Cooties on a Plane —
Estimating the Risk of Exposure to COVID-19 When Flying
A year ago, my niece bought a lovely house on the west coast of Ireland. She and her family are all traveling there this summer, and they have invited my family to join them. Naturally, after months of COVID-19 sequestration, my wife and daughter have been pleading for a chance to get out of town.
Widely considered to be the worst of several hypochondriacs and germaphobes in the extended family, I pointed out that there might be significant risks. How could traveling in the confines of an airplane possibly be safe? An infected person might be sitting right there next to you, breathing his or her cooties all over you and your family for the entire flight. And seven hours is a long time to hold your breath! When I pointed this out to my niece, she flashed me a rather zany photo of herself wearing a snazzy blue mask and a plastic shield that she’d just bought online. That’s plenty of protection, she assured me.
I saw no reason to take her word for it. As part of my career transition into data science, I only recently completed a rather grueling course in probability, and this seemed like a great chance to put my new skills to the test. It should be possible to compute just how high the risk of flying is.
Asking the obvious…
At first I thought this should be pretty simple. Just find out how many people have the virus, and compute the likelihood of (i) infected people being on the plane, and (ii) infected people being close enough to you on the plane to be a major risk factor. With all the COVID-19 data floating around the Internet, I reckoned this should just take a minute.
Apparently not. You can find out how many new cases were reported yesterday — approximately 41,073 as of August 18. You can find the number of newly reported deaths (654). And you can find the cumulative figures on infection (5,354,013) and death (41,073) since the outbreak began. But not how many people have it now.
So to answer the question of air travel safety, we first need to estimate how many people are carrying the virus based on the available data, and then we need to figure out how many of those people are are likely to be out and about. Some of those people just might step onto the same plane as you!
Typical disease outcomes…
To estimate the number of people who are possibly going to step onto your plane, you first have to reason out what — statistically — happens to the typical person who gets infected. These things are best done using percentages, and so let’s assume we have 100 people.
A Sample of 100 People…
Imagine that on any given day 100 people get infected. For roughly five days, those five people are going to be walking around symptom-free. These presymptomatic people are apparently infectious. Then, out of those 100 people, studies show that for the next 2 weeks or so roughly 45 of the original 100 are going to continue their lives oblivious to the fact that they are carrying the virus. These are the asymptomatic carriers … they never get sick at all! Meanwhile, the other 55 people are going to feel distinctly unwell and will continue to do so for about 14 days. Unfortunately, 20 of those 55 who get noticeable symptoms may end up in the hospital, and only 15 or 16 of those will ever recover.
Worry about ghost infections, the oblivious people with no symptoms…
What this snapshot tells you is that for our hypothetical sample, you have to worry about 100 people during their presymptomatic phase and 45 people during their asymptomatic phase. You don’t really have to worry about the people who get noticeably sick. Most of them will stay home, and even if they do come to the airport, they should show a fever and get turned away on screening. The 20 or so who go to the hospital are even less of a concern as they will definitely not be getting on your plane.
There are more ghost infections than you’d expect…
OK, so we know a little bit about the kind of people who pose a risk. They either just got exposed, or they’ve been carrying the virus for 6–19 days. But our tiny model of 100 people only talks about the people who were infected on a given day. What about 100 people infected the day before? Or the day after? Or the day after that? In our hypothetical universe, the same branching of outcomes is going to happen to all those people too. This mixture of presymptomatic and asymptomatic people keeps accumulating every single day as more and more people catch the virus, joining a pool of “ghost infections” — infections that are out there that you just can’t see. They’re in your park, on your playground, in the supermarket, and at the gas station. They look totally normal. They themselves feel totally normal. But they can get you sick.
Just how many of them are there?
To figure what percentage of the population is a walking ghost infection, we need to do a bit more math. We also need to make some more assumptions, because the available data doesn’t tell the whole story.
It actually matters when people get tested — something the data simply doesn’t tell us. What we can do is assume that people get tested because they feel sick, not because someone is forcing them to test. Sure, some people have to test periodically because of their profession, but we don’t know what that percentage is. So to be conservative let’s assume the vast majority are testing because they feel unwell, meaning they are in the symptomatic phase.
Now imagine that on a given day 55 people test positive. What does that tell you? If you’re not sure, look back at the flowchart. It means that 100 people were initially infected, and these are the 55 that actually developed symptoms of some kind.
