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PROFESSOR: So now, let's start
talking about airborne disease

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transmission, thinking
especially of viral diseases

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where we expect the
transmission to occur

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through aerosol droplets or
at least smaller droplets,

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which may sediment
but are going to be

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suspended in the air for a
significant amount of time.

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So the first approximation
of such a situation

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is to assume a
well-mixed room, meaning

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that the air in the
room is well mixed.

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And even if there is, let's
say, one person who's infected

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and that infected person is
breathing, talking, singing,

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respiring, and exhaling
infected aerosol droplets,

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then these droplets start
to spread around the room.

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And they do so
because the room is--

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and we assume that the
room is well mixed.

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So even though,
as I've sketched,

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there's certainly going to
be a higher concentration

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of the droplets near the
infected person, that

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is there are significant
airflows in the room, which

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induce mixing.

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That is partly driven
by the flow of fresh air

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through the room.

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Typically, we may have
some forced ventilation

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coming through ducts with fans.

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There could be open windows.

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And even when a
room is not open,

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meaning that the windows
are closed, maybe even

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the door is closed,
there's always

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some leakage and exchange of
air with the outside, which

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typically happens on
the order of hours.

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So there's always at least
some kind of flow rate.

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Also, there's a movement
of people in the room,

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and there's also breathing
itself, which basically

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imparts momentum to
the fluid and causes

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swirling motions of the fluid.

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So the situation actually
is more like this

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where there's these flows
that are generated either

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by the ventilation,
by the movement

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and respiration of the
people, also by thermal flows.

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So when you're breathing, the
air coming out of your lungs

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is warmer, and
that tends to rise.

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But, also, if it's very
humid air and very warm air,

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then actually the weight of
all the humidity and droplets

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that come with
air could actually

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cause some settling as well.

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So you have all these
different processes going on.

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And your first
approximation, just

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assume that those lead to
a well-mixed situation.

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And so we will proceed to
analyze disease transmission

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from that assumption,
and then we'll

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come back at the
end to consider what

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would happen if there
are actually fluctuations

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and think about the distance
from an infected person

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and how that might play
a role in departures

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from the predictions of
a well mixed room.

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OK.

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So defining some
variables, then.

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We'll let C of t
be the, basically,

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infectiousness of
the air, if you will.

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But more specifically,
if we think of virus,

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it'll be the virions
per air volume.

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So they're contained
in droplets.

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And later we will consider
what size droplets

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they may be contained
in, but for now, let's

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just average over
everything and say there's

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some concentration in the air.

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We have a production rate of
these infectious droplets,

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which can be broken
down into many terms.

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So this is the
production rate, so this

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is the number of virions
per time that are produced.

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So that would be Qb is
our breathing flow rate.

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So that's basically the volume
of exhaled air per breath

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cycle, so that's how much
air is being pushed out

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in your breathing.

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And this is typically--

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ranges from 0.5 up to around
3 meters cubed per hour.

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So 0.5 is a typical
resting breathing rate.

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And, in fact, even
if you're just

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kind of calmly speaking and
sitting in a room breathing

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through your nose, that'll be
your typical breathing rate.

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But if you start exercising
and start exerting yourself,

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then that could go
up to maybe around 3.

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And so that's the typical
range of the breathing rate.

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The next thing we need is Nd
is the droplet concentration,

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so this is the number
of drops per air volume.

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So that's sort of the
number density of drops.

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So if I have various
visualization techniques,

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I can actually see all the
droplets, and I can count them.

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I can say how many droplets
are in a given volume.

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The next thing is I need
the volume of a drop.

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So as I mentioned,
of course there

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are different drop
volumes, and we'll

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come back to assessing the
effects of a droplet size

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distribution.

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But for simplicity,
why don't we just take

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these sort of average size
drops that we discussed

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coming from respiration,

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And that might be a number
around 1 micron in size.

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And there's a volume that
corresponds with that, which,

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of course, is the
4/3 pi r cubed where

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we assume for now only one
size r for all drops just

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for the moment.

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So now we have the
volume here, so now

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we sort of what is the
total amount of drops.

