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PROFESSOR: So now that we've
discussed prevalence and risk

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scenarios, we come to
a very important topic,

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which is, when should
you impose the guideline?

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Obviously, when the epidemic
is raging at a very high rate,

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it feels out of
control, it's spreading,

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we want to impose the
guideline that we've discussed,

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which limits the indoor
reproductive number

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for each space within
some tolerance.

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And that would, for example,
give us some number N1

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from the safety
guideline, R N less than

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epsilon, which is
going to be smaller

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than the normal occupancy
of that room for a given

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time and all the other factors
that we've been discussing.

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But what happens as the
prevalence of infection

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goes down.

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Generically, there must
be some curve like this.

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And we'd like to understand
what is the simplest

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reasonable model we can come
up with that can tell us

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how to relax the approximation?

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So for example, let's think
of a school or a business,

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where we typically have a
lot of the same people there

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every day, but some others
are coming and going,

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or maybe going home and
getting infected and coming in.

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So the rate of infection is low.

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So we expect typically the
number of infected people

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is zero or occasionally one.

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And as the prevalence
goes down, then we

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start to see that the situation
is getting safer and safer.

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So that at a certain point
P1, where we start to say,

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you know what we can
actually increase

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the occupancy of the space
with everything else held

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fixed while still wearing masks.

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And then we hit the
normal occupancy.

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And you might call that
the new normal, where

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we're going about our
business, the room

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is filled with the
typical number of people.

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Let's say, the classroom
is back to its normal size.

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We don't have any remote
teaching going on.

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But we're wearing masks
or taking other factors

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into account, such as higher
ventilation rates, let's say,

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open windows.

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But then we continue
lowering the prevalence.

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There's a certain
point where we get rid

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of those other precautions.

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So extra open your windows.

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Or more importantly,
the dominant effect

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is the removal of the
mask because we know

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that's a significant factor.

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And then you might call that
back to the real normal,

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actually, not the new normal.

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Where we are back to full
occupancy and really not

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taking any precautions.

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That's going to happen at
a rather low prevalence.

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But we hope that that
time will eventually come

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and hopefully not so
far in the future.

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And I mention here a
very important point

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we've not talked about
yet in this class

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but we should keep in
the back of our minds.

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When I talk about
occupancy I'm really

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talking about the number
of susceptible people.

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We just saw that in the
last couple of boards.

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But the number of
susceptible people

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are really only those that
are not immune to the disease,

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for example, by vaccination.

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So as more and more
people become vaccinated,

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then this occupancy number
might-- for example,

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let's say, as a
typical occupancy

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will be at 25 people in a class.

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As more and more
people are vaccinated,

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the number that we
plug in this formula

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here might actually be lower
when we just make decisions

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because there are fewer
and fewer susceptible

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people that are left.

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So that's another very important
factor to keep in mind.

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Of course, also, vaccination
has the indirect effect

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of lowering the prevalence
that is seen in the population

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as we start to stamp
out the epidemics.

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So we also have that effect.

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So I won't talk about
that any further now

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but come back to this
calculation based on the risk

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scenario, the second one that
I talked about last time, where

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we take into account prevalence

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So let's look at the three
different cases here.

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So the first one is where we
have restricted occupancy.

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This is where we've
decided there's

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an N1, which is epsilon
over average beta tau.

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So all the physical parameters
are buried in there.

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And this is going
to be something

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less than the normal
occupancy, N0.

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Now what is the tau we
want to think about?

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Well, if we have a
school or a business,

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this would be the
cumulative time

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that people spend
together to the point

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where the number of days they
spend together is, let's say,

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on the order of a week
would be a reasonable number

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to think about.

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Because if we write tau is
the typical hours per day.

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Time is some kind of
maximum number of days.

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This maximum number
of days could

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be set by, for example,
the testing frequency.

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For example, here at MIT, we are
testing our entire population

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at least once a week
in order for anyone,

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including myself, to be
admitted to the campus.

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And so we are definitely
testing within a week

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and catching new
infections at that rate.

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It could also be motivated
by the incubation time, which

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is the time to show symptoms.

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And most people will
remove themselves.

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And we know that's around 5.5
days, a typically reported

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value.

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So again, on the
order of a week.

