[David Cooper (OP)]

I don't know what the risk of transmission is if two people are two metres apart, but it must increase in proportion to the amount of time that they maintain that proximity. It must also increase depending on how many people are that distance away, and on how many are further away in outer "rings".

If you're 2m away from one person and the risk is n, the risk is going to be 4n if you're 2m away from 4 people. If you're in the middle of a block of nine people, your risk must go up again, but how much? What happens if it's a grid of 10 by 10? How can the risk be calculated?

The air breathed out by someone will presumably spread out in three dimensions, but one of those is blocked by the ground and may also be blocked by a ceiling if you're indoors, and because the air touching objects doesn't move as much as the air further away, you can't assume there will be efficient removal of the virus from the air onto surfaces, but it will certainly get sucked into lungs easily. What prompted this question is the thought that the risk might go up more than people expect as the grid size increases. Clearly there's also another important factor involving the likely number of infected people in the grid, so the risk varies again with that and will grow as we move up the curve.

If we think of the virus radiating out from an infected person and passing through the space that another person is breathing air from, then it the risk should follow the inverse square law with distance for spreading out in three dimensions, but because of the ground and possible ceiling (which I think very little of the virus will contact), it may not thin out so quickly at greater distance from the source and may end up being closer to halving with distance after it's equalised through the vertical space, so that could lead to the risk going up substantially more than expected in a place with many people in it even if they're well spaced out. They're now talking about putting up plastic screens to protect people working at the checkout in supermarkets, but the risk from the immediate proximity of the customers there may not be as high as it is from the general background level of virus in the air.

[Bored chemist]

The best guess is that the risk of infection at two metres is "small enough"- whatever value that might be.
It's also plausible (if simplistic) to assume an inverse square law for infection. That's not unreasonable if you consider that infective particles get spread out over the area, and the area rises as the square of the radius .

But, superimposed on that is the fact that most particles land "near" the source.

So the actual form of the distribution is that it falls off faster than inverse square

Now imagine many people standing on a square 2 m grid.

The Guy in the middle is infectious.
The folks N, S, E and W of him are "safe enough" because they are 2 metres away. Then there's the folk who are further away- within some radius. Well, the number of people within that circle increases with the square of the radius.
 If the infection fell off as an inverse square law then (paradoxical) the number of people at risk rises as the integral of that combined number of people and the distribution.
That number rises without bound as the group gets bigger. If I got the arithmetic right, it's proportional to the radius of the crowd.

But we know that the actual distribution falls faster than that. So there's a point at which the size of the crowd doesn't matter. Most people are "far enough" away. to be safe.


It would be interesting to model this on a spreadsheet.

[David Cooper (OP)]

I'd be very surprised if much of the virus manages to fall out of the air within two metres. If you spray tiny droplets of water into the air, they just float about for ages without falling much at all. Viruses are much smaller than that, so I can't imagine them falling out either. I suspect that it's going to be close to inverse square law.

[hamdani yusuf]

Height, wind, temperature, air pressure and humidity can all contribute to the risk of transmission. But the rule of thumb must be simple enough to be followed by lay people practically within reasonable level.

[David Cooper (OP)]

Yes, but the 2 metre rule is too simplistic - it can put lots of people close enough together for a lot of spreading if there are a few infectious people there; particularly if they're there in such numbers for a long time. Our top politicians are now learning that from direct experience.

[Bored chemist]

Yes, but the 2 metre rule is too simplistic - it can put lots of people close enough together for a lot of spreading if there are a few infectious people there; particularly if they're there in such numbers for a long time. Our top politicians are now learning that from direct experience.

That's why they also banned mass gatherings.

[evan_au]

It's also plausible (if simplistic) to assume an inverse square law for infection.

We are familiar with the inverse-square law in physics because phenomena like light and gravity radiate out as a sphere in 3 dimensions.
- If a light ray travels 2 x Distance, it spreads out over 4 x Area.
- At 2 x Distance, the intensity is Intensity/4
- This is an inverse-square law
- The inverse-square law still works if you emit light from a point on a flat surface, so it only fills a hemisphere, rather than a full sphere (eg an FM radio transmitter on "flat" ground).

However, if you constrain that third dimension with floor and ceiling, the spread is not like a 3D sphere (Area α Radius²), but like a 2D circle (Circumference α Radius).
- This suggests an Inverse Distance law, rather than an Inverse Distance² law.

