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PROFESSOR: In most of our
calculations of safety,

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we're going to be interested
in the steady state

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average transmission rate
between individuals in a room,

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after the aerosol particles
have built up to a steady state.

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But let's briefly talk
about the transient buildup,

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and how to take
that into account,

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just as an aside
in this board here.

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So here is the
general expression

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for the transmission
rate that we describe,

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which depends on the
breathing rate squared,

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the volume of the room.

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It's an integral over
all the different drop

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sizes, where NQ is a
lumped distribution

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of the number of infection
quanta per volume per radius.

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And then PM is the mass
transmission factor,

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which depends on radius.

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And lambda C is the
total relaxation rate

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involving sedimentation
or settling and viral

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deactivation and filtration.

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And that also depends,
of course, on R.

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And that lambda C of R also
ends up in the exponent here.

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So that actually, that
rate of relaxation

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of the concentration
in the air, is also

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setting the time scale,
lambda C inverse,

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for the buildup of those
aerosol droplets in the air.

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And so that's this factor here.

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This is basically the
transient term is this term.

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And then this term, the
one, is the steady term.

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So we're interested now what's
the effect of the transient.

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Now, before we get to that, if
we forget about the transient

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now, and we just have
the steady state,

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then we introduce
beta bar as the sort

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of constant steady state
value transmission.

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And through this definition
here, by doing these integrals,

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we have defined an
effective radius R bar,

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which is sort of where you
evaluate the mass transmission

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factor, and also the
filtration-- or the relaxation

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rate in order to make
these two values equal.

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So that's actually our
definition of effective radius.

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And so now, looking
at the transient term,

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let's ask ourselves, what
is the average transmission

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rate up to a certain time tau.

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So that would be, we
divide by a time tau,

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and we ask ourselves
up to that time,

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what is the average
transmission rate?

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So we integrate beta dT from
0 to tau, then divide by tau.

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So what would that be?

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Well, we can take
this time integral

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and bring it inside
the radius integral,

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and write this as QB
squared over B integral 0

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to infinity of PM
squared NQ of lambda C.

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Keep in mind all those
factors depend on R. Times,

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now, an integral from 0 to tau--

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so I'll put this in brackets--

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of 1 minus e to the minus
lambda C, which depends on R,

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times T divided by
divided by tau dT.

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And then dR.

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So switching the order of
integration, where we're going

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to do the time integral first.

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And so what we have here,
if we just look only

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at this expression
right here, we

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can write this as a sum
of a steady state term.

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So when it's just 1, this is
the integral 1 over tau from 0

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to tau, so that's just 1.

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So that's the steady
state contribution.

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1 plus, and there's a
transient contribution where I

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have to do this integral here.

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So that's e to the
minus lambda C of T

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over lambda C tau,
evaluated from 0 to tau.

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And so we'll come back
to this in just a moment

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and evaluate that.

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But just to draw a
picture maybe first

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of what we're looking at here.

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The average transmission rate
as a function of this averaging

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time tau, well,
eventually of course,

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it tends to the
steady state value.

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But it does so in a certain
way we're going to calculate,

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like that, where the time
for that transmission--

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or for that transition, is the
inverse of the relaxation time.

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Although there's not a
precise value of that.

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But if we want to keep
actually a scale for it,

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it's going to be evaluate at that
value R bar that I mentioned.

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That gives you a rough sense
of the overall relaxation.

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So this build up of the aerosol
concentration in the room

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once the infected
person has entered,

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and eventually, there's sort
of a steady transmission rate

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to everyone else in the room.

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So let's continue calculating
this right here now.

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So this is the transient.

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And I can write this as 1.

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And if I evaluate here, I can
put it this way, as minus,

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and then I evaluate first at
the lower limit, which gives me

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another 1, minus,
and then evaluating

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at the upper limit, which is
tau, e to the minus lambda

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C tau over lambda C tau.

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And now, I'll use
an approximation

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that helps me get a
simple analytical results.

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So I should mention,
as soon as we

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have exponential and
polynomial factors,

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it can be difficult
to solve equations.

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For example, what is the
bound on the occupancy

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or the time in the room,
or the ventilation.

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We like to get a simple formula.

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And so if there's a nice
approximation I can make,

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which is that 1 minus
e to the minus x over x

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is not too far off from 1
over 1 plus x, it turns out.

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So it's not a perfect match.