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And there's also, of
course, other ways

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that people are removed
or they recover.

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So there's removal
and recovery, which

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is another way that if
you start to go more than,

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let's say, two weeks,
we start to think

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an infected person
that didn't get

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removed then end up in
the hospital has probably

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recovered.

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So if we think of a certain
number of days and hours

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per day, that
gives a tau that is

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going to go into this formula.

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And actually, I should also
mention that for simplicity

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here, technically this
should be N1 minus 1.

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And I can either include that or
not when I do this calculation.

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But I'm generally
thinking of N1, which

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is going to be bigger than 1.

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So think of an
occupancy of 10 people

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in a classroom, or
something, might be a limit

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that we would be
interested in considering.

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But certainly we can put the
1 in there if we want to.

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So now let's ask
ourselves, how would we

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start to reopen the
space once we've

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decided on a safe occupancy
during the greatest

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level of restrictions?

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So that would then
lead us into a phase

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of relaxing restrictions.

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And this would
still be with masks.

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So keeping in mind that
masks are an essential part

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of achieving a
reasonable occupancy when

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the pandemic is high and
there's a lot of prevalence.

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And that we would only start
to relax occupancy first

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before we take away the
suggestion to wear masks.

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And that would be
then the last step.

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So for relaxing
restrictions, we're

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then going to be interested in
the indoor reproductive number

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being less than now a
rescaled value, which

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would be epsilon over P i Q i
N. And the indoor reproductive

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number, remember, is N minus 1.

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But it's approximately
N times beta tau.

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So I've replaced
again N minus 1 with N

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just to get a simpler formula.

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And so notice now here, N is
in both sides of the equation.

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If I want to solve for
the value N2, which

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is this yellow curve
here, I'm actually

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going to have to put
the N's on one side

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and take a square root.

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So my N2 then would
be the square root

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of epsilon over beta
tau times P i times Q i.

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And remember also,
another approximation

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here is that P i is
definitely much less than 1.

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We're looking to limit
a very low prevalence.

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And so also therefore, Q i
is basically tending to 1

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because it's 1 minus P i.

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And so that factor is
really not that important.

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And notice also, epsilon
over beta tau, that's N1.

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So N3 is approximately related
to N1 by the square root of N1

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divided by P i.

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That's this number here.

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So the function prevalence,
this yellow curve,

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is 1 over square
root of prevalence.

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And one nice thing about
writing it this way

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is that I can decide
on a reopening protocol

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without actually
redoing my calculation

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with all those complicated
variables, including the risk

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tolerance epsilon, and
all the factors that

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go into beta because
I've lumped them into N1.

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What I'm saying here
is that we've already

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done a calculation and decided
to impose a certain occupancy

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restriction on certain
space based on principles

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that we've been
discussing in this course.

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But now as prevalence
goes down, according

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to the simple formula,
whenever N2 is bigger than 1--

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we would use this if
N2 is bigger than N1.

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So basically, when
these two curves cross,

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as you get a lower prevalence,
you now switch to two

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and you'll start increasing.

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And you will do that
until you get to N0.

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This is the relaxing
restrictions.

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So basically, when N2 is bigger,
than we allow this until N2

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equals N1.

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And that's this
time here, P, P2.

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So basically, we start
imposing restrictions at P1.

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So basically, we start
relaxing restrictions

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when the prevalence equals P1.

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That would be when
N2 is equal to N1.

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And so that would be
when P1 is 1 over N1.

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So essentially,
that's when you expect

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to find one infected person.

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So this is
approximately 1 over N1.

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Up here, when you go
below this, you're saying,

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well, it's actually unlikely
that during the time tau

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we'll even get one
infected person.

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And that's when
we start to relax.

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So that's the first
crossover point.

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And there's a second
crossover point

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when this is equal to P2--

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sorry, is equal to N0.

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00:10:04,480 --> 00:10:06,070
That's [INAUDIBLE] P2 is.

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00:10:06,070 --> 00:10:09,130
And so basically,
P2, which is when

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we would hit the
saturation point and that's

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when we've reopened
in some sense

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to the full normal
situation, that would

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be when N2 is equal to N0.

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Wait, sorry.