Of course, the range of light and gravity is infinite (in a vacuum), but other physics is at play with virus-laden water droplets...

If you spray tiny droplets of water into the air, they just float about for ages without falling much at all.

I heard about some tests done with lifetime of virus-laden water droplets. Apparently, the lifetime depends on the size of the droplets.
- Large droplets(as big as raindrops) fall down immediately
- Small droplets (such as you might see in a backlit sneeze) fall down in seconds
- Very tiny droplets can linger for an hour in the air

The size of these small droplets depends critically on the humidity
- These droplets have large surface area for their volume
- So they exchange water molecules easily with the surrounding air
- If the humidity is around 40%, these droplets will stay small, and linger longer in the air. This is a problem for dry, air-conditioned air in the winter months.
- If the humidity is 60% or above, these droplets will grow, gravity will dominate over random Brownian motion, and they will fall out of the air more quickly.

Perhaps humidity is a useful factor to consider in air conditioning for hospitals, supermarkets and cruise ships?

[Bored chemist]

There's another  bit of maths to account for.
Is someone's saliva has (and this is a made up number)  a million virions per ml, and you convert it into a spray with particles 1 micron across then very nearly all droplets won't contain any virions.
If some of the droplets are bigger- say 100 micron, there will be more bugs in them, but they won't get far before gravity deals with them.

So this depends critically on the particle size distribution too.

[alancalverd]

But beware of sneezing. Whilst normal exhalation is turbulent and droplets settle fairly close to the source, a good sneeze can produce a vortex ring that travels a long way before dispersing.

[chris]

- If the humidity is 60% or above, these droplets will grow, gravity will dominate over random Brownian motion, and they will fall out of the air more quickly.

Good point. I'd not thought of it like that before. We do find that higher humidity tends to result in a more rapid drop off in flu infectivity / spread, so that would fit with the point you make: the virus aerosol drops out of the air more quickly and is trodden underfoot (so to speak).

[vhfpmr]

I think the point of the two metres is not that it reduces the R0 value to zero, it's that it reduces the R0 value sufficiently to stop the virus spreading. Fog stays in suspension in the air indefinitely, so a virus isn't going to drop on the floor with a thud within two metres of the person who exhaled it.

[set fair]

Our top politicians

This is a local forum, we'll have no oxymorons here.

[David Cooper (OP)]

Well, I've now seen a few things saying the virus particles are unusually large (much bigger than with flu viruses) and do fall out of the air fairly quickly. The only significant danger is thus from close proximity (which must include walking along two metres behind someone where you're continually within the cloud of air they've just exhaled) and of course from touching surfaces that it might have landed on or which are touched by lots of people.

[Bored chemist]

I've now seen a few things saying the virus particles are unusually large

Did they say so with any authority?

[set fair]

For a video showing the particles we exhale start at 15:00 minutes

[Bored chemist]

Why should we listen to "a man in a youtube video"?

[David Cooper (OP)]

I've now seen a few things saying the virus particles are unusually large

Did they say so with any authority?

Not fully adequate authority, but not attracting any objections from Quora's multitude of genuine experts, and it does fit with the two metre rule which the government's experts have bought into, so it sounds plausible.

Incidentally, I've noticed at the end of the Downing Street briefings the three bigwigs march out of there two metres apart, travelling within each other's gas cloud - it's no great surprise that they catch it from each other. It might be a good idea to have a ten second rule where you stay that time behind the person walking ahead of you.

[Bored chemist]

The 2 metre distance, if it's anything but a random number, should be based on the particle size distribution generated by coughing and sneezing.
That's pretty much independent of the virus.

The estimates I have seen of the virus diameter seem to be pretty much the same as flu.

Each SARS-CoV-2 virion is approximately 50–200 nanometres in diameter.[62]
from
https://en.wikipedia.org/wiki/Severe_acute_respiratory_syndrome_coronavirus_2

vs

"The virion is pleomorphic; the envelope can occur in spherical and filamentous forms. In general, the virus's morphology is ellipsoidal with particles 80 to 120 nm in diameter, or filamentous virions 80–120 nm in diameter and up to 20 µm long.[34"
from
https://en.wikipedia.org/wiki/Orthomyxoviridae