I’m concerned about the relatively large number of SARS-CoV-2 infections that do not present with the usual symptoms. If these cases are viewed as benign, then we’re going to have a tough row to hoe. We are limited as to what can be done about cases that are entirely without symptoms, but there are probably a fair number of cases that can be detected beyond what we’re now doing.

Testing is important. However, I suspect that most of us tend to view the test results as definitive. By “definitive,” I mean that it’s a straightforward fact, to be relied on as a guide to how we should act. There are no assays (for anything) that are *that* good. Unfortunately, the rt-PCR tests (now, in many different versions) that are used to detect the presence of viral RNA seem to be really good in the lab, but give false negatives about 25% of the time (I checked this out recently with folks at NIH in Bethesda) when used with the usual nasopharyngeal swab (in the field).

In case you think that this is just a cranky old former math prof getting agitated over nothing, consider what happens if you rely on the current tests to screen airline passengers for Covid-19. A not unrealistic thought experiment would have 200 passengers show up for a flight. Say there are 100 uninfected (in epidemiology these are said to be “susceptible”), 100 infected. With the field assay sensitivity at 75% (i.e., 25% false negatives), the test results, on average, should find that there are 75 positive, and the rest negative. Alas, that means that, on average, you expect to find that 25 of the 100 infected passengers have “passed” your screening test. So, you wind up telling (on average) 75 people that they can’t fly today, and you put 125 people (on average) on the plane. Notice that these passengers all have the same test result. The problem is that your wonderful idea of screening passengers has put 125 people on the flight, including 25 people who are actually infected. Would you like to fly on that plane? Starting with the reasonable intentions of allowing people to return to air travel we have created a sort of petri dish of a flight.

My sister-in-law asked about what can be said mathematically about what happens after the airplane lands and the 25 infected passengers interact with family, friends, etc. It’s a good question. What are the expected numerical consequences of loading a plane with 125 passengers of whom 25 are infected with SARS-CoV-2? There are two separate things happening. First, during the plane flight, some fraction of the 100 susceptible people get infected. We can take a very rough guess that this fraction might be around *half*, so, around 50 more people who’d get infected. That’s just guesswork, but our guess is based on a case where lots of folks became infected at a church choir practice, even though they were careful about maintaining distance between people and about how the music sheets were handled. See: https://www.cnn.com/2020/04/01/us/washington-choir-practice-coronavirus-deaths/index.html/.

Thus, when the plane lands, we picture a group of 125 people divided into 3 subgroups: subgroup (A) consists of 25 people who are infectious (shedding virus); subgroup (B) consists of 50 newly infected people who will not start shedding virus for several days (these people are sometimes said to be “exposed” rather than “infected”); and 50 susceptible (i.e., as yet uninfected) people. How much time does it take for the newly infected people to start shedding virus? I don’t know, but there have been estimates ranging from 3 days up to 21 days. It is sure to be different for different people among the 50. For simplicity, we take a single number for incubation for *everybody* in the newly infected group to start shedding virus, 6 days. Note: for those who are familiar with probability theory, the usual procedure would be to take a random variable as the “value” for the number of days . . .

To understand what might happen in the 30 days, say, after the plane lands, one needs to have some familiarity with exponential growth. There are many definitions of exponential growth. We choose one that is convenient. A quantity is said to grow exponentially if there is length of time, say D, the “doubling time,” for which the quantity at the end of any time interval of length D is always twice the quantity at the beginning of the interval. For example, if the period is 3 days, then every 3 days, the quantity would double. In detail, at the end of 3 days, you’d have twice the starting quantity; at the end of 6 days (that is, 3 more days), you’d have 2 times 2, which is 4, times the starting quantity, and so on. At the end of 12 days, you’d thus have 16 times the starting quantity. The idea is illustrated by the fable “The Rice and the Chess Board.” See https://medium.com/@purposefocuscommitment/the-rice-and-the-chess-board-story-the-power-of-exponential-growth-b1f7bd70aaca/

After a bit of thought about the 25 original infections and the resulting 50 immediate infections, we’ll be able to analyze the likely result. My wife and I took a walk one afternoon and considered what might happen over a period of a month, say, 30 days; we came up with a figure of over 35,000 infections. Here are the reasoning and calculations. Look at one of the original 25 infected passengers (the 25% false negatives). This population of 1 undergoes doubling every 3 days, for a period of 30 days, so 10 doublings, which results in a population of 1024 infected individuals. We have 25 such populations, and (ignoring overlaps among the 25 populations) combining these gives a population of (25)(1024) = 25,600. In addition to the original 25 infected passengers, we should account for the 50 people who were infected when the plane landed. We start with one individual among the 50 who were newly infected, and figure out how the population of people infected by that one person grows. That population doubles every 3 days, starting 6 days after the plane lands, so it has 30-6=24 days, doubling only 8 times, so at the end of the 30 days, it contains 256 infected people. We have 50 such populations, and (ignoring overlapping among the 50 populations) these combine to give a population of (50)(256) = 12,800 infected people. To finish, we combine the populations of 25,600 and 12,800 (again ignoring the overlap of these two populations) to estimate a total of 37,400 infections that resulted in 30 days from the original 25 infected people. Notice the assumption that the subsequently infected populations from different passengers do not overlap. That may be a reasonably good assumption in a big city in which there is a large population of susceptible people, but not at all in a small town (where the population of infected people tracing back to one person might include a large part of the population of the town).

