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PROFESSOR: So now let's add
to our theory ways

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in which droplets
can be removed,

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or infectious virions
can be removed,

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within the room in addition
to ventilation and filtration

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effects that we've
already described.

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So the primary
way that can occur

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is simply by settling
of the droplets.

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So we've already talked
about the Stokes settling

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speed, which scales as the
radius of the droplets squared.

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So basically, larger
droplets fall fairly quickly.

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And in fact, we've
already discussed

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how they can fall to the
ground in a fairly short time.

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It could be on the
order of minutes or less

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for large droplets,
such as those

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which are spewing out of
your throat when you cough

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or when you sneeze--

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out of your nose.

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But also, the smaller
aerosol droplets

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we've already
calculated can stay

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in the air for a very long time.

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So they settle much more slowly,
but they do still settle.

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So we can try to see
how this enters in.

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And the second topic
is also deactivation.

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We've mentioned that the
virus doesn't live forever.

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So the virions in these droplets
do need to find a target

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and get out of those droplets
and into some healthy tissue

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to infect it within a
certain period of time.

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So there's a notion of
a viral deactivation

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rate which can also be a
parameter in our models.

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So if we now add those two
effects to our existing model--

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I'll just keep rewriting
our mass balance equation.

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And this is the mass balance
for the concentration

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of virions in the air.

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I've also used the terminology
in chemical engineering.

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We're going to call this kind
of approximation the CSTR,

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or the continuously stirred-tank
reactor approximation.

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And now with all the
effects we're including,

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it's starting to look more like
actual modeling of chemical

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reactors and chemical
plants by this method.

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So the mass balance tells me
that the volume of the room

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times dc/dt--

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or again, c is the
virion concentration

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per volume in the air--

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is the production
rate, P, minus--

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and then we have a
flux, which is the flow

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rate times the concentration.

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And there are several
flow rates here.

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There's Q, which is ventilation.

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There is filtration,
which is PF QF.

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And then we have now
a new term, which

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I'll write in another color--

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plus vsA.

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So that's the settling here.

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So that idea is, in
a well-mixed room,

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there is a complex flow profile
which is leading to the mixing

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by convection of the air.

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And you might say, well, OK.

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That flow is very quickly
carrying the particles up,

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carrying them down--

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but on average, the particles
go down just as much

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as they go up.

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And if it's well-mixed, then
the particles essentially

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are sampling the whole space.

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And relative to that well-mixed
flow, which averages to 0,

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they are slowly settling.

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And so a reasonable
approximation is to say,

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well, the removal is
basically happening

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with a flux rate, which is
that velocity of falling times

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the area.

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That's how quickly
those particles

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are falling through
any horizontal surfaces

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make relative to their average
0 motion from convective mixing.

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So this is the new
effect of sedimentation.

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And then actually, I will also
add to that another new term,

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which is the deactivation.

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And so here, we will also add
lambda v times the volume.

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So this is just saying that
throughout the whole volume

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of the room, there is a rate
at which every virion is

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just slowly deactivating.

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That will be lambda
v. Also, if you

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have any volumetric
treatments of the air,

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such as chemical disinfectants
or even UV light,

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that may also slowly deactivate
the virus or the virions

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in the air with a term
that goes like this.

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So I'll just mention
that maybe briefly here.

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So lambda v is the
virion deactivation rate.

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And well, if we look
at tv which we've

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talked about before,
which is lambda v inverse,

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this is the deactivation time.

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This thing has been
measured to be of order

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of 1 hour in some studies.

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But also , even greater than 16
hours in aerosol form in other

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studies for SARS-CoV-2.

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So it could be potentially long.

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Also, this could include
effects such as I mentioned--

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UV light treatments,
which might be operating

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in a certain part of the room,
but then the air circulates

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and we're essentially treating a
significant part of the volume.

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It could also be
chemical disinfectants.

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So there are various
chemicals that

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can be sprayed in the air which
are believed to essentially

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kill the virus or
deactivate the virions,

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although they may cause
other harmful effects,

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and so it's not so widely used.

