# Viruses need to survive a large but slow shock, to spread across a room

Someone infected with COVID-19, flu, measles, maybe mpox, or any one of a number of a bunch of viruses, breathes out the virus in tiny (micrometre) droplets of mucus. The droplets start in the infected person’s breath, which is saturated with water, i.e., at 100 % relative humidity (RH), because the breath has come straight from the infected person’s lungs. As the breath mixes with the air of a room (or of the surrounding air if they are outdoors), the humidity drops from 100% to the typical 40 to 60% of the air in and outdoors. This mixing takes a few seconds, and driven by this change of humidity the droplet dries out. The virus needs to survive this sudden drying intact, in order to go on to infect a new person. Drying in a few seconds may sound quick, but viruses are small so this may actually be felt as slow drying by the virus.

The argument for a few seconds being a long time for a virus is as follows. Surrounding the virus is an envelope that is two back-to-back layers of fatty molecules, aka a lipid bilayer. This has a permeability p of water of order 10-6 m/s*. Then the flux, j, of water molecules across the viral envelope when the difference in concentration of water molecules is Δc, is j ~ p Δc**. This is a number of molecules crossing unit area of the envelope per second.

Viruses have a radius R ~ 100 nanometres, and so an area of order R2 and a volume of order R3. Combining these factors of R we get the differential equation for the concentration difference across the membrane (usually called the envelope) of flu or SARS-CoV-2 or …:

d Δ c / d t = – p Δ c / R

This is just the standard equation that gives as a solution exponential decay with a

time constant = R/p ~ 10-7 m / 10-6 m/s = 0.1 s

about ten times smaller than the timescale for drying of the droplet the virus is in. So the drying of the droplet of mucus you breath out, is sufficiently slow for the water to move across the virus’s envelope. This means that drying is not a sudden shock, so its speed is not a problem for survival of the virus, but this loss of water – this dehydration – may still destroy some fraction of the viral particles.

* This value is the estimate for the permeability of flu virus envelopes by Choi and coworkers. NB The units of permeability are the same units as speed which looks odd until you realise that water diffuses across the envelope which has a a thickness of a few nanometres and so the permeability is just a diffusion constant (units of m2/s) divided by the thickness (units of m), which gives units of m/s. For a thickness of order one nanometre, this permeability corresponds to an effective diffusion constant of order 10-15 m2/s inside the envelope. This permeability is comparable to that of simple back-to-back layers of lipids (see also here)but at least some virsues, eg SARS-CoV-2 also has pores in the membrane formed by proteins embedded in this envelope, and this will contribute to the permeability to water.