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PROFESSOR: So now that we've
done a very simple calculation

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of C of t, the
concentration of virions

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in the air per volume,
which is coming

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from just a mass balance
in a well-mixed room,

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let's zoom in, now, and
think about transmission

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between the infected person,
or an infected person

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and then a susceptible person
who is not yet infected.

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And the transmission
is occurring

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by breathing the
infected droplets in,

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and then the virus has to
get out of those droplets

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and interact with
the host tissues.

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So if we let beta be
the transmission rate,

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so this is basically
new infection per time,

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so basically, it's the rate
at which another person may

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become infected.

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Then how will we write this?

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Well, we could write it as--

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so we're, now, forgetting
about the infected person.

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We care about their
breathing only

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because it's producing this
concentration C that we just

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talked about.

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But now we're going to focus
on the susceptible person.

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The susceptible person is now
breathing in at the same flow

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rate Qb because the breathing in
and breathing out are the same.

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And so Qb is the volume
per time around, let's say,

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0.5 meters cubed per hour with
which they're sampling the air.

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And the air comes
in, and then C of t

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is the concentration
of virions per volume.

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So this combination now tells
me how many virions per time

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they're actually taking in.

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There'll be a quantity Ci, which
we mentioned earlier, which

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is the infectivity, and
that's the probability

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that an individual
virion actually

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causes this person to
get sick and to become

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infected themselves.

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So this is the infectivity
of an individual virion.

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And then on top of that,
we also want to put in--

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I'll just change the
color to highlight it--

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the mask factor that we
talked about earlier.

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So that's the transmission
or penetration probability

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for the droplets of
interest through the mask.

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These quantities, many of
them will depend on size,

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and we'll come to that,
size of the droplet,

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but just as a rough
approximation,

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this is the starting point.

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So this is the number of,
basically, new infections

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per time, and there
is a useful notion

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in epidemiology, which is
that of the infection quantum.

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So transmission rates
are often written

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as infection quanta
per time, and that

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is the rate at which a
person, which is susceptible,

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will get infected.

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What we have not yet
captured is if you

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have a finite number
of people in the room,

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when someone gets infected,
they can't get infected again.

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So the numbers of susceptible
people is changing.

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So we have to model the
progression of the disease

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in the room, which
we have not done yet.

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So that's why the beta is not
the number of infected people

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cause eventually you run
out of people to infect.

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So we have to account
for that later.

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But a useful way
to think of it is

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that beta is sort of the rate
at which this person is sending

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infection quanta over here.

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Those quanta may not
actually lead to an infection

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because they might already
have been infected.

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But if they're
susceptible, that tells you

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the rate at which that
person would become infected.

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So that's the notion
of infection quanta

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is essentially defined by
the transmission rate beta.

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Now, this infectivity is
something we'll come back to.

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We will actually go through
the calculation for SARS-CoV-2,

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but it's been estimated
before to be at about 2%

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from the original
SARS virus in 2003.

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And in fact, I will argue
that it's greater than 10%

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for SARS-CoV-2.

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And we'll do that by analyzing
spreading data with the model.

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And, of course, that
helps to explain

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why SARS-CoV-2 has led
to the COVID-19 pandemic,

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and SARS-CoV-1 was not
able to spread as much.

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So now we have here
our transmission rate.

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And we can ask ourselves, this
is a transmission rate, which

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is time-dependent,
but what if now

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we calculate the steady-state
transmission rate?

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So the transient would be
when the infected person first

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enters the room.

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The concentration is
changing in time in the air,

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but eventually, there's kind
of a steady-state where there's

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a balance of the production
and then the flow rate

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through the room of refreshing
the air with outdoor air.

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And in steady-state, we
have the transmission rate

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is going to go to
a constant value,

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which I'll call
beta bar, and that

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is given by the
steady-state concentrations.

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So here I'll just write it--

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I'll rewrite this expression
here Qb Ci Pm and then

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the steady-state C, which is
the production rate P divided

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by the outdoor airflow rate Q.

