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PROFESSOR: So the next important
part of turbulent plume theory

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that we need is the distribution
of concentration of particles

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or droplets, in this
case, that are injected

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with the fluid at the source.

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So as we've just arrived, the
concentration C in this case,

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we could have referred
to infection quanta

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and infectious aerosols
relative to that leaving

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the mouth, which is
we've called C_q, scales

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as square root of area of the
mouth divided by alpha x where

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alpha is the turbulent
entrainment

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coefficient, around 0.1 or 0.15.

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And that leads to a
jet which grows in size

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and grows in fluctuations as
you see more and more eddies,

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and eventually might
even bend due to flows

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in the room or thermal
buoyancy effects.

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And I'd like to talk about the
difference between short range

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transmission due to really
placing yourself in this jet

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and breathing that
air directly, which

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is more concentrated
than the background,

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and then compare that
with the transmission

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in the well-mixed
room that we've

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been talking about all along.

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So obviously if
we're at position 0

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here right at the mouth, this
is the worst case scenario.

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So if we're here, let's
say we're only 1 inch

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or 1 centimeter away.

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And we put our mouth on top of
the other person's mouth, that

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is the worst case scenario
of short range transmission.

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That is, relative to
the background room,

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we're getting a much
worse situation there.

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But if we ask ourselves,
how much worse is it,

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so we know that fd is
the dilution factor, that

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is the concentration kind of
at the source relative to C

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sort of at infinity and
the well-mixed room,

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so let's just say far away.

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Actually instead of C infinity,
I should call that C average.

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Because it is the average
concentration in the room.

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And we've seen that
the dilution factor

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can be written as the flow
rate of the breath divided

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by the sort of decay rate
of the concentration field

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at the average--

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at the appropriately
defined mean radius

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divided by the
volume of the room.

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So that is telling us how
much more concentrated

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the infection quanta
or viruses are here

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versus the well-mixed room
where they're really spread out.

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How big is this factor?

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So this factor for the Skagit
choir, which we've analyzed,

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the Skagit Valley Chorale.

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So that was a fairly large
room, 4.5 meter ceiling.

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But it didn't have
very good ventilation.

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So that's sort of
similar to smaller rooms

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with better ventilation.

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And in that case, this
number was 10 to the minus 3.

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And in general, fd for
typical indoor spaces--

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for offices,
classrooms, and homes--

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is on the order of 10
to the minus 2 to the 10

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to the minus 4.

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So there's quite a
significant difference.

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If you are right at the
point of somebody's mouth

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and breathing in their
air versus being far away,

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there really is
a big difference.

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To put it in perspective,
we can ask ourselves, well,

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how long would you have to
stay in the well-mixed room,

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far away, breathing the air
to have the same exposure

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and dose of infection quanta
as if you put your mouth on top

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of somebody else and breathed
in one lung-full of air.

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So if we just say that if we
calculate it as a timescale,

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taking in the volume
of a single breath

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and then dividing that by Q
breath and the dilution factor,

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that is the time you
would have to spend

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breathing the background air in
order to achieve the same dose.

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And this quantity ends up being
for the Skagit choir that we

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calculated around one hour.

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So if 63 people were in
the room and 53 or so

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were infected, then
that most likely

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happened through the airborne
route, as we've discussed.

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Because 53 people were mostly
breathing the background air

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that was, perhaps, well-mixed.

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On the other hand, you
also could infect 53 people

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by taking turns one
at a time putting

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their lips against the
other person who's infected

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and just breathing in one
full lung-full of air.

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Then you would get
a similar number

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of people infected on the
order of an hour or two,

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which is the length of time
of that choir practice.

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But we know that didn't happen.

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So that already tells us that
short range transmission really

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can't explain what happened
in the Skagit choir.

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And as we've discussed,
it has to be longer range

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airborne aerosol transmission.

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But how much longer range?

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We can also sample at
different positions here.

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So that's the absolute
worst case scenario.

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So let's consider as important
numbers 3 feet and 6 feet.

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So 3 feet corresponds
to 1 meter,

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which is the social distancing
guideline of the World Health

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Organization today.

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I would also argue
that 3 or maybe 2 feet

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is kind of close
to what you might

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call natural social distancing.

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So if you don't impose
social distancing,

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most people prefer to
have a little space

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bubble around them.

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They don't want to be
right up against somebody

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if they don't have to.

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And they tend to stand
2 or 3 feet apart.

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So somewhere in
here is what I would

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call natural social
distancing except in cases

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where you're in a crowd.