Similar reasoning would apply to screening procedures for folks entering a long-term care facility.

And on April 5, USA Today, page 4C, had a piece, “Efficient testing could be key for NFL”. The company Bodysphere based in L. A. claims to have received FDA “emergency use approval for a rapid Covid-19 test that can deliver results in two minutes. The assay . . . uses antibodies in blood to test for current or past infections with, according to the company, a 91% clinical specificity rate. Of course, in a bigger picture, this is viewed as a game-changer for a nation grappling with the coronavirus pandemic.”

The NFL hopes to use such a test to screen players and staff so that they can play football in the fall. If somebody you know were on the staff or on the team, would you like to see them back to work? Although 91% specificity (meaning that 91% of the actual *negatives* are correctly identified by the test) is great, the problem of false negatives (lack of sensitivity), illustrated above for screening airline passengers, doesn’t go away: you are certain of getting some infected people who “pass” your test. I compare the idea to playing Russian roulette. We’d all like to get back to the status quo ante, to be able to work as before, shop as before, and travel as before. However, the seemingly reasonable, attractive idea of using screening to “help us” get back to business more or less as usual is really dangerous!!

Before you get back to whatever we were doing before you started reading, let’s return to the question, “What fraction of susceptible passengers become infected on the flight?” We guessed that a half might become infected. It’s wise to be skeptical of guesswork, so let’s try to double check the guess by trying to calculate probabilities, as best we can. To calculate, we give each of the 25 infected passengers a red ball, and each of the susceptible passengers a white ball. Put all the balls in an urn. Now think of yourself as a susceptible passenger.

Let’s suppose that if you are within 6 feet of an infected passenger, then you become infected, and suppose that if you use the rest room soon after an infected passenger uses it, you become infected. Imagine a sort of “bubble” (in space-time) that you need to avoid to stay uninfected. If you manage to maintain your bubble (keep your distance, etc.), then you won’t become infected during the flight; if one other passenger enters your bubble, then (knowing only what has been determined by the test), you calculate your probability of being infected as the probability of that one person being infected, that is, the probability of drawing a red ball from the urn after having taken out your ball, which we supposed to be white. After removing that white ball from the urn, we have an urn containing 124 balls, of which 25 are red, so your probability of getting infected is the probability of drawing a red ball, which is 25/124. If the first person who enters your bubble is uninfected (susceptible), but a second person enters your bubble, then the probability of your getting infected from the second person is (99/124)(25/123), and your probability of getting infected by these first two people who entered your bubble comes to 25/124 + (99/124)(25/123), which is

25 * 222 / (124 * 123)= 0.3638867033831629 (approximately).

This probability is more than one-third. Clearly, there is also the possibility that you’d have 3 people in your bubble, and so on, so it’s not so unreasonable to guess that the probability of getting infected would exceed 50%. It would depend on how much space was left empty on the aircraft and on the behavior of the passengers.

[AN ASIDE] Notice that we’ve done the calculation with full knowledge of which passengers are susceptible and which are infected, since we created the thought experiment. One could also do the calculation from the point-of-view of a passenger, who doesn’t know for sure whether s/he is infected or not. Think of the test result as a certificate that shows either 4 red balls (positive for the coronavirus, i.e., we’re sure that you are infected) or 1 red ball and 3 white balls (negative for the coronavirus, i.e., we believe that you are *not* infected, but there is a one in four chance that you *are* infected). Then it should be a do-able exercise to calculate the probabilities of transmission from the point-of-view of a passenger (“the subject”), keeping in mind that the calculation should be broken into two cases: (a) the subject is susceptible (where there is a chance of transmission to the subject); and (b) the subject is already infected (where the subject’s status does not change). I guess that, once the subject takes account of the case where s/he would have no change in status, the probabilities are what we calculated, but I haven’t carried out the details.

The bottom line: worry about the symptoms, and self-isolate if you think there’s any chance that you are infected. Covid-19 is amazingly transmissible, and the course of a case is unpredictable.

I hope that by taking the virus seriously, we can avoid the worst consequences. My wife says that I should have stated, “self-isolate as much as possible even if you are free of symptoms.”

Besides the 2 well-known symptoms, cough and fever? There are others. See this: https://www.womenshealthmag.com/uk/health/a31986965/mild-coronavirus-symptoms/