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00:10:22,430 --> 00:10:24,390
I wrote here N2 equals N1.

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00:10:24,390 --> 00:10:26,210
Sorry, I meant
when N2 equals N0.

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00:10:26,210 --> 00:10:27,590
Sorry.

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00:10:27,590 --> 00:10:30,140
That's the time P2 here,
when N2 is equal to N0,

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that's when we cut off.

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So this is N0.

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And we solve for P i.

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You can see that we
get N1 over N0 squared.

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So that is the place
where I then switch.

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And now I'm going to
cap the occupancy at N0.

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So maybe to summarize
here, what I would say

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is that the occupancy
should be less

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than or equal to N1 for
P i greater than P1.

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It'll be N2, which depends
on P i for P i between P1

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or P2 and P1.

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And then as the
prevalence gets lower,

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we go to full occupancy, N0,
when P i is less than P2.

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So this is basically this
full curve of reopening.

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And then the final
decision to make

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is, when do we return
completely to normal

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and take away certain
restrictions we've done?

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So here I mentioned masks.

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We could also include in
this calculation relaxing

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other restrictions,
such as maybe not having

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00:11:47,000 --> 00:11:49,160
the ventilation
on quite so high.

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00:11:49,160 --> 00:11:51,740
So that would be when
we finally go back

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00:11:51,740 --> 00:11:54,350
to no restrictions of any kind.

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00:11:54,350 --> 00:11:57,030
We're back to normal.

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00:11:57,030 --> 00:12:03,800
So this means no masks, no other
precautions, full occupancy.

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00:12:03,800 --> 00:12:10,880
So in this case, R N is going
to be less than epsilon over--

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00:12:14,030 --> 00:12:14,910
well, let's see here.

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It's the same as before.

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We have this P i, N
that we just looked at.

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Or technically. times Q i.

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00:12:22,920 --> 00:12:26,730
But now we have another
factor, P M squared.

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Because compared to
the case with no masks,

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we know the bound
in the guidelines.

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00:12:30,220 --> 00:12:34,750
So the effect of beta just
gets rescaled by P M squared.

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00:12:34,750 --> 00:12:38,490
So technically
that's in R N here.

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00:12:38,490 --> 00:12:40,470
But now we have
this extra factor.

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00:12:40,470 --> 00:12:43,770
You think of it like a
rescaling of epsilon.

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00:12:43,770 --> 00:12:46,710
And so what that's
going to do for us then

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00:12:46,710 --> 00:12:57,980
is that there's another curve
that goes like this, which

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00:12:57,980 --> 00:13:02,000
is just like this one but it's
shifted by a factor P M cubed--

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or P M squared, sorry.

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00:13:03,110 --> 00:13:09,790
Which is like the case where you
had-- so this is like no masks.

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It's like the N2 with no masks.

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00:13:11,710 --> 00:13:17,600
And the other curve
is with masks.

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00:13:17,600 --> 00:13:19,160
And there's a
rescaling factor, which

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really has to do with the
remediation that you've done.

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And in this case, P3 would then
just be P M squared times P2.

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So if our P M is
a factor of 10%,

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let's say, masks are letting
10% of infectious droplets

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get through, the P M squared
might be a factor of 100.

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00:13:39,740 --> 00:13:41,510
So then we would wait
to the prevalence

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is 100 times smaller before we
finally allow people to remove

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masks and be at full occupancy.

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00:13:47,510 --> 00:13:49,370
And you could make a
similar calculation

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00:13:49,370 --> 00:13:51,460
for other types of restrictions.

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And in fact, you can
calculate such a curve

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00:13:54,590 --> 00:13:59,000
for a given room, given
scenario of human behavior

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and interventions, such as
filtration or ventilation.

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00:14:03,590 --> 00:14:06,290
And what the theory allows
you to do is to, of course,

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recalculate N1.

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00:14:07,790 --> 00:14:10,950
And then you can
recalculate N2 as well.

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00:14:10,950 --> 00:14:13,550
And so you can say, well,
I don't like this curve.

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I would like to try to
reopen my school sooner.

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00:14:16,610 --> 00:14:18,210
How would I do that?

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00:14:18,210 --> 00:14:21,650
Well, I know that if I
make various interventions,

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00:14:21,650 --> 00:14:23,060
I can raise the pink curve.