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So if you're in a
nightclub or a bar

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or some crowded
space where you're

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starting to press
against people,

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then you come closer you
might be 1 foot or 1/2 foot,

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and you might start to get
closer to this worst case.

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But people tend to be
about 3 feet apart.

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So we can ask ourselves
what happens there.

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We can also look
a little further.

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The Center for Disease
Control the United States

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has imposed a 6 foot rule
as we have discussed.

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In fact, it's been interpreted
so strictly in the United

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States that you can find floor
stickers exactly 6 feet apart

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in all sorts of indoor
spaces even when

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people are wearing
masks and when

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we aren't sure exactly
what the flows are like.

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And we can ask ourselves,
what is the sort of level

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of concentration there.

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It's also worth noting
that we can also

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look at a negative value.

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How about minus 3 feet?

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Because it's important to note
that respiratory jets do not

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only increase your risk relative
to the well-mixed ambient,

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there must be regions where
the risk is actually lower

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than the well-mixed ambient.

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Because in the end,
the well-mixed solution

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was obtained by mass balance.

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So that means we've essentially
counted all the infection

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quanta or virions in the room.

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And if there is a higher
concentration here,

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there must be a lower
concentration somewhere else

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to account for that.

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So perhaps if you're
standing behind somebody

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at a reasonable
distance, you actually

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have a lower concentration,
although there is still

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mixing going on.

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And you will be exposed
to the well-mixed room.

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Which then brings me to the
way that we should really

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think about the role of
short range transmission

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versus long range transmission.

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And that is to compare
the concentration

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of the respiratory jet to
that of the well-mixed ambient

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that we calculated.

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That's essentially
the definition

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of when there's a transition
from short range to long range

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behavior.

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So where will that occur?

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It will typically
be somewhere here.

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So there's a certain
position, which we'll call xC,

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and this will be
defined by saying

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that C over the initial value
is equal to the dilution factor.

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So this is the point where if
I follow this orange curve,

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I've just hit the
concentration that

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is predicted for the background
well-mixed room, which we've

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already calculated before.

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So if you use our
formula for C there,

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you then get a formula
for this xC, which

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is the square root of the mouth
area times lambda C of r bar,

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so the decay rate of
the concentration field,

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the volume of the room,
and then alpha QB.

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So I would argue
that this is really

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the boundary which
separates long range

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airborne transmission by
aerosols with short range

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transmission, which also
includes aerosols by the way.

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So some of these
aerosol droplets,

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you definitely could be
inhaling anywhere in here.

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But if you go long
range, you're only

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talking about aerosol droplets.

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So this is really where
the dividing line is.

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And if you plug in numbers
for different settings,

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you'll find this is
often larger than 6 feet.

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In fact, in some
cases, many cases,

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it's actually larger
than the room.

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It could be tens of meters even,
because this dilution factor

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can be of order a
factor 100 or even

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10,000 lower concentration
in the well-mixed room

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than at the person's mouth.

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So you have to go pretty
far away to see it actually

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drop back down.

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What that means is that when
people are breathing in a room

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these breaths are crossing
all over the place.

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They're mixing.

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And you really can't think about
a turbulent plume lasting out

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to infinity because there is
another person standing here,

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there's ventilation flow,
there are thermal flows.

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All of the mechanisms we talked
about mixing will take this jet

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and start to mix
it around the room

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so it won't look like a
perfect jet all the way.

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But at least this
gives us an estimate

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for these closer
spaces here that's

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sort of what our risk might be.

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So using this concept, I'll
just mention this is typically

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much bigger than 6 feet, we
could estimate how much worse

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is it to be 6 feet
or 3 feet away

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with yourself in a worst
case scenario perfectly

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placed in a respiratory jet.

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So what we're
asking here is what

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if we were unlucky enough to
be right here and breathing

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in for a long period of time.

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So not just for a
fleeting second but you're

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sitting there and
just over and over

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you're just sitting
here breathing

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in that person's breath.

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It's not an
unreasonable situation.

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You could imagine
at a meeting when

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two people are
sitting at a table

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or maybe they're having dinner.

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Masks are off, they
are facing each other,

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talking to each other.

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And in fact that is
a high risk situation

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in terms of this
kind of transmission.

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So let's see how much
worse it actually could be.

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Well, if you plug in the numbers
then using a typical mouth

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area, in this case
the concentration

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drops to about 6%.

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In this case, the concentration
since it's what goes as 1

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over distance, is more like 3%.

