Spherical cows #1: Face masks edition

Let's talk a little about the effectiveness of face masks and the way levels of participation in waering them change the overall outlook of a epidemic. I want to go there because I hear some complaints hear and there that cloths masks don't do "much" and so there is no point in waering them, and I don't think that position is terribly well thought out.

This is the first of a couple of articles I intend to write applying simple models to look for insight into policy choices related to infectous disease in general and aimed at understnding a little about Covid19 in particular. Thought I won't try to make the models numerically faithful to Covid19.

Basic reproduction number

In the jargon of epidemiology the "basic reproduction number" (givern the symbol $R_0$) is the number of people, on average, who any given carrier gives the disease to. In simple models, if this number is larger than one then the disease spreads throughout the susceptible population, if it is lower than one the disease fades away. And we can (again in simple models) further break this number into a product of the avearge number of people a carrier interacts with ($N$) and the average chance that they will transmit the disease in their interactions with a single person ($F$).

For our purposes we can say that masks reduce the chance of tranmission¹ Which means that the effectiveness of using masks translates directly into a reduction in $R_0$.

Guessing at the effectiveness of a properly worn mask

Here we consider "non medical" cloths masks that you make yourself or buy at retail. Now a mask could help in two ways: it can reduce the amount of virus a carrier puts into the environment and it can reduce the fraction of environmental virus that a un-infected wearer takes in. For simplcity I'm going to model these as each generating a equal reduction in the chance of transmission modeled by a multiplicative factor $f$ that is less than one (so it models a reduction) and greater than zero (which would be perfect effectiveness.

So, if a carrier has a single "interaction" with a uninfected but susceptible person the probability of transmission is:

• $F$ if neither is wearing a mask
• $fF$ is one (either one) is wearing a mask
• $f^2F$ if both are wearing masks

But what should $f$ be? I don't really know, so I'm going to guess. To be conservative I'm going to guess that they are not terribly effective. I'm arbitrarily assigning $f = 0.8$. That is, a single mask reduces your chances of getting infected by only one fifth, and both parties being masked gets you down to 64 percent.

What about compliance

Not everybody wears masks. This may be due to a oversight such a failing to grab one on the way out the door, lack of access, or by choice. It dosen't matter to the model. But what does that mean for the overall effectivenes of a masking policy. Not all iteractions will be between two mask wearers. Some will involve one mask and others no masks at all.

Call the rate of mask wearing $r$. If everyone wears them $r$ is one, if only people who were burned by acid wear them $r$ is essentially zero.

Sticking with our "do everything on average figures" approachand assuming that everyone has the smae chance of meeting everyone else² we can compute the overall fraction $\Gamma$ in a simple way \begin{align} \Gamma(f,r) &= \left[ (1-r) + f r \right]^2 \\ &= (1-r)^2 + 2f (1-r) r + f^2 r^2 ] \end{align} And that's the whole basic calculation. It's quadradic in $r$ and concave up. Done and dusted.

Except that we might want to manipulate the math a little in search of further insight. So we continue \begin{align} \Gamma(f,r) &= (1 - 2r + r^2) + f(r - r^2) + f^2 r^2 \\ &= r^2 (1 - f + f^2) + 2 r(f - 1) + 1 \end{align} Also quadratic and concave up in $f$.

Sticking some numbers in

Just looking around me I think it's little optimistic to expect more than 90% mask compliance, so a optimistic number of the overall effect would be $$ \Gamma(0.8,0.9) = 0.672 \;, $$ or a little better than a 30% reduction. That's not great but it is significant. If $R_0$ was as small as 1.45 that alone would be enought to squeeze off the epidemic.³

A more pesimistic value of compliance might be around 60%, leading to $$ \Gamma(0.8,0.6) = 0.774 \,. $$ which is to say less than a 25% reduction in the tranmition proability. For that to be enough we'd need to have started with $R_0$ less than 1.3.

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¹ We could also say that stay-at-home orders and the like reduce the number of interactions. Social distancing could be modeled as a little of each. I'll say more about this in the next post.

² I'd guess that this is the worst assumption in this calculation, but making it better requires a huge increase in the complexity of the problem and probably necessitates a Monte Carlo (randomized simulation) approach. I'm doing a back-of-the-envelope calculation, so this is what we get.

³ I haven't seen a number recently but early reports suggested something larger than 2.0, and I suspect it might be a little higher than that simple because early on there wasn't much awareness of how many people get it but remain asymptomatic.