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PROFESSOR: So we've just
discussed the strategies

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for reopening schools
and businesses based

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on the indoor safety
guideline for the given

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space and the various physical
parameters in that space.

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And the new concept we added
was a prevalence of infection.

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So we would know, on average,
how many infected people

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we might expect in a room.

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And as that number goes to
zero, as the pandemic subsides,

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we can switch from a
more restricted situation

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with an occupancy
N1, prescribed

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by the original guideline based
on the indoor reproductive

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number, to kind of
increasing occupancy up

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to N0, which is the
normal original occupancy,

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initially with masks.

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And as the prevalence
goes down, we

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switch to taking the
masks off and really

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returning back to normal.

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So I'd like to take this a
little further now and ask,

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how would we change our
policies or this discussion

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here if we not only
have information

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about the prevalence
of infection

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but we also have an
understanding of immunity,

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which could be acquired
through vaccination

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or through previous exposure?

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And especially right
now as I'm recording

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this lecture in early
2021 and several vaccines

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have rolled out
for COVID-19, this

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is a topic of great interest.

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So let's think
about, how would we

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take into account
susceptibility?

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So, in some sense, this
was a conservative estimate

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of the true risk of
transmission because we've

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assumed that everyone who's
uninfected is susceptible.

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But, of course, as immunity is
increased in the population,

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we're going to have
to modify that.

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So instead of having
our populations

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of susceptible and
infected persons

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being sampled from a two-state
or two-category process,

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we can think of
three categories.

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There can be--

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PI is the probability
that a person is infected,

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which means really, again,
that they're infectious

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and they can affect
other people.

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And this is coming from
the local population that

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is entering that indoor space.

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And now we're going to add
PS, which is the probability

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that a person is susceptible.

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And then the third category is
PM, which is the probability

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that the person is immune.

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And so we have a
three-category process,

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so those three
should add up to 1.

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So this is 1 minus PI plus PS.

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And we can also
further write this

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as PVAC, the probability that
a person has been successfully

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vaccinated and actually
has acquired immunity,

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plus the probability
of previous exposure.

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We'll call that PX.

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So this would be
vaccination, and this

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would be previous exposure
if that previous exposure has

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actually led to immunity.

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And that's a controversial
topic, still, under research

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and may depend on the
specific population at hand.

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But let's imagine
that we have subsets

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of what these
numbers are and then

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we'd like to see how to
adjust our thinking here.

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So we're still going
to base our guideline

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on saying that the expected
number of transmissions

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is the expected number of
infected time susceptible,

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so the expected number of pairs,
times the average transmission

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rate, average of beta
times tau, the time,

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and that that expected
number of transmissions

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should be less than
our tolerance epsilon.

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So that's still our guideline.

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So what we're really
trying to consider,

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now, are different assumptions
about this expected number

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of infected susceptible
pairs that are in the room.

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And we've broken that down
into three risk scenarios.

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And let's revisit that, now,
with our three-category model.

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So the first risk scenario
was describing a desire

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to limit spreading
of the disease

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through this indoor space.

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This is our original goal.

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And by that, we mean, if
an infected person enters

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the room, then we would like
to make sure that it's unlikely

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that a new case would
emerge from transmission

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from that person.

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So in that case, we
have I is equal to 1.

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And then, now, I is known.

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So this is just the
expected value of S.

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And the expected value of S,
though, in this new model,

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is the number of other people in
the room, n minus 1, times PS.

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So you see, now, when I define
my indoor reproductive number

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as N minus 1 times
beta tau and I

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want to bound that to
be less than epsilon--

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that's my typical guideline--
there's this extra factor, PS,

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which could be moved
to the other side.

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So one way to think about it
is, since PS is less than 1,

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we are increasing that
tolerance because there

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are fewer susceptible people.

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So we're allowed to
stay in the room longer,

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have a higher occupancy,
lower ventilation, et cetera.

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So this is one case.

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The next case is to
limit transmission.

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So here we're not
going to assume

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that an infected person
actually is there,

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but we are going to consider
the possibility that there

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is an infected person there.

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So that makes transmission
potentially a lot less likely.

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And so what we'd
like to do here is

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to look at the expected
value of I times S.

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And I won't go
through the details.

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But for the trinomial
distribution

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with three independent
possibilities,

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with these
probabilities-- and you're

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making N samples from
that distribution--

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you can show that the
expected value of this product

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is actually N minus
1 times PI times PS,

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by very similar
arguments as we have

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done for the binomial case.

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One way to think about
this is that N minus 1

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is the number of permutations
of two people that

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can be made in that room.

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So if you pick one person
to be first the infected

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and the other one to
be the susceptible,

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this is the number
of such pairs.

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And PI PS is the probability
of each of those instances.

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So this is the expected
number of I to S pairs.

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And I put a directionality here
because we are distinguishing

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each individual person.

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So if I take two people,
I am counting differently.

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If one is infected, the
other one's susceptible

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or the reverse situation since
everyone's a unique individual.

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OK, so if we then substitute
into this formula,

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then, notice, now, we've
picked up some extra factors.

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So now the guideline
would read that RN

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will be less than epsilon over
N times PI, which is something

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we already had before.

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But now there's also a PS.

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So that's modified.

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And then, finally, our
third risk scenario

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was to limit personal risk.

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So this is the case
where S is equal to 1.

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I'm only worried about
one susceptible person,

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and that's me.

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And I'm then-- if
S is known, then

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we just have the
expected value of I,

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which is just N minus 1, all
the other people, times PI.

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And if you plug that
into the formula,

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then you find that
RN is now bounded

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by epsilon divided by PI.

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So take into account both
prevalence of infection

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and susceptibility, at least to
these somewhat modified bounds.

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And let's focus on where
the changes took place.

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So first of all, in
the original guideline

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for limiting spreading, we can
be a little bit more lenient.

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So as there's more
vaccination and more immunity,

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we don't need to keep
holding that guideline

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at the same level,
even in the most

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sort of conservative stance
of trying to limit spreading.

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What's more interesting for this
plot here is the middle one.

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So now, we again think
up a factor of PS.

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But everything else is the same.

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So what it means is that,
relative to the calculation

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that I showed here,
I should actually

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make the very same
plot where I don't just

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plot PI in this
axis, but actually

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plot PI times PS, where PS
is the probability of being

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susceptible, which is
related to vaccination

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and previous exposure rates.

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So that actually does bring this
down and, hence, make it easier

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to make the decision to
relax restrictions and even

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ultimately take off the mask,
because as a combination

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of these two factors, we're
getting even more safe.

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Interestingly, down
here for personal risk,

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we don't really care about the
probability of susceptibles

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because the only person I
care about in that situation

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is myself.

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And so I don't have any
effect of susceptibility,

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only the effect of infection.