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But in principle,
that would also

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appear in our simple model,
lumped into lambda v.

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So let's put all these
effects together now.

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So again, we haven't really
changed the calculation much.

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We're just building it up
and making it a little more

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complicated each time.

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So let's see here.

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So one thing we did
with this equation

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is we divided both
sides by v. And so let

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me write this equation
again after such a division.

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That would be
dc/dt is equal to--

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well there's P/v. But
then we have over here--

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Q/v is our lambda.

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But notice, all these
things are essentially

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giving us a correction to
lambda, the relaxation time.

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And actually, I should
say this is a minus sign.

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So we get minus lambda.

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And I'll say lambda c, just
for the relaxation rate

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of the concentration field.

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So we can lump all
these parameters

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and we can write lambda c is--

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Well, from the first
one, Q/v is lambda a.

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That's the air change
rate of outdoor fresh air.

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There is PF lambda F, which is
the rate of filtration times

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the filtration efficiency, PF.

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And then we have another term
which we can write as-- well,

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we have lambda v.
That's an easy one.

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And then the sedimentation
term is the one

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I want to focus on right now.

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That is vs. We can write it as
vs/h, where I've divided by v

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and I'm writing v/a is h.

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So I'm writing h equals v/a.

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So if we have a
rectangular box of a room,

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then h is the ceiling height.

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But this is some kind of
effective ceiling height

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if it's not a perfect box shape.

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But if you take the volume
and divide by the projected

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area of horizontal
surfaces, then that's

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giving you a sense of
the typical height.

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And that's the typical
distance by which

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particles have to fall.

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And notice, velocity
is distance per time.

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So when I do vs,
and divide by h I

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am getting something with
units of inverse time.

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So it's just like all
the other lambdas.

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It is basically a rate--

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something per time.

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So this is the concentration
relaxation rate.

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I guess it would be the
theory on concentration

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in the air, which is relaxing
at this rate, lambda c.

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And then we come back to
solving the same simple order

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of differential equation
that we've done all along.

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And the solution's just c
of t is a steady state value

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times 1 minus e to
the minus lambda ct,

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assuming that this thing is
a constant for the moment.

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And also, we know that the
cs is p over v lambda c.

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So basically, if
lambda c is high,

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if all these removal rates are
high, then that makes cs low.

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So the background
concentration of the room

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is much smaller if these
lambda rates are all high.

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Also, if the lambda
rates are high,

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then the relaxation
is very fast,

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so you very quickly
get to the final value.

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And so that's actually worth
sketching what that looks like.

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So if I plot what is the
concentration, c, as a function

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of time, we have
here this lambda c

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inverse is the overall
concentration relaxation time.

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So it looks like an exponential
relaxation to a value cs.

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But I just want to
emphasize what I just

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said verbally, looking
at these equations,

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is that as I vary lambda c--

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so if I have a fast relaxation.

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Let's say lambda c
is a large value.

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Then I start out
at the same rate,

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but I saturate a lot faster
and at a lower value.

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So if it's here, this
is fast lambda c.

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So now if I increase lambda
c relative to the blue curve,

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the whole thing comes down.

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But also, it relaxes
more quickly.

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On the other hand, if
I have slow relaxation,

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because any of these
processes here are slow,

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then I get something which
relaxes much more slowly

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and ends up at a higher value.

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So if someone is breathing
infectious air out,

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exhaling, the infected
person, and there's

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only very slow processes
in the room which

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are removing that
infectious air,

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then it's slow to
build up, but it

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keeps building and building
and building until it finally

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saturates.

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So this is basically,
whenever lambda

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c, is not a large
value or slower,

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you have a slow process, but
it also builds up a lot higher.

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So this effect of lambda_c is
very important to keep in mind,

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especially because
these parameters here

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are not necessarily constants.

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In particular, v_S has a very
strong dependence on size.

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So these different kind
of saturation curves

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are, at the very least,
dependent on the size

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of the droplets that
we're talking about.

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And there's not just one size,
so we will come to that point.