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And another way we
can write that is

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that remember Q we can
write as lambda a times

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V. So this is Qb, and in
fact, let me-- well, here,

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I'll write it one more time.

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That's my Ci Pm capital
P over lambda A V.

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Now, recall that the
P we had written as,

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that's the production
rate, also depends

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on Qb, that's the rate at which
the infected person is exhaling

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infected air.

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So that was Qb times nd, the
number of droplets per volume,

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Vd, the volume of
liquid in a droplet,

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so this nd Vd's the
volume fraction of liquid.

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And then we needed Cv, which
was the concentration of virions

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in the liquid or in the fluid.

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And there's also a factor
of Pm if that person

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is wearing a mask.

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So we put all this together, we
get an important result here,

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which is that the steady-state
transmission rate can

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be written, when I plug-in here,
as Qb squared times Pm squared.

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So the mask factor
comes in twice

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because if they are wearing
masks, there's two masks.

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You have a mask at the source.

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You also mask at the
target, and the fluid

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has to go through
both of those filters.

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So that's one reason, as
we'll see, that masks can be,

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actually, very effective.

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And then we'll lump all
the parameters in something

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I'll call Cq, which
I'll come back to,

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and then we'll leave lambda
a V in the denominator.

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So this is the main
transmission rate

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where I've defined this
important parameter Cq which

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has all the information
about the specific disease,

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and what is it?

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It's everything else is left.

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It's nd Vd, so that has
to do with respiration,

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so the distribution
of droplet sizes

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and the size of the
droplets is something

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that's coming from the
type of respiration

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that the infected
person is engaging in.

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Cv is their viral
load, so it has

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to do with the progression
of their disease

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and how many viruses
or virions are found.

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And then we have Ci,
which is this infectivity,

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the probability that
any one of those virions

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will actually infect
this susceptible person

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if it manages to get in
there and diffuse out

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of the droplets.

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So coming back to this
notion of infection quanta,

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while beta is an infection
quanta per time, which

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are being kind of transmitted
from one infected person to one

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susceptible person, the
way I've written it here

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is I've reexpressed it as
infection quanta per volume

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of air exhaled.

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So while C is the concentration
of virions in the background,

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there is this Cq, which is
essentially the infection

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quanta that are being released,
and the factor of CvCi

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is actually what
connects those two.

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In fact, sometimes we
write Cq little cq CvCi,

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this is infection quanta
per liquid volume in a drop.

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So from the mucus or material
that's being released,

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there is a certain concentration
of infection quanta, which

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is the physical concentration
of the virions Cv

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times the probability
that if they

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were to be exposed to the
susceptible person's cells,

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that they would actually
infect those cells

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and cause a transmission
of the infection.

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So that's another
important quantity,

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and this here is really
the primary sort of lumped

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or combined disease
and physiological

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parameter in the model.

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So what's nice about
separating this way is

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that Qb is something which has
to do with people's activity,

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it's how fast they're
breathing, and that's

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something we know very easily
whether they're at rest

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or they're exercising.

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Pm is also known if we know
the kinds of masks people

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are wearing.

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There's various studies
of transmission factors

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and filtration
efficiencies of masks.

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And so we can put
reasonable estimates there.

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And then here we
see the importance

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of lambda a, which is
the air exchange rate.

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So how quickly is fresh
air coming in the room?

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That's a physical
parameter of the room, has

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nothing to do with the disease.

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And V, of course, is a
geometrical parameter,

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the room, which is the volume.

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And so all the disease aspects
are kind of lumped into the Cq.

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So if I want to apply this to
actual spreading of COVID-19,

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I have all these
parameters that I

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know that come from
the physics and fluid

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mechanics of the room.

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And then, I have a
single parameter Cq

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that I need to obtain from
an understanding of disease

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transmission and looking
at spreading events

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in indoor situations.