only two have been tested.)
3. Contact tracing. South Korea was aggressive about contact tracing, even using a smartphone app to help people identify nearby cases of live infection – and avoid them.

Even now, we have yet to catch up. A big reason has been the failed response at a national level, and lack of relevant or accurate guidance from the President. That leaves containment to the states. And as of April 2, only 39 out of 50 governors have implemented shelter-at-home measures.

Now let’s look at regions:

The only region to get the spread under control is the Pacific states (Washington, California, and Oregon combined). Other regions are still experiencing runaway spread – though overall, trends are encouraging. Since it takes 5 days for coronavirus infection to turn symptomatic on average, and about 14 days to kill a person, it will take about 2-3 weeks for social distancing and sheltering measures to put a dent in the runaway growth.

Here is the doubling time on a state-by-state basis:

Again, measures haven’t kicked in yet, even in places like Ohio where the governor worked aggressively and early.

Don’t get too excited, Floridians. You still have few cases, so your curve is noisy. Many thousands of cases are incubating now, and that curve will probably come right down to the pink zone of runaway spread in the coming days.

>>

Now, why am I not talking about confirmed cases? Because they can contain terrible biases.

Here, the growth appears to be slowing down. But that is probably because of limited availability and use of testing. For example, the number of tests run per day in New Jersey has flattened, despite the fact that we’re crashing into disaster.

So although we are calculating these curves too, we’re not emphasizing them.

I will post more on this as conditions change.

Thanks to Ben Deverett for designing the plots, Lucas Manning for implementing the pipeline, and Johns Hopkins University and health authorities worldwide for the data. GitHub: https://github.com/Princeton-Election-Consortium/covid-19

Tags: Redistricting

### 21 Comments so far ↓

• This analysis by Mark Sumner at Daily Kos has persuaded me to view such charts with extreme doubt. As he says,

The problem with the United States, far more than any other country on the chart, is that the results are constrained by testing. The numbers we’re seeing aren’t a measure of the spread of COVID-19 in this country; they’re a measure of our ability to detect COVID-19.

Not only are the numbers of cases strongly influenced by how much testing is available in a given area, so are the numbers of *deaths*. I’ve seen numerous anecdotal reports from hospital workers of deaths they’re pretty sure are due to COVID-19 but aren’t recorded as such–because they don’t want to use one of their few available tests to check.

Why should we trust analysis of the numbers, when we can’t trust the numbers?

• This is why we’re tracking deaths, not cases. So yes, many other people are doing this wrong. The writeup above has fewer problems.

• I’ve read that even deaths are under recorded at this point. Their may be some swabs laying around that get tested eventually that will alter the current daily figures. Many are still under the guidelines that you only test if a positive result will change the course of treatment so we can only speculate.

• It would change a little but not that much.

For example, imagine we start off catching 100% of deaths, and six days later we catch 80% of deaths. That would change the doubling time from 3 days to 3.5 days. And it’s a one-time difference – over a period of weeks, the error in the doubling time would be smaller.

• 538_Refugee

I’ve been keeping an eye on cleveland dot com for a more local perspective on the numbers. They include hospitalizations and ICU cases also. Locally and at the Cleveland Clinic I can tell you they didn’t seem to have a lot of headroom in the ICU units under ‘normal’ circumstances.

Maybe it is time to come to the realization that strict capitalism doesn’t work for everything? The over riding mantra that business hides behind is ‘We have a fiduciary responsibility to our share holders (to make them as much money as we can because our compensation packages depend on it)’. That’s fine for sourcing paper clips but profitability should not be the first objective of the health care industry. Maybe that should be the over riding issue in the up coming election?

• I saw a trend earlier in the month where the number of tests tracked the number of cases very closely (about 9x higher). Number of cases was smoothly exponential, as expected, but it puzzled me that the number of tests was too.

My conclusion was that doctors were strictly following rules to only test patients with certain severe symptoms. As a result, cases drove tests, not the other way around.

A further conclusion was that we never actually had a test shortage per se, given that restriction on when to give a test. We might have come close, but if we’d really been short, the curve of number of tests would have plateaued.

• 538_Refugee

Well this is disturbing.

https://www.nytimes.com/2020/04/03/nyregion/coronavirus-new-york-update.html

N.Y. Virus Deaths Double in Three Days to Almost 3,000

• Kurtis Behn

Have these charts been updated (automatically?) since this was posted? The text seems to refer to data through about 4/1 (which would make sense given the date of the post) but the charts go further which is a little confusing when reading. I would definitely welcome an update of this analysis with new data!

• Surely if the testing ratio remains a constant (proportion of the total population) then estimations of the case count doubling time will be independent of the testing ratio, and comparable between countries?

