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PROFESSOR: So now
let's use the results

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of our probabilistic
model of transmission

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to see how we can
modify our safety

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guideline to take into account
different risk scenarios.

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So our general result is
that the expected number

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of transmissions in the
room in a given time

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is given by the expected
number of infected times

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susceptible times
the mean transmission

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rate, which is the average
beta times the time.

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And we want this less to
be than some tolerance.

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And also, given the
various approximations

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we've made where,
for example, we

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have not let the number
of susceptibles change--

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people can be infected
more than once,

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and we've neglected
various aspects

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of the model in that way--
this should typically

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be assumed less than 1.

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So we're always kind of
thinking of transmission

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of being a rare event.

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If we have lots
of infected people

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in the room, lots of
transmission going on,

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that's a more
complicated situation

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which requires more
sophisticated models.

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But for purpose of
guidelines, this is the limit

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that we want to think about.

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But the different
scenarios correspond

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to our different
assumptions about I and S.

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So first, I'll just
remind you of what

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we've been doing until now,
which is to limit spreading

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of the epidemic as a whole.

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Thinking what if every
indoor space were

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to impose the guideline, we're
really thinking of the case

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where I is 1, and
S is N minus 1,

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and then this expected
number of transmissions

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is just the indoor reproductive
number, which is, of course,

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just N minus 1 times beta
tau is less than epsilon.

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So that's the guideline
that we've already

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been talking about because the I
and the S here are now actually

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no longer random.

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We're just saying let's just
consider that situation.

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And if everybody
does that, then we

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are limiting the spread
of the disease overall

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and should be
hopefully fighting it.

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What we'd like to
talk about here

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is how to start with this kind
of a restriction which gives us

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certain bounds on occupancy,
ventilation, and other factors

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and think about,
well, how would we

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actually remove that restriction
as the prevalence of infection

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goes down?

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So that's a bit of a
different question, which

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is to limit transmission--

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or maybe another way of
saying that more precisely

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is new cases that are going
to arise in this indoor space.

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So now we're not just saying
if an infected person enters,

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we don't want any new cases.

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What if we just don't want
new cases at all, including

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taking into account the low
probability that somebody

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actually does enter this
room who is infected?

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So now I'll just remind
you of the results

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from the last board
for that situation

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with all the assumptions
of the previous model.

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So the expected number
of I is now pIN.

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The expected number
of susceptibles N

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minus the expected
number, which is qIN.

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And importantly then, the
expected value of I times S

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is pIqI times N times N
minus 1, which was our result

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from the end of the board.

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And so now when we write that
we like the expected number

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to be much less than
epsilon, expected

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total indoor transmissions,
now notice instead of just

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an N minus 1 like
we had before, we

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have these additional
factors pQN.

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And so we effectively
divide by that.

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So in this case, we can
write our safety guideline

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taking into account
the prevalence

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as epsilon divided by pIqIN.

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And so this allows
us as pI goes to 0

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and qI goes to 1,
so in other words,

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as the infection
becomes less prevalent,

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then we can start
modifying our guideline

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to increase this bound and,
for example, a lot more

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people to enter the room or to
increase their time in the room

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or to maybe turn down the
ventilation a little bit.

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And we can make
changes like that.

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It's typically considered
to be a high prevalence

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infection when we're
getting, let's say,

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in the range of
maybe 100 to 1,000

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infected per 100,000
people in the population.

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So that would be
0.1 to 1.0 percent.

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This is really considered
usually quite high prevalence,

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actually.

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So there's quite a few
infected people around.

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But in that case, if--

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let's just say we
had a situation

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with 10 people in the room
just to give an example then.

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What would this tell us in
terms of increasing our time

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in the room or our occupancy?

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Well, occupancy is here fixed.

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But let's say time in
the room or ventilation

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or other factors.

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We can basically
increase our N minus 1

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tau, our cumulative
exposure time,

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which is basically this
indoor reported number

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that we're bounding.

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This bound will increase or
increases by 10 to 100 times

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because it's basically-- yeah.

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This is an extra factor there.

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So that means that if
the thing was telling us

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that we could be in the room
for five hours, maybe now

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it'll be 50 hours or even
500 hours actually depending

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on how low the
prevalence actually gets.

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And of course, as the
prevalence goes down further,

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and the epidemic disappears,
we start to completely relax

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our assumptions.

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And we'll talk
about that shortly.

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There's a third
risk scenario that

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is also of interest, which
is to limit my personal risk

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for a given individual.

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So in this case,
we have a situation

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where I only care
about myself, one

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particular person in the room.

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So the number of susceptibles
is now fixed at 1.

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And the number infected
people is potentially

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anyone else in the room.

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So I'm worried about
attending event or being

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in an office or some
situation where there's

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a certain number of people.

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And the number of infected
that I would expect would be--

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expected number infected
would be-- well,

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it'd be exactly equal
to pI times N minus 1.

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So, basically, any other person
than myself could be infected.

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And so that's the
expected number.

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And of course, if S is fixed
at 1, then this expected value

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I is also the
expected value of IS.

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So my transmission rate
now has this factor.

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And notice in this case, we get
the same N minus 1 as before.

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But there's this
new factor PI here.

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So that then tells me I
could express the bound

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as the indoor reported
number is less

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than epsilon divided by pI.

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So, basically, we
have these factors

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that come in when we talk
about prevalence that take

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our previous bound
that brings in all

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of the physical
quantities related

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to the room, its ventilation,
filtration, viral deactivation,

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time in the room, occupancy.

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And we take those bounds.

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And we can essentially
rescale them with these values

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depending on how we are
using the guideline.

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Now, this is a very
simple model but at least

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gives us a sense of how
to make those decisions.

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So, for example,
let's consider a case

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like we did here where
if the prevalence is

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in the range of 0.1% to
1.0%, which is actually

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a fairly high
prevalence, then we

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could increase the
bound on N minus 1 tau,

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our cumulative exposure time, by
100 to 1,000 times as a factor.

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So if the guideline
is telling us

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that you have five
hours in this room,

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it might actually be more
like 500 or even 5,000

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for one particular susceptible
person given the prevalence

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in the population.

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So, basically, just
want you to keep in mind

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that when applying
the guideline,

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the basic ideas don't change.

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We start with a bound on
this reproductive number that

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brings in all of the physical
quantities and disease

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quantities that we've
been talking about.

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But we also may modify
that bound a bit depending

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on our risk scenario.