Minimal model of corona virus exposure

Transmission of the corona virus (aka SARS-CoV-2) is very complex, which is basically why it is so poorly understood. But in true theoretical-physicist style, a minimal model has been developed, by a guy called Roland Netz (who is a theoretical physicist in Berlin). It makes a lot of assumptions, and it is clear that there is lot of variability, between one infected individual and another and between one situation and another, so its predictions should be taken with a large pinch of salt. But in this post I will outline this minimal model.

Transmission is via droplets exhaled by an infected person*. We all exhale droplets when we breathe and especially when we speak. Netz notes that when we speak a lot of the volume is typically in droplets a few tens of micrometres across**, and that we emit about a 1000 of these droplets per second.

Droplets of this size can hang around for very roughly a 1000 seconds***. As someone speaking emits around a 1000 droplets per second, and as each droplet lives in the air for about a 1000 seconds, this means that someone speaking constantly (something we academics are prone to do) generates a steady state of about a million droplets — in this size range — in the room they are in.

The volume of a droplet a few tens of micrometres across is about 10-14 m3, so the million droplets have a total volume of around 10-8 m3.

Measurements of how much virus**** there is per unit volume of the mucus lining our throat cover a huge range, from less than 1010 to more than 1014 m-3. Let’s take a middle value of 1012 m-3. Then these million droplets have about 10,000 virus particles.

So, with all these assumptions if you are sharing a room with an infected person (who is not wearing a mask) then you are sharing the room with 10,000 airborne virus particles. Not a reassuring thought — that is why we should be wearing masks!

Say you are sharing a small room say a couple of metres across, then the volume of the room is maybe 10 m3 and so the 10,000 virus particles are present at an average***** concentration of 1000 m-3.

If you are just sitting around then you breathing in about 0.1 litres/second or about 10-4 m3/s. So in this small room this equates to breathing a few viruses a minute, if you are not wearing a mask.

This estimate of breathing in a few virus particles per minute when in a small room with an infected person who is speaking, relies on a lot of assumptions. And it also applies when neither the infected person nor the person vulnerable to infection is wearing a mask.

We are also missing the final step, as we have a very poor idea of how many viruses you may need to breath in, before say the probability that you become infected reaches 50%. For SARS-CoV-2 I am not aware of any number here.

But if you are going to quantify transmission, you have to start somewhere. I am a big fan of the statistician George Box’s quote: “All models are wrong, but some are useful.” I think it applies here.

* Here I only look at direct transmission via the air. Transmission via a droplet hitting a surface and then being picked up, eg on someone’s hands is another transmission mechanism. This may be less common or a lot less common than transmission via air.

** There is a fair bit of volume in larger droplets, when we are speaking, but these sediment out quite quickly so should be less of a problem unless you are not doing any social distancing. Also, the volume we breathe out is smaller if we are not speaking (or coughing). It looks like it is less dangerous to share a room with a quiet infected person than with one who is speaking.

*** This time is increased by about a factor of ten by the fact that the droplets evaporate in around a second (they are well over 90% water) and so shrink, which slows down sedimentation. As they shrink they grow lighter.

**** For technical reasons what is measured is not actually the number of viruses but the number of viral genomes. Basically we can relatively easily measure how much viral RNA there is, but counting viable viruses is harder-.

***** This assumes the air is well mixed, which may not be true.

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