Only if the testing rate is continuously varying in time will there be a potential bias in the estimate. It cannot be chance that most countries showed a similar doubling rate of 3 days in the early phase of the epidemic, even though testing rates vary between countries by easily a factor four? I imagine a similar bias could also affect doubling time estimates made using deaths data, if diagnosis policy or available hospital resources are varying with time?

• Yes, but what if the testing capacity is finite? In any event, we have those plots too, will link later today.

• Alan Cobo-Lewis

Agreed about doubling time as simple metric. That’s what I’ve been calculating for Maine, for example at https://www.facebook.com/photo.php?fbid=767625053644268

I’ve had to live with analyzing Maine’s positive cases, however, because while you’re absolutely correct that deaths are much less subject to bias, they (thank goodness) have a small-n issue in Maine.

• Partha Neogy

Doubling times seem to have dipped a bit (except for the Rockies and the mid-Atlantic) after April 13th. Does that bother you, or is it within expected variations?

• Kurtis Behn

I’ve noticed the same but I don’t see any general concern over it. Maybe this is just better counting? I know Colorado added a chunk of older deaths into their totals. And NYC obviously revised their counts. (Starting to happen abroad too.)

• In the case of New York, we think there were hiccups in the JHU data feed – not errors, but just delays in reporting that led to odd estimates.

We have now switched to NYT data, which we think will be more suitable for this purpose. For example, they graph their data in a related way to what we’re doing, and therefore have to inspect it. Also, the NYT database seems to have more deaths than the JHU database, which makes us believe that it is more comprehensive.

• PJC

This may be a late comment to your write up, but this chart probably shows the 3-day effect better than any other.

https://aatishb.com/covidtrends/

You have to select for deaths vs cases and increment the doubling time line from 2 to 3 days.

Once you set the doubling time to three days it does indeed become clear that nearly all nations follow this line identically … until they don’t. Some fall right off a cliff.

The underlying data is from Johns Hopkins.

If there is any question about the axes etc there is a minutephysics video on youtube that explains the entire chart.

• PJC

To the hygiene protocol you should add

Wash your hands *after* touching things that other people touch.

Since hands are a bi-directional vector they require a before and after.

Realistically we can’t always wash before and after touching everything (think grocery store), so hand sanitizer can be a short term substitute between last touching a vector point and washing.

• Ruhrfisch

Thanks so much again for these doubling times, which are very useful. I have several suggestions which I hope may improve them and their utility.

1) The y-axes of the graphs are currently scaled so that almost all of the plots are off scale (not visible) and should be re-scaled.

2) The link says “Data for all 50 states” but only 43 states and DC are shown. My guess is the states not displayed may have very few cases (HI) and/or extremely long doubling times, but there should be something shown or explanatory text for the missing states.

3) The doubling times (DT) are currently updated twice a day (AM and PM) and may be more consistent if only updated once a day. The five most recent US DTs are as follows:
The June 8 PM DT was 101.9 days with 110,422 dead, compared to a DT of 77.7 days a week ago.
The June 9 AM DT was 204.3 days for the same dead, compared to a DT of 93.9 days a week ago.
The June 9 PM DT was 137.8 days with 110,966 dead and same DT a week ago.
The June 10 AM DT was 246.0 days for the same death count, compared to 90.2 days a week ago.
The June 10 PM DT is 107.9 days with 112,174 dead and DT a week ago.
I believe the AM DT still uses the previous day’s death count, but recalculates based on the new number of days, and compares to the PM DT from a week ago. The PM DT has the new death count, but the same days and week ago DT. Since each day has just one death count and one week ago comparisons seem to always be to the same (PM?) DT, I think it would make sense to also have only one DT per day (and this would avoid the large swings in the values, where the AM seems to generally be much higher, i.e. 246.0 d to 107.9 d today).

4) It is clear that the death data reported is lower on weekends and early in the week, and higher later in the week. Would it make any sense to use some sort of moving average to calculate the doubling times? http://www.danreichart.com/covid19-reports does some smoothing (albeit for diagnoses DT, not deaths).

Thanks again for a very useful metric for (most) states, and I hope these suggestions may be useful.

• Robert wills

Is there any place that true doubling times are reported with rate of change of doubling information, ie, are things getting better or worse and how fast?
I agree on bias in number of cases bias, but isn’t it relatively related to true number?
Updates would be appreciated, thanks for your effort and the web site.

• Robert Wills

Also, are you worried that better clinical practices are changing the death rate and that you are not comparing same “death rate” from early treatment methods to better current treatment methods?
I have a friend who works in a hospital and care has changed and outcomes are better.

• Henry Lowendorf

Please comment on the apparently regular 7-day cycling of doubling times. If there is less reporting on weekends, then presumably deaths on weekends appear the next week. How can you systematically fix this systematic error?