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You can try plotting
these two functions.

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But it's a reasonable
approximation,

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given that everything we're
doing in this calculation,

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when applied to
a real situation,

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is going to be off by
some uncertainty, which

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could be a factor of 2
or 3, this is actually

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going to be more
than good enough

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of an approximation for us.

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So if I make that approximation,
then what I have here

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is that this thing is
1 over 1 plus x here.

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And so we end up with 1 minus
1 over 1 plus lambda C tau.

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And when I combine
those two terms,

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I end up with lambda C tau
over 1 plus lambda C tau.

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So this is my approximation.

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In fact, I can
further then write

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that as 1 over 1 plus
lambda C tau inverse,

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dividing the numerator and
denominator by lambda C tau.

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So I'm just making some
approximations here

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that allow me to get a
very simple expression

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in the end for my
safety guideline,

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taking into account this
transient build up here.

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So remember that
the bound we have

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is on the indoor
reproductive number, which

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is N minus 1 times the
integral to tau of beta dT.

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So what is that?

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That's just the sort of
time average beta times tau.

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So this bound is
actually N minus 1,

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time average beta times tau.

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And then our
guideline, of course,

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is to make this less than
our tolerance, epsilon.

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And so what that means
then is using this result,

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you can see that I
just get the rest--

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so if I look at the
expression for beta bracket,

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it's just the steady state
expression times this factor.

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So basically, this is
kind of the factor that

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corrects for transient effects.

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Again, with just a
simple approximation.

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So I can then write
that my guideline now

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has a modified form, which
is that N minus 1 times tau

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is less than epsilon
over the time average

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beta up to time tau.

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And this is approximately
equal to epsilon over

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beta steady state
times this factor here.

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If I multiply that to the
other side, I just get 1 plus 1

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over lambda C of R bar tau.

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So basically, this right here
is the transient correction,

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or modification.

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And this is the
steady state formula,

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which we will more
typically be using.

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Now, why do we care
about the transient?

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Well, first of all, you can see
that by using the transient,

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we are being less conservative.

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So if we want to be
very conservative,

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we can say, you know
what, let's just

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assume the second that
the infected person enters

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the room, boom,
the transition rate

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goes right to the maximum value.

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That's the most conservative.

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So generally, using the steady
state is more conservative.

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So that's one reason
we like to use it.

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Also, it gives you
a simpler formula.

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Why add a bunch of
factors to a formula that

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only make it less conservative.

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And we've made a lot of
assumptions in this model,

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so it makes sense, maybe
let's not worry about it.

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However, I do actually
like to include it

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for certain examples,
because it allows

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you to capture the
intuition we all have

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that when the time goes
to 0, the risk also

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has to go to 0, which you
don't get from a steady state.

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If this bumps up
right away, then you

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could be spending like
2 seconds in a room,

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and you have a chance of getting
infected right away, which

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is actually not right.

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There has to be some time
physically for transmission

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to happen through these droplets
from one person to another.

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So the effect of the transient
correction, as you can see here

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is, when lambda C tau
is larger than 1--

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so that's times that are kind
of out here-- that term is gone.

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But when you get to these
earlier times, or very short

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times, where there
hasn't been time yet

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for the build up of the airborne
concentration, then as you see,

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as tau goes is 0 actually,
this term diverges.

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So what actually happens
is that if I calculate sort

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of, for example, one thing you
can get from this guideline

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is what is the sort of maximum
occupancy, or versus time.

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Or it's sort of the maximum
time in the room for a given

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occupancy.

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We're going to be looking
at lots of plots like this.

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I would have something
that might look like this

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for steady transmission.

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And when this time
gets larger then lambda

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C of R bar inverse--

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there's some critical
time scale there,

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which is this one right here--

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then when you're
past that time scale,

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you've got the steady state.

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But if you go to smaller times,
then what happens at this thing

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can sort of blow
up a lot faster.

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So what it kind of
helps to capture

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is, again, this intuition
that if I put in like tau

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is extremely small, then of
course, the risk goes away,

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and I can have larger numbers
of people in the room,

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or I can tolerate smaller
times and actually be safe.

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So anyway, that's one
reason we do that.

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00:12:15,250 --> 00:12:17,250
On the other hand, for a
conservative guideline,

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and the most important
message of this course,

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really, is to think about
that steady state transmission

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rate, which we will
mainly be focusing

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on in all of our examples.