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So it's a very
rough calculation.

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But relative, and I should
say, so relative to the highest

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concentration of
reaching the mouth,

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the plume has been
diluted down to 6% or 3%

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as you go these distances.

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Notice there's not a massive
difference between 3 and 6.

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And it does make a
very big difference

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in terms of decisions
to reopen spaces

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and how people interact,
whether you're strictly

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6 feet apart or
perhaps you can be 3.

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So keep in mind that
3% to 6% dilution.

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But what we're really
interested in here

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is now, how do these
fractions compare

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with the well-mixed room.

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And so for that I
actually would like

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to do a different comparison
as I'd like to do C over fd.

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Because fd, relative to here,
is the concentration far away.

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So I want C over C_q,
I should say, over fd.

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And so this also can be
written as xC over x.

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So it's basically how much
farther is that crossover point

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00:12:33,060 --> 00:12:34,480
relative to where
you're standing.

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That's another way
to think about it.

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And this is then six to 600.

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So this is the excess risk.

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If you were to
stand 3 feet away,

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that's the excess risk you face
from short range transmission

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if you are 100% of
the time breathing

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in the jet of the infected
person pointing right at you

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all the time.

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If you take this 6 foot rule,
it's a little bit less strong.

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You have a C over
C_q divided by fd.

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It is more like three to 300.

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00:13:13,540 --> 00:13:16,220
It, again, depends on
the details of the room.

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But it still can be
a significant factor.

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So we will come back
to thinking about how

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to handle short transmission
in light of its interaction

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00:13:25,720 --> 00:13:27,820
with long range.

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00:13:27,820 --> 00:13:29,680
But let me make
one final comment

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on this discussion, which is
where is this crossover point.

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So notice it scales with volume.

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00:13:37,580 --> 00:13:39,770
So something you
may be wondering

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is, don't we have
airborne transmission

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and shortwave transmission
outdoors also.

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Yes.

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And that is actually covered
by the theoretic arguments

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00:13:49,340 --> 00:13:50,300
that we're making here.

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Although, as I said, these
jets really can't last forever.

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But we can at least
say to ourselves, what

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happens as v goes to infinity.

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So I like to call this
the outdoor limit.

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If you apply our safety
criterion for a well-mixed room

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to very big room, think of,
for example, a gymnasium

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or a sports stadium,
very, very big room, then

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00:14:12,780 --> 00:14:14,880
if there's only one
infected person,

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then the dilution of those
droplets in a massive space

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is like being outside.

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00:14:20,650 --> 00:14:22,140
You might as well be outside.

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00:14:22,140 --> 00:14:23,790
And what you find
that is the crossover

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is very far away, which means
the long range analysis is not

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00:14:27,270 --> 00:14:27,900
even helpful.

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00:14:27,900 --> 00:14:29,460
It's not even valid.

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00:14:29,460 --> 00:14:31,170
Really, it's the
short range analysis

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00:14:31,170 --> 00:14:32,760
you'd be thinking about.

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00:14:32,760 --> 00:14:34,560
Now unfortunately, the
short range analysis

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00:14:34,560 --> 00:14:36,180
depends on many
assumptions about how

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00:14:36,180 --> 00:14:38,130
people are positioned,
how they're interacting,

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00:14:38,130 --> 00:14:39,600
what are the local flow fields.

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00:14:39,600 --> 00:14:41,720
But you have to deal with
short range transmission.

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00:14:41,720 --> 00:14:43,530
And social distancing
can play a role

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00:14:43,530 --> 00:14:45,420
in how you address that threat.

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00:14:45,420 --> 00:14:48,480
But once you get to smaller
rooms and longer periods

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00:14:48,480 --> 00:14:53,220
of time then you actually
find that this xC is

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00:14:53,220 --> 00:14:54,480
on the order of the room size.

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00:14:54,480 --> 00:14:58,260
And then you really are going
to be having significant effects

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00:14:58,260 --> 00:14:59,910
of long range transmission.

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00:14:59,910 --> 00:15:03,810
And I will argue that
for most typical rooms

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00:15:03,810 --> 00:15:09,060
that we inhabit indoors, that
the long range airborne risk is

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00:15:09,060 --> 00:15:12,180
the leading order first
approximation and best

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00:15:12,180 --> 00:15:13,350
approximation.

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00:15:13,350 --> 00:15:16,380
And the short range
risk must be considered,

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00:15:16,380 --> 00:15:19,070
but it's a correction to that.