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PROFESSOR: So now let's focus
on the steady-state transmission

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rate, which is really the most
useful in designing a safety

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guideline.

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It's also the most
conservative because

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the transient transmission
rate is always

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smaller than a steady state.

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So, our formula for the
steady state transmission rate

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is shown here, in
terms of the relaxation

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rate, lambda_c(r), of
the aerosol concentration

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in the air, and
also n_q(r), which

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is the density of infection
quanta in the air, per radius.

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So let's sketch some of
the important functions

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here as a function
of radius and try

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to get a sense of how we can
maybe simplify this expression.

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First of all, we've
already defined

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C_q, which is the
integral of n_q(r),

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dr. This is the critical
disease parameter, which

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is the infection quanta
per exhaled air volume.

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I don't mean to cross that out
but rather just to do this.

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So C_q is a very
important quantity for us

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and we will return to that.

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That's the quality
that we're going

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to want to fit to disease data
for COVID-19, specifically.

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And this will be the exhaled
infection quanta per air volume

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and we'll typically
want to measure

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that sort of peak
infectivity of an individual

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in order to design the
conservative criteria.

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So what is C_q?

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If I plot this n_q, it has
a bunch of factors in it.

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So it has the roughly
constant assumed viral load

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per liquid volume, it
has the infectivity,

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which we've already
argued should be smaller

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in larger droplets because it's
more difficult for the variant

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to diffuse out of those droplets
once you get above, say, 5,

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10 microns, if not less.

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There's the droplet
distribution itself,

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which depends on the
type of respiration

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but often has a peak
which is submicron

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and then sort of a fairly
broad tail at the higher end,

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with smaller amounts
of larger droplets,

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and then V_d is 4
pi r^3, which

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is just the volume of a drop.

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So this net quantity,
n_q has some kind

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of peak around 1 micron
or less and then a tail.

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And then in the
integral here, we

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have the integral of n_q over r
is c_q, so that's very important.

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But there is these other
factors, p_m and lambda_c,

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or 1/lambda_c.

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Each of those
quantities gives us

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a cutoff which makes
the larger droplets less

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important for this problem
of airborne transmission

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in a well-mixed room.

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So lambda_c, as you
can see, is a bunch

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of constant factors except
for the sedimentation rate.

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So this lambda_a times
(r/rc)^2,

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that is the sort of
radius-dependent change

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of the sedimentation rate
relative to the ventilation

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rate, lambda_a.

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So as you can see, this goes
like 1 plus a constant plus,

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r^2.

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So as you go to large
r, the inverse of that

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is 1 over a constant plus
r^2, so it goes to 0.

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So it provides a cutoff and
the scale for that cutoff

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is what we called r_c, that's
the critical size of a droplet,

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which is just
sedimentary at a rate

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comparable to the
ventilation rate

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because, really,
this is ventilation

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and sedimentation which are
compared when you define r_c.

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In addition to that,
we have (p_m)^2,

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which is the max penetration,
or transmission factor.

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So while masks are
100%, or very efficient,

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filtering large
droplets which don't

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fit through the
fabric or the mesh,

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they're not as good
as filtering

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smaller droplets.

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So if you look at the
transmission probability

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when you're down
well below micron,

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most masks are not doing
a great job filtering,

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they may get 5%,
10% if you're lucky,

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depending the
quality of the mask.

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But then it comes
down because you

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start to have better and
better blockage of particles

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by the masks.

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All these factors serve to
cut off this distribution so

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that we're not worried
about the large drops,

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and we're interested
in aerosols.

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But it does so, here, in a
way which is quantitative.

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So we're not just
arbitrarily saying,

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as it's sometimes said
in the field, that

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say 10 microns or 5 microns
is the limit of the aerosols,

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but rather, we actually have
a well-defined characteristic

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size that can emerge here.

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And the way we can define
that is by taking the full

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expression for the steady
state transmission that is has

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the radius-dependent terms in it
and write this as (Q_b)^2/V

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times--

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and then we'll keep the C_q from
the integral of n_r-- C_q times--

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and then we'll imagine that
the remaining radius-dependent

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factors, p_m and
lambda_c, are sampled

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at a certain value, r-bar.

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So what is r-bar?

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Well if you know the
function is p_m and lambda_c

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is a function of
r, there is a value

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of r, which we call r-bar,
which is when you actually

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do this full integral,
you would get that value.

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So that has to be determined.

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It can be done numerically but
you can kind of see graphically

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where it ends up.

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What we're asking
here is what is

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the typical value of
the mask penetration

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factor and the relaxation time?

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Well, it's going to be where
the most weight is here,

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keeping in mind,
also, that there's

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more volume at the higher
side than at the lower side.

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So if we look at how
much activity there is,

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we might want to emphasize that.

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So depending on the details
here, somewhere over here

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is going to be r-bar.

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What we're saying here is that
even though our theory has

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all of the radius
dependents in it,

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so if you know exactly
the type of masks

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you have and you know p_m(r)
from experimental measurements,

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maybe a lot about the
virion and how infectious

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it is in different-sized
droplets,

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or you studied sedimentation--
you have all these functions.

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There's a well-defined r-bar
at which you can just

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use this simple expression
in place of actually doing

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those integrals.

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So that's actually
useful simplification.

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And in addition to
that, we can also

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write this another way,
which can be useful,

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is to take the mask
factor out and write it

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as a quanta emission
rate, lambda_q, where

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lambda_q is Q_b*C_q.

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This is the quanta emission
rate by an infector.

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So if you don't like this notion
infection quanta per volume,

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when you multiply by
the breathing flow rate,

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you're actually getting
how many quanta per time

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are being emitted
by the infector.

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And then what's leftover
is another factor,

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which I'll call f_d,
which is something

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we'll come back to later, which
is what I call the dilution

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factors.

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If we take the breath of
an infected individual

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and then it ends
up being diluted

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into the room, the ratio of
the concentration of infection

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quanta, or virions in
the breath compared

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to that which emerges
in the well-mixed room,

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that's the dilution factor.

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This will become important
later when we look more closely

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at respiratory fluid
mechanics and we

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look at the plumes,
or clouds, of droplets

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that are being emitted by a
person when they're breathing,

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very close to the
person's mouth it's

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a much higher concentration
and eventually it

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gets sort of swirled around
and mixed in the room,

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and it reaches the steady
state values that we calculate.

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This f_d gives you that
ratio in some sense

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and gives you a sense
of how bad the risk is

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from short-range transmission
versus the well-mixed room,

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so we'll come back to that.

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But this is a nice
simplification

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for how we can think about
the steady transmission

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rate in terms of several
key variables, which

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I've boxed here.

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And so we will now move on
to applying this to COVID-19.