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So the next thing
we need now is Cv.

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Cv will be the number of virions
per liquid volume or per drop

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volume, so there's certain-- so
if I could take the pure liquid

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in the drop-- let's
say it's mucus--

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and it's been coming
out of your pharynx

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and has sort of fragmented and
taken some virions with it,

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then that would be the
viral load, essentially.

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Sometimes that's
another word for this.

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And the viral load
varies with time.

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So when you first get infected,
at first the viral load

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is very low in the fluids
that you're breathing out.

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And then that raises up.

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And during the period when
you're most infectious,

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which for COVID-19
and SARS-CoV-2 virus,

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that time ends up being
around a week or so when you--

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well, within a few days, you
reach the peak infectiousness.

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And then when you're
at the peak viral load,

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this ends up being about
10 to the ninth virions

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per milliliter of
fluid when you're

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at your peak infectiousness
for SARS-CoV-2.

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OK.

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So just to give you a sense,
but of course, a lot of times

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it might be less than that.

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But if you're very
infected individual,

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that's kind of a worst case.

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And we're going to be
interested in calculating

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safety guidelines and
probabilities of transmission,

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so to be conservative,
it's good to have

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an idea of how big this
number can actually be.

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So what we just
calculate here-- so,

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basically, this Nd, Vd
is the amount of drop

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or liquid volume per air
volume, so that's essentially

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the volume fraction of liquid.

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When we go times Cv,
we're essentially

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getting the number of virions
in the air per volume of air.

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And then Qv is the
volume per time,

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so this is basically
virions per time.

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And then one other factor
that we should also

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consider is what
if we're actually

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filtering those droplets
right at the source?

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And that'll be the case
if you're wearing a mask.

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So this is an important
quantity we'll come back to.

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This will be the mask
penetration probability

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for a droplet.

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So this is-- of course
is size dependent,

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but we'll come back to that.

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But for the moment, we're just
saying it's one drop size.

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And so for the size of drops
of interest, we're asking,

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do they go through the mask?

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So 1 minus Pm is also called
the filtration efficiency.

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Yes.

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So a very good mask might be
99% of droplets are filtered.

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A very poor cloth
covering might be

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10% of droplets are
filtered, and we'll come back

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to that in just a moment.

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So this here is our
production rate capital P.

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And already with the variables
that we've written down here,

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we can write down a mass
balance for the virions.

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So virions are being produced.

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They end up in these droplets.

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The droplets are being swept out
of the room at a flow rate Q,

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and the room has a volume
V. So if I write down

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just the conservation of
mass, making sure that I'm not

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losing any virus yet--

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I'm not allowing them
to stick to the walls

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or do anything else
just yet but just

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looking at the mass balance of
one infected person breathing

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out.

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This is, I should say,
production rate per infector,

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or an infected person.

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So if there are more
infected people,

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then you'll have this
production rate for each person.

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They also might be at
different stage of the disease,

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so maybe the viral
load will be a little

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different for each person.

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But let's not worry about
such details right now.

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We want to keep things general.

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So for the mass
balance, we write down

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the total number of
virions in the room

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and how that changes in time.

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So that'll be the concentration
per room air volume, which

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is well-mixed times the
volume of the room per time.

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So this is the change in the
number of virions per time,

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and that can change in two ways.

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One, we have the production P.
So for every infected person,

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we have production
P, but this will be--

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I should say virions per
air volume per infector.

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So if we want to think about
having multiple infected people

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in the room, we can always
just basically increase

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this concentration.

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A well-mixed room, it doesn't
matter where people are placed.

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You're just getting
more and more droplets,

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and it's assumed to be mixed.

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So we produce at a rate P,
but then the outdoor flow

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is taking away
droplets and, hence,

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removing virions at a rate
Q. So we have a Q times

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C removal rate.

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So this is our equation.

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So let's divide through by V.

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So we can write this
as dC dt is P/V minus,

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and then Q/V I'll
write as lambda a C.

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So lambda a, which is Q/V,
this is the outdoor air change

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or exchange rate.

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So that is the rate at which
the entire volume of the room

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is replaced with outdoor air.