Calculating backwards like this is the magic trick that allows us to figure out how many people are getting infected, as well as how many ghost infections are out there. Some people are getting tested on day 1 of symptoms (day 6 of infection), some on day 5 (day 10 of infection). There is no way for anyone to know what day of infection people are getting tested on. But no matter what it is, there is going to be some average, and whatever average that is, it means that 5 days (the presymptomatic phase) plus that average post-symptom testing delay (call it 3) will end up being (on average) the number of days backwards that we need to count to figure out when the original 100 people got infected.
Does it matter whether the average test day is day 6 or day 8? A little, because when dealing with real numbers your “100-people” extrapolation will point to a different starting point depending on the post-symptom testing delay you chose. But we are in the land of guesstimates anyway, and daily new test cases tend to be in a general ballpark from one day to the next, so no matter how long after feeling sick it takes for people to go get a test, we should get numbers that come out pretty close.
Crunching the numbers…
The computation is that simple. If we assume testing occurs on day 8, for example, then if 55 test positive today, that means there’s currently a complement of 45 people running around asymptomatic. It also means that 100 people were in their presymptomatic phase between 8 and 3 days ago (days 1–5 of infection). Once you make that realization, it’s just number-crunching, only you have to work with real daily test numbers (not 55). Summing up all the people in the current ghost infection window produces a spreadsheet with a running tally of offset values going back 19 days from each test date. Why 19? Because on day 19 day the asymptomatics exit the infected pool.
To see how quickly this scales, imagine 100 people are getting infected everyday for 19 days in a row. Because of the way the numbers accumulate and transform from 100% to 45% (with one day’s offset for each new batch of people), the tally of ghost infections after 19 days would come to a whopping 1130 people! (To see what such a spreadsheet would look like, go to the second page of this website.) This means that, in a pinch, you could multiply yesterday’s new infections tally by a magic number like 11 and you’d have a first-order approximation of the number of walking ghost infections. (The correct value right now is actually 15, and this value has varied between 8 and 15 over the past two months.)
Back in late June when I first tried this approach, my numbers showed that nearly 874,000 ghost-infections were walking around the U.S. oblivious to that fact that they could be putting people at risk. That’s roughly 0.27% of the population. Right now, ghost infections tally nearly 1,100,000, or 0.33% of the population. That’s a bit down from a peak of 1,352,000 (0.41% of the population) on July 27, but it’s still substantial. And taking the U.S. population to be 328 million, that means roughly 1 in every 300 people you see on the street is a potential risk.
Yeah, but what about the plane?
This is where that grueling probability course comes in handy. While you could argue for a number of different models, this is really what they call a binomial distribution problem, the kind of problem that answers questions like how many heads do you get if you flip a (highly biased) coin 20 times. Now, to compute the chance of being totally safe on your plane (meaning no one on board has it) will depend on how many people get on in the first place. Using my current 1-in-300 infection estimate, the numbers look like this.
These calculations for the U.S. show that the chance that no one on the plane is a walking ghost infection ranges from 23% — 85%, depending on how many passengers there are. So you really want to hope you get an empty flight.
Other factors
There are a huge number of variables and parameters to account for when talking about safety during air travel. My niece ended up taking a plane that was only 5% full. Between that and her goofy plastic shield, you can be sure that her risk of exposure was quite low. My nephew, going to the same destination, took a flight that was almost three-quarters full. That’s not good, and he didn’t even have a shield, though so far he hasn’t gotten sick.
All this time, we have only been talking about the possibility of an infected person being onboard the plane — that is, the possibility of exposure. There is also the question of the likelihood of a COVID-19 ghost sitting close enough to you to matter. Proximity to an infected person alters the likelihood of actually catching the disease. Even if no one is close to you, studies show that by just walking down the aisle a passenger can spread the virus further afield. Then there’s the toilets, where just about everyone leaves some microbes behind! Oh, and we also didn’t talk about all the people you will pass by at the airport. If you’re a germaphobe, the list goes on and on.
I’m going to leave these additional considerations for another day. But the numbers above are at least a start. For now, just remember if you choose to fly, bring a good mask, plenty of wipes, and maybe even a face shield. But above all … look for an empty flight!
