Tuesday, February 02, 2021

Masks and distance and time

Disclaimer: The numbers are for illustration only. Getting the real values for some of these things, such as the number of infective particles exhaled per minute, is difficult and expensive.

Simple model: Your body, breathing in an airborne virus, has some probability of fighting it off using the standard defenses. This varies with your condition, obviously. The probability can be pretty high—some strains need tens of thousands of viruses to reliably infect you, others only a few. Luck plays a role—this is probabilistic—but for the moment assume you need to breathe in some number N of viruses, plus or minus something for luck, in 8 hours.

How you get to that total isn't obviously relevant. It could be a minute face to face with 30 infected people or 30 minutes face to face with one--just so long as you get the dose quicker than your natural defenses can cope with the alien invasion.

How much can get through your mask? I can't say. More than you'd like. Call the fraction that gets through F.

How much can get through the other guy's mask? Since some of the viruses piggyback on droplets,, and masks are pretty good at filtering droplets, some of them get blocked. Call the fraction that gets through G, and I strongly suspect that F > G; the mask is more useful at the other guy's end.

Unfortunately I don't have any definitive numbers on what F and G are. Nor do I know what N is, nor the number V of how many viruses an infected man will be dispersing per hour. All I can do is make some statements about relative rates.

When you're close and downwind of the infected man, the amount of his exhaled plume that your face is intersecting is roughly proportional to 1/R^2. In this example at about 2 feet away, you'd be intersecting maybe 1/10 of all his plume. At 4 feet, you'd only intersect 1/40 of it. In this toy model, you intersect (1/10) * (1/R^2) with the distance R measured in feet.

If you are outside in still air, the exhaled plume will be diluted quite a bit all around and above him, and the rate of dropoff would still go as 1/R^2. However, thanks to dilution, the constant term would be a lot lower. I'll use a toy model of (1/1000) (1/R^2). If he's facing you, more, facing away, less.

If you're downwind in a slight breeze, the rate is bigger, of course. The breeze reduces the time it takes the plume to reach you, and thus it doesn't have time to diffuse as far and mix with clean air. A stiff breeze will induce more mixing--its hard to model.

If you're indoors, the plume can't expand upwards. The drop-off is proportional to 1/R instead—dropping off more slowly. I'll use a toy model: (1/1000) (1/R). Indoor breezes aren't usually very dramatic, so I don't expect as much mixing of the polluted with clean air—so if you're downwind of the plume you're getting a bigger dose than the toy model suggests. If you're off to the side—maybe quite a bit less. When I can, I try to find the air return and stay far away from that. Usually I can't find it.


(1/R^2) gets small faster

So let me pull some numbers out of the air. If G=.5 and F=.7, then if both of you wear masks the fraction getting through .35. That's not exciting, but every bit helps. If you're downwind indoors, 10 feet away, my little model says you get (1/1000)*(1/10) * 2(downwind factor) * 0.35 (mask) of whatever the exhalation rate was. If (pulling another number out of the air), the exhalation rate is a million viruses a minute, that's about 70/minute you'd breathe in. In half an hour that's about 2000 that you breathe in. You'll breathe a bunch of them out, too, but never mind that. (Did I mention that modeling this is complicated? It is.)

My equation with its imaginary numbers is (1/1000)*(1/R) *2 * 0.35 * 10^6 (#/min)* Time(minutes)

Let's play relative games with these imaginary numbers. If you skip your mask, you get 2900 instead. If he skips his too, make that 5700. If instead you're 5 feet away, your masked number goes to 4000. If you hang around for 5 minutes instead of 30, you get about 330 instead of 2000. If you're wearing a mask that's 95% efficient instead of the 50% I assumed, your dose drops to 200 viruses.

So, the dose you get is proportional to the total time you spend, roughly inversely proportional to the distance away (with all sorts of caveats!), and proportional to 1-{mask efficiency}. (e.g. a 95% efficient mask lets through the fraction (1-.95) =.05 of particles). And it depends on how much virus the guy is pumping out.

As another for-instance, if you're wearing a 95% mask and you spend 1 minute 1 foot away from a guy's face while you're trying to put a tube in him, you get about 20,000 viruses. That plastic shield might help a little, if the exposure time is short, since it takes a little while for stuff to diffuse around it--but it doesn't take very long.

Repeating the disclaimer--this is all about relative doses, and does not purport to have real numbers. (That's what you expect when things are complex...)


Unknown said...

so 6 variables:

• Viral load of the infected person
• effectiveness of that person's mask
• effectiveness of your mask
• distance between you
• time spent in proximity
• airflow/dilution

Over the past few days, I've had to go into my workplace a few times and buy parts locally and get fuel, and the same sort of people that had ill-fitting too-small masks now have double ill-fitting too-small masks. The second mask has, if anything, made the possibility of leakage greater.
I wish instead of suggesting double-masks (on the more-is-better philosophy) Fauchi had emphasized mask fit and use of better masks -- I don't think rarity of the supply of KN95 and KF94 is an issue any more.

james said...

And clean masks are important--I breathe lots of stuff out into it, which is bound to make a nice growth medium for something that I don't want to breathe the spores of.