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So the outdoor air
is refreshing the air

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in the room at
this rate Q/V. OK.

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And so that's what appears here.

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And if you compare these two,
you can see that this is dC dt.

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This is C times lambda, so
lambda is units of 1 over time.

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It's a t.

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This is also sometimes called--

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the ACH is air changes per hour
if you write it in per hour.

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So that's a typical way
that this is written.

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And, in fact, while
we're just talking about,

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this is a very
important concept.

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So lambda a is around 0.3 per
hour, so roughly every 3 hours

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for a closed room.

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So closed room or what you might
call natural ventilation where

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there's no attempt to
deliver air to that room.

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Of course, this number
depends on the tightness

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of the construction and whether
there is cracks in the windows,

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whether doors are being
opened to the hallway.

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So, of course, that's
not a perfect number,

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but that's a rough estimate
how quickly air is escaping

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from typical construction.

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But then it can be, also,
in a different range.

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And this a very important
parameter for the theory,

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so let's pause just to look
at some of the numbers.

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So it's in the range--
it's typically 3 to 8

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per hour for
mechanical ventilation.

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So this could be--

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it could be open windows
with fans blowing in and out,

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which might give you 3
or even 6 on this number.

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It could also be a
ventilation system,

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which is delivering
fresh air to the space.

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And for typical classrooms,
offices, and even homes,

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this is a typical range.

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So for example, if
it's 3, then that

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would be every 20 minutes
the room gets its air

245
00:12:47,100 --> 00:12:49,110
to be fully exchanged.

246
00:12:49,110 --> 00:12:51,630
But of course, if
you have situations

247
00:12:51,630 --> 00:12:53,670
where you need to have
better air quality

248
00:12:53,670 --> 00:12:55,920
and you have more risk
of, say, transmission

249
00:12:55,920 --> 00:13:00,030
of disease or passage of
pollutants or contaminants,

250
00:13:00,030 --> 00:13:02,010
then we need higher values.

251
00:13:02,010 --> 00:13:04,890
A typical number in the
United States for hospitals

252
00:13:04,890 --> 00:13:06,810
is 18 air changes per hour.

253
00:13:10,730 --> 00:13:12,030
And then it can be even larger.

254
00:13:12,030 --> 00:13:13,440
So if you have a
laboratory, which

255
00:13:13,440 --> 00:13:16,100
is dealing with toxic
chemicals or even,

256
00:13:16,100 --> 00:13:19,790
let's say, virus and
pathogens, then you

257
00:13:19,790 --> 00:13:21,830
need even higher air changes.

258
00:13:21,830 --> 00:13:26,600
And typical rates, then,
can be as high as 20 to 30

259
00:13:26,600 --> 00:13:33,180
for labs that are dealing
with toxins of various types

260
00:13:33,180 --> 00:13:33,680
because--

261
00:13:33,680 --> 00:13:36,390
and any airborne toxins
have to be quickly removed

262
00:13:36,390 --> 00:13:37,850
so that if they
happen to be leaked

263
00:13:37,850 --> 00:13:41,620
into the air from your
experiment or from your hood,

264
00:13:41,620 --> 00:13:43,970
they need to be
quickly sucked out.

265
00:13:43,970 --> 00:13:45,500
Also, even parking
lots where you

266
00:13:45,500 --> 00:13:47,980
have cars in an enclosed space
that are generating carbon

267
00:13:47,980 --> 00:13:50,570
monoxide and other fumes, which
have to be quickly rushed out,

268
00:13:50,570 --> 00:13:52,640
parking lots tend to have
this number around 30.

269
00:13:52,640 --> 00:13:56,690
So that's a full air change of
the entire room in 2 minutes.

270
00:13:56,690 --> 00:13:58,020
That's a very fast flow rate.

271
00:13:58,020 --> 00:14:00,350
So this is kind of the
range of this lambda a.

272
00:14:00,350 --> 00:14:03,420
That's, obviously, a
very important parameter.

273
00:14:03,420 --> 00:14:06,300
So let's now solve
this equation here.

