Mask-wearing and COVID-19 is complicated. And not just because it’s become politically fraught, not just because there are many different kinds of masks of varying levels of effectiveness, not just because there’s been a shortage of medical-grade masks so authorities were trying to convince the public not to buy them, not just because there’s tons of malicious or simply misguided misinformation flying around, not just because our understanding of how COVID spreads has been changing, and not just because countries in the West ignored the lessons learned by Asian countries that faced SARS (though all of these things are true).
In addition to all this, masks are also complicated because they fly in the face of our mathematical intuition. The good news is that when you do the math (and we’re going to), you find that masks are much more effective than you might think. Say you have a mask that cuts in half the chance a contagious person will infect a nearby susceptible person. In other words, this mask is 50% effective. Except, this mask is way more than 50% effective because as we’ll see, when many people wear even just a 50% effective mask, you end up with way more than 50% protection (both to the wearer and to society at large). It seems obvious that if no one wears a mask, then no one gets any benefit – and that’s true.
And you might assume that if everyone wears a 50% effective mask, there’d be a 50% benefit – that is, a 50% drop in disease transmission. But that’s not how the math of masks works! When everyone wears a 50% effective mask, disease transmission actually drops by 75% — much better than 50%. Masks break our intuition because we’re used to thinking about masks as single-directional, only protecting the wearer. But masks can protect in both directions, when you breath in through them, and when you breath out. This means that when everyone is wearing masks, there are in fact two masks between any two people. If we assume for simplicity that masks are equally effective in either direction – and if this assumption bothers you, stick around till the end of the video – if masks are equally effective in either direction, then the first mask cuts disease transmission in half, and the second mask cuts it in half again. So overall, you end up with a 75% drop in disease transmission, not 50%. In this scenario, masks do double duty! But in reality, not everyone will wear a mask.
So when a contagious person encounters a susceptible person, there are in fact FOUR possible routes of infection. In the first route neither person is wearing a mask, which means there’s no reduction in disease transmission. In the second route, only the contagious person is wearing a mask, and so for a 50% effective mask, disease transmission drops by 50%. In the third route, only the susceptible person is wearing a mask, and again disease transmission drops by 50%. And in the final route where both the contagious person and the susceptible person are wearing masks, disease transmission gets cut in half twice – aka it drops by 75%. What does this mean for society overall? Well, it depends on what fraction of people wear masks. As we’ve seen, if no one wears masks then no interactions involve any masks and the overall drop in disease transmission is 0%. And if 100% of people wear masks, then all interactions involve two masks and the overall drop in disease transmission is 75%. But if 50% of people wear masks?
Then on average – assuming that people interact randomly – a quarter of all interactions will involve no masks, a quarter will have the contagious person masked, a quarter will have the susceptible person masked, and a quarter will have two masks. So even when just half of people wear masks, three-quarters of interactions involve masks (and a significant portion of those involve two masks). Do you see the magic math of masks yet? Your first guess might have been that if 50% of people wore 50% effective masks, you’d get a 25% drop in disease transmission because 50% of 50% is 25%. In fact, this intuition would be true if masks were only effective one-way (like on exhalation only) – then there’d just be two routes: either the contagious person wears a mask, or they don’t, and these average to 25%. BUT when we take into account the two-way nature of masks and average over all four possible mask combinations, the overall drop in disease transmission becomes almost twice as good! Masks Work Better Than You’d Think. And this is true in general – no matter what numbers you choose for mask effectiveness and usage, the overall drop in disease transmission is always better than the intuitive guess from just multiplying those numbers together.
So what does this mean for the 2020 pandemic? Well, for COVID-19, epidemiology suggests that each contagious person infects on average 2.5 other people. If you could drop that number to below one, a drop of just over 60%, then each contagious person would infect fewer than one other person on average, which would be enough to swiftly halt the spread of COVID-19. So what would it take to drop disease transmission by 60%? Well, there are many options, but a particularly cost effective and arithmetically satisfying one is this — if 60% of people wore 60% effective masks, disease transmission would drop by 60%! And if we did that, we would beat COVID – the mask math shows us how. Specifically, it shows us that masks are more effective than you’d think for two reasons: first, they do double duty when both people wear them, and second, the fraction of interactions involving masks is always much more than the fraction of people who wear masks. This is the magic multiplicative power of masks –– even partially effective masks, partially adopted, can extinguish an epidemic, as long as enough people wear them.
Ok, some caveats to all this: We’ve been pretty vague about what it actually means for a mask to be X% effective — for the purposes of the math in this video, all that matters is that disease transmission drops by X%, irrespective of how the mask actually achieves this drop. In reality, masks reduce disease transmission through a combination of filtering and redirecting air, and they vary a lot in effectiveness depending on their filtration, how tightly they fit, if they have an exhalation valve, etc. So it’s hard to give exact numbers; a 50% effective mask could be something like an N95 worn poorly (or incorrectly decontaminated) or a cloth mask worn well. We’ve assumed that masks provide equivalent protection upon inhalation and exhalation. Aatish put together an interactive essay where you can see what happens when inhalation and exhalation effectivenesses differ, what happens when more (or less) of the population uses masks, and more. For simplicity we’ve assumed that contagious people are just as likely to wear masks as non-contagious people. We also assumed that people mix randomly, which isn’t necessarily true. For various reasons, people who wear masks may be more likely to interact with other mask wearers, and less likely to interact with those who don’t wear masks (and vice versa). Clustering non-mask users together diminishes the overall protective power of masks and means you need more people to wear masks to achieve the same drop in transmission. Again, if you’re interested in more details and references, definitely check out the interactive companion essay at aatishb.com/howmaskswork. This video was made with the generous support of the Heising-Simons Foundation, which normally works with MinutePhysics to help communicate about fundamental physics research, but this year they’re providing additional funding to focus on the response to COVID-19. That means they’ve supported research – including some of the N95 mask decontamination work I mentioned in my video on the physics of N95s – they’ve supported hospitals and remote learning, they’ve helped low-income households maintain access to utilities, and they’re funding COVID science communication like this video! A big thanks to Heising-Simons for their support of science – both fundamental and applied – as well as science communication.