Covid-19 and exponential growth: Actual cases = 22 × Reported cases

Covid-19 is growing exponentially. Exponential growth is funny (counterintuitive). For a while, nothing seems to be happening. Then very quickly “everything explodes.” (Also strange: this visual behavior repeats at different scales.) For example, here is a chart of reported cases in Italy. (Source: dev_thetromboneguy )

Exponentially growing data looks small at first; then it appears to zoom upwards

As an applied mathematician (among other things), I’ve worked with lots of exponential growth. I fear that reporters and others are not understanding the implications.)

  • Covid-19 infections are growing exponentially, with a doubling time of 5 days. This is based on Italy, and on China before they took drastic measures to limit spread. (On the Italian chart above, it’s even faster.)
  • The incubation period between getting sick and knowing you are sick averages 14 days.
  • In the US, it is taking multiple days to get tested when you ask for it. Optimistically, a 4 day delay on average.
  • The test results are apparently taking days to come back. Approximately 3 days
  • So total lag between being infected, and being reported in the statistics as a “confirmed new case” is 14 + 4 + 3 = 21 days.
  • This is 4.2 doubling intervals = 21/5
  • 2^4.2 = 22.6 ;2 raised to the 4.2 power is 22
  • So for every confirmed case, there are about 21 more that are sick but not yet reported.
  • This assumes everyone with the disease is eventually tested. If half of people who are sick are never tested, double this number.

This is why we each need to take personal defensive action before there are many reported cases in our city. By the time we know there are many cases near us, it is 3 weeks later and the real number is 22 times higher.

Of course there are many approximations in this model. If you know anywhere that has done similar analysis more rigorously, please post it in the comments or tweet it to me, @RogerBohn

Update March 17: The 20:1 calculation is driven by the 21 days assumed between being infected, and getting positive test results. The average may be lower, and it will fall as more tests become available. On the other hand Italy reports a case growth rate of 35% per day! So the ratio could be as low as 8.

John IOANNIDIS calls for random testing of entire population, which would give an unbiased estimate. That’s not yet feasible for the U.S. due to CDC’s inexplicable foot dragging on test development and procurement. But Korea could do it. The required sample size is large due to uneven incidence in different parts of a country. But a running sample e.g. 1000 per day would give very useful numbers.
If the test is sensitive enough, random sampling would also show how many people are sick with few or no symptoms.

One Day, a Machine Will Smell Whether You’re Sick – The New York Times

Sniffing disease markers is a fundamentally promising concept. We know that dogs have very good smell, so that is an existence proof that something interesting can be detected in the air. (In my family’s experience, human smell can also become amazingly good, at least for pregnant women!) In fact, if B.F. Skinner were still alive, I wonder if he would be training pigeons to sniff out disease?

But although air is feasible, it does seem like blood is a better choice because it is likely to have stronger signals and lower noise. Air-based sensors would be non-invasive, so perhaps that is why some groups are pursuing air.

…a team of researchers from the ..Monell Chemical Senses Center and the University of Pennsylvania [are working] on a prototype odor sensor that detects ovarian cancer in samples of blood plasma.

The team chose plasma because it is somewhat less likely than breath or urine to be corrupted by confounding factors like diet or environmental chemicals, including cleaning products or pollution. Instead of ligands, their sensors rely on snippets of single-strand DNA to do the work of latching onto odor particles.

“We are trying to make the device work the way we understand mammalian olfaction works,” … “DNA gives unique characteristics for this process.”

Judging by research at UCSD and elsewhere, I envision tests like this eventually be run as add-on modules to smartphones. Buy a module for $100 (single molecule, home use) up to $5000 (multiple molecules, ambulance use), and plug it into your phone. Above $5000, you will probably use a dedicated electronics package. But that package might be based on Android OS.

This is also another example of Big Data science. It could be done before, but it will be a lot easier now. Blood collected for other purposes from “known sick” patients could be used to create a 50,000 person training set. (The biggest problem might be getting informed consent.)