274
00:14:06,300 --> 00:14:09,450
So, first of all, you
can see when dC dt is 0,

275
00:14:09,450 --> 00:14:13,190
then the steady state is
just P over lambda a V.

276
00:14:13,190 --> 00:14:16,400
But if I write Q
equals lambda a V,

277
00:14:16,400 --> 00:14:20,350
I can see the steady
state is just P/Q.

278
00:14:20,350 --> 00:14:22,320
So I can write the
solution like this--

279
00:14:22,320 --> 00:14:27,740
that if we're given time
dependence, that the P/Q is

280
00:14:27,740 --> 00:14:29,230
the steady state.

281
00:14:29,230 --> 00:14:32,270
And if I started out with
my initial condition,

282
00:14:32,270 --> 00:14:36,260
was that C of 0 equals 0.

283
00:14:36,260 --> 00:14:39,710
So let's say time equals 0 is
when the infected person enters

284
00:14:39,710 --> 00:14:42,110
the room and starts breathing.

285
00:14:42,110 --> 00:14:43,720
And then there's
a mixing process

286
00:14:43,720 --> 00:14:45,350
and there's a build-up
of concentration

287
00:14:45,350 --> 00:14:49,700
until there's a balance between
the production of virus,

288
00:14:49,700 --> 00:14:53,660
virions, or infectious air and
the removal of infectious air

289
00:14:53,660 --> 00:14:55,310
by the ventilation.

290
00:14:55,310 --> 00:14:58,580
And that gives you this ratio
P/Q. But the way the relaxation

291
00:14:58,580 --> 00:14:59,720
happens, though--

292
00:14:59,720 --> 00:15:01,700
if you balance these
two terms here,

293
00:15:01,700 --> 00:15:06,500
that's just an exponential
decay with a decay rate lambda

294
00:15:06,500 --> 00:15:10,180
a, so e to the minus lambda a t.

295
00:15:10,180 --> 00:15:12,560
So this is basically the way
the concentration builds up.

296
00:15:12,560 --> 00:15:17,150
If I plot this, then at a
certain time here, which

297
00:15:17,150 --> 00:15:18,590
is sometimes called Tres--

298
00:15:18,590 --> 00:15:21,320
in chemical engineering,
these kinds of models

299
00:15:21,320 --> 00:15:23,480
are commonly used to
design chemical reactors.

300
00:15:23,480 --> 00:15:26,150
In fact, this kind of model
is in chemical engineering

301
00:15:26,150 --> 00:15:29,200
called a continuous
stirred-tank reactor.

302
00:15:29,200 --> 00:15:30,750
We don't worry
about the details,

303
00:15:30,750 --> 00:15:33,110
but we have a flow
of some reactants

304
00:15:33,110 --> 00:15:35,700
and various chemical
species going into a tank.

305
00:15:35,700 --> 00:15:37,880
We assume it's well
mixed and then it leaves,

306
00:15:37,880 --> 00:15:40,690
and this is the mass
balance that we use.

307
00:15:40,690 --> 00:15:45,500
And the residence time,
Tres, is the inverse

308
00:15:45,500 --> 00:15:47,040
of lambda a in this case.

309
00:15:47,040 --> 00:15:48,270
So that is the--

310
00:15:48,270 --> 00:15:50,830
as soon as you know the volume
and the ventilation flow rate,

311
00:15:50,830 --> 00:15:52,640
there's a typical speed
going through here.

312
00:15:52,640 --> 00:15:55,070
And the time that
fresh air spends

313
00:15:55,070 --> 00:15:57,560
in the room interacting with
all the droplets and people

314
00:15:57,560 --> 00:16:01,070
and then leaving, carrying some
of those droplets, is this.

315
00:16:01,070 --> 00:16:03,200
And in a simple model
like this, it's just

316
00:16:03,200 --> 00:16:10,070
an exponential relaxation at
that time scale approaching

317
00:16:10,070 --> 00:16:17,720
the steady state, which is P/Q.
So there's always that kind

318
00:16:17,720 --> 00:16:21,030
of balance which is reached.