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00:14:23,060 --> 00:14:26,120
So I could end up somewhere
here, let's just say.

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00:14:26,120 --> 00:14:32,190
This might be with
safety interventions.

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00:14:35,660 --> 00:14:37,860
Actually, one such
intervention, by the way,

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00:14:37,860 --> 00:14:38,870
is masks themselves.

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00:14:38,870 --> 00:14:40,670
Because if I follow this
curve all the way down here,

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00:14:40,670 --> 00:14:42,120
there's some curve
down here, which

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00:14:42,120 --> 00:14:45,780
is no masks, which is the
safety guideline with no masks.

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00:14:45,780 --> 00:14:48,720
And if I turn on masks, I go up.

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00:14:48,720 --> 00:14:52,020
But when I make that
intervention, also P2, notice,

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00:14:52,020 --> 00:14:55,200
scales also like N1.

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00:14:55,200 --> 00:14:57,550
And so that's actually
moving in this direction.

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00:14:57,550 --> 00:15:00,160
And so I essentially
move this yellow curve.

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00:15:00,160 --> 00:15:02,880
And so I'm now
going to say, well,

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00:15:02,880 --> 00:15:07,020
with a different
set of interventions

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00:15:07,020 --> 00:15:08,920
I can make the room safer.

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00:15:08,920 --> 00:15:11,920
And what that does, it gives
me more people in the room.

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00:15:11,920 --> 00:15:15,220
But it also means
that I change when

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00:15:15,220 --> 00:15:16,470
I make the decision to reopen.

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00:15:16,470 --> 00:15:18,210
And in particular,
I can get myself

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00:15:18,210 --> 00:15:21,030
to full occupancy at
a higher prevalence

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00:15:21,030 --> 00:15:24,200
because the room is
actually now made safer.

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00:15:24,200 --> 00:15:26,760
So I note it.

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00:15:26,760 --> 00:15:29,630
But on the other hand, this
switch here was at 1 over N1.

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00:15:29,630 --> 00:15:31,910
So this part
shrinks a little bit

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00:15:31,910 --> 00:15:35,520
as this ultimately goes
up to full occupancy.

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00:15:35,520 --> 00:15:40,520
So basically, I think compared
to the current situation

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00:15:40,520 --> 00:15:43,520
or the typical situation
where policymakers are making

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00:15:43,520 --> 00:15:46,640
decisions based on something
like the six-foot rule

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00:15:46,640 --> 00:15:49,910
and a somewhat arbitrary
feeling about what is a high

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00:15:49,910 --> 00:15:52,760
prevalence-- is it 1%, is 0.1%--

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00:15:52,760 --> 00:15:55,460
and we decide, OK, now
we can close our schools

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00:15:55,460 --> 00:15:58,610
or reopen our schools or
set the occupancy at half,

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00:15:58,610 --> 00:16:01,220
the guideline tells you
how to set occupancy

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00:16:01,220 --> 00:16:06,530
for of the worst case
scenario, when the pandemic is

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00:16:06,530 --> 00:16:09,200
very prevalent in society.

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00:16:09,200 --> 00:16:11,420
But now also, through these
kinds of calculations,

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00:16:11,420 --> 00:16:14,240
we can make rational
decisions about how to reopen.

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00:16:14,240 --> 00:16:17,050
I'm not advocating necessarily
for the exact formulas

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00:16:17,050 --> 00:16:18,310
we find on the board here.

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00:16:18,310 --> 00:16:19,730
But the principles
I'm showing you

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00:16:19,730 --> 00:16:23,630
could lead to quantitative
and scientifically justifiable

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00:16:23,630 --> 00:16:27,650
ways of taking a specific
space and a specific usage

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00:16:27,650 --> 00:16:30,680
of that space and deciding how
to close, as prevalence goes

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00:16:30,680 --> 00:16:33,320
up, and reopen, as
prevalence goes down,

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00:16:33,320 --> 00:16:35,510
including ultimately
returning to normal

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00:16:35,510 --> 00:16:39,110
and removing masks and all
other forms of precaution

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00:16:39,110 --> 00:16:42,400
as the epidemic disappears.