319
00:16:21,030 --> 00:16:23,720
Now let's also ask
ourselves, briefly at

320
00:16:23,720 --> 00:16:27,320
this point, how reasonable is
the well-mixed approximation?

321
00:16:27,320 --> 00:16:29,870
We will come back to this and
analyze it much more carefully,

322
00:16:29,870 --> 00:16:32,320
taking into account all the
different processes I described

323
00:16:32,320 --> 00:16:35,330
at the beginning, including
breathing, motion of people.

324
00:16:35,330 --> 00:16:39,350
But let's just think about the
motion caused by the airflow

325
00:16:39,350 --> 00:16:39,930
itself.

326
00:16:39,930 --> 00:16:42,830
So in the case of
mechanical ventilation

327
00:16:42,830 --> 00:16:46,280
where that flow rate can
be sometimes rather high,

328
00:16:46,280 --> 00:16:49,150
that can be a significant
source of mixing in the system.

329
00:16:49,150 --> 00:16:50,570
And the way we'll
think about that

330
00:16:50,570 --> 00:16:53,360
is by writing down the
typical velocity of the air

331
00:16:53,360 --> 00:16:57,080
due to the outdoor air flow.

332
00:16:57,080 --> 00:17:01,230
That is Q. We can write it as
either Q divided by the area,

333
00:17:01,230 --> 00:17:03,260
so some kind of representative
area of the room.

334
00:17:03,260 --> 00:17:04,960
It could be, let's
say, the floor area,

335
00:17:04,960 --> 00:17:06,589
but depending on the
shape of the room,

336
00:17:06,589 --> 00:17:09,810
it might be a little bit
different value than that.

337
00:17:09,810 --> 00:17:14,030
We can also write this as
Q times H over V, where

338
00:17:14,030 --> 00:17:15,680
V is the volume
of the room and H

339
00:17:15,680 --> 00:17:18,140
is some characteristic height,
like, for example, a ceiling

340
00:17:18,140 --> 00:17:19,470
height of the room.

341
00:17:19,470 --> 00:17:20,240
OK.

342
00:17:20,240 --> 00:17:26,900
So this is the mean
airspeed due to ventilation.

343
00:17:32,070 --> 00:17:34,530
And we'll come back to a
more deeper investigation

344
00:17:34,530 --> 00:17:37,770
of these fluid
mechanics of the room.

345
00:17:37,770 --> 00:17:40,950
But as a first example of what
we'll be interested in is we'd

346
00:17:40,950 --> 00:17:44,870
like to calculate the Reynolds
number due to the airflow.

347
00:17:44,870 --> 00:17:47,490
So I'll put a subscript a there.

348
00:17:47,490 --> 00:17:52,380
And that is the typical velocity
times a length scale, which

349
00:17:52,380 --> 00:17:54,270
could be, let's say,
the height of the room

350
00:17:54,270 --> 00:17:56,130
or some linear length
scale of the room,

351
00:17:56,130 --> 00:17:58,290
depending on the
direction of the airflow.

352
00:17:58,290 --> 00:18:02,250
And then the kinematic
viscosity of air,

353
00:18:02,250 --> 00:18:04,500
so I'll write that as new a.

354
00:18:04,500 --> 00:18:06,000
So this here is the
Reynolds number.

355
00:18:12,540 --> 00:18:17,020
This is basically telling us how
important inertia of the fluid

356
00:18:17,020 --> 00:18:21,760
is compared to viscous stresses
that slow the fluid down.

357
00:18:21,760 --> 00:18:24,540
So, basically, it
tells us how quickly--

358
00:18:24,540 --> 00:18:26,380
how much of a tendency
there is for momentum

359
00:18:26,380 --> 00:18:27,960
to be carried in the
fluid, which then

360
00:18:27,960 --> 00:18:30,610
leads to sort of swirling
motions and complex flows,

361
00:18:30,610 --> 00:18:32,260
as I've sketched here.

362
00:18:32,260 --> 00:18:35,650
And this new a is the
viscosity of the air

363
00:18:35,650 --> 00:18:38,560
that we've already talked about
when looking at Stokes flow

364
00:18:38,560 --> 00:18:41,390
but divided by the
density of the air.

365
00:18:41,390 --> 00:18:41,890
OK.

366
00:18:41,890 --> 00:18:43,660
So that's the
kinematic viscosity.

367
00:18:43,660 --> 00:18:46,840
And for air, the
kinematic viscosity

368
00:18:46,840 --> 00:18:54,360
is 1.5 times 10 to the minus
5 meters squared per second.

369
00:18:54,360 --> 00:18:57,710
And so if I plug these
numbers in and I pick,

370
00:18:57,710 --> 00:19:03,700
let's say, a typical ceiling
height of maybe a few--

371
00:19:04,960 --> 00:19:09,070
I think this-- if the H
is of order 3 meters--

372
00:19:09,070 --> 00:19:12,160
or 2 meters might
be a typical scale.

373
00:19:12,160 --> 00:19:16,120
Just to get an approximate
sense of the scale here,

374
00:19:16,120 --> 00:19:21,670
the Reynolds number will be
varying from around 50 or tens,

375
00:19:21,670 --> 00:19:25,420
up to 5,000, which would
be in the case of very fast

376
00:19:25,420 --> 00:19:28,300
ventilation like the 30 ACH.

377
00:19:28,300 --> 00:19:37,340
So this would be if we have
0.3 ACH up to around 30 ACH.

378
00:19:37,340 --> 00:19:39,640
That's just a rough number.

379
00:19:39,640 --> 00:19:41,740
And from fluid
mechanics, we know

380
00:19:41,740 --> 00:19:44,200
the significance of these
large Reynolds numbers

381
00:19:44,200 --> 00:19:47,940
is that the flows really do
look a bit like I've shown here.

382
00:19:47,940 --> 00:19:50,110
So, basically, when the air
is sitting in the room--

383
00:19:50,110 --> 00:19:51,730
you know this from
looking at the smoke

384
00:19:51,730 --> 00:19:54,460
from a candle or other flows
that you can visualize--

385
00:19:54,460 --> 00:19:58,270
it's not just a sort
of uniform flow.

386
00:19:58,270 --> 00:20:00,910
But instead there are all
these sort of plumes and swirls

387
00:20:00,910 --> 00:20:02,260
and vortices.

388
00:20:02,260 --> 00:20:04,900
And when the Reynolds number
is on the order of tens,

389
00:20:04,900 --> 00:20:07,570
when there is a motion,
it tends to lead

390
00:20:07,570 --> 00:20:11,460
to a shedding of a vortex and
to some kind of swirling flows.

391
00:20:11,460 --> 00:20:13,300
And when the Reynolds
number gets up as high

392
00:20:13,300 --> 00:20:15,940
as several thousand,
then in most geometries

393
00:20:15,940 --> 00:20:18,190
that starts to lead to a
transition to turbulent flow.

394
00:20:18,190 --> 00:20:19,640
And that's when
the flow is getting

395
00:20:19,640 --> 00:20:21,910
so complicated there are
eddies of different sizes

396
00:20:21,910 --> 00:20:23,980
and very rapid mixing.

397
00:20:23,980 --> 00:20:25,780
So I just wanted
to show this right

398
00:20:25,780 --> 00:20:27,430
at the beginning
of the discussion

399
00:20:27,430 --> 00:20:29,260
to point out that
for typical flows

400
00:20:29,260 --> 00:20:31,120
we would expect
that there is going

401
00:20:31,120 --> 00:20:33,160
to be a decent amount
of mixing occurring just

402
00:20:33,160 --> 00:20:34,680
because of the ventilation.

403
00:20:34,680 --> 00:20:37,060
And now if you add to the fact
that people are breathing,

404
00:20:37,060 --> 00:20:38,860
imparting momentum
to the fluid, we

405
00:20:38,860 --> 00:20:42,970
have people moving and other
kinds of activities in a room,

406
00:20:42,970 --> 00:20:45,940
that all of those
processes lead to giving us

407
00:20:45,940 --> 00:20:50,400
a reasonable assumption
of a well-mixed room.