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PROFESSOR: So now
let's synthesize

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all of our models of aerosol
generation and dynamics

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in a well-mixed indoor space
and epidemiological models

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of spreading and transfer
of disease in that space

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and formulate a safety
guideline for COVID-19

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assuming indoor
airborne transmission

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in a well-mixed room.

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And so there's a number of
ways we could go about this.

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And the approach that
I would like to propose

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is that we require that the
indoor reproductive number is

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less than some tolerance.

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So this epsilon here
is the tolerance.

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So typically something
might be a lot less

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than 1 or maybe you get to
allow it to be closer to 1.

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And so that is essentially
the probability

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of a transmission
from one infector that

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enters the room at t equals
0 and stays for a time

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tau in the presence of a
total number of n people

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in the room, which is the
occupancy of the room.

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So obviously, this
doesn't handle

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every possible situation.

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It is possible, if there is
a very prevalent infection

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rate in the community
that the room may

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find several infectors entering
at once, in which case,

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we could certainly
increase this number.

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And then the 1 here
would be replaced by i0

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as we've seen before.

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On the other hand, if you're
interested in controlling

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the spread of the
disease and you're not

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worried about a specific person,
but you just want to say,

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I would like to make
this room not contribute

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to spreading of the disease--

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and imagine if
everybody did that,

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then no rooms would contribute
to spreading the disease--

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so then this criteria would
say that if two infectors came,

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they might infect two
people, or three infectors,

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you're worried
about three people.

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Well we want to make sure is
that the number of infectors

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is not growing.

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So in other words, if
there's one infector,

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they don't infect another one.

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So the fact that there
are more infectors

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gives a chance of at
least some infection,

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but this seems like a more
general and simpler criterion.

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It's also a little bit like
when the fire department

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looks at a room and
decides on occupancy

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limit for an indoor space.

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There's no fire going on.

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They're not estimating what's
the probability of a fire.

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It's a conditional probability.

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They're saying,
if there's a fire,

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we want to make sure
these people can get out.

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And so they think about
where the fire might start,

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where the smoke is,
where are exits,

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and how many people
can realistically

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get out of that room before
the fire and the smoke

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reach a dangerous level.

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So this indoor safety
guideline has a similar flavor,

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where we're essentially defining
here the number of people

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and also the amount of occupancy
time that would be allowable.

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So that it'd be unlikely
for one infector

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to create a new infection
and hence spread the disease.

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So that is the basic thinking.

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So this it's basically the
conditional probability

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of spreading given this, that
an infector enters at times 0

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and stays for a time t.

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So that's basically how we
formulate the guideline.

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And now let's think about--

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already at this
level without getting

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into the details of what goes
into beta, which we've already

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talked about, there's a
very important concept here,

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which is that if I define
beta with brackets here to be

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the average transmission rate.

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So the transmission rate
may be changing in time,

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for example, as droplets
are building up in the room

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as we've discussed.

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But let's just think of kind of
an average transmission rate.

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Then this integral in time
is just beta brackets,

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beta average times tau.

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So if I put the beta
bracket on the other side,

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I arrive at a very
fundamental result,

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which is that the number of
susceptibles in the room,

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which is roughly the occupancy--

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although, if you're
very low occupancy,

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it's everybody but the infected
person, so it's n minus 1--

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times the typical time spent in
the room by the infected person

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is less than the
tolerance divided

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by the average
transmission rate.

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This is very simple
and very general

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relationship, which does not
depend on the details of beta.

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But what it's telling you is
something which already you

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may recognize is very
different than any

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of the existing official
safety guidelines.

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Official safety guidelines
always limit one quantity.

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For example, they might limit
the number of people in a room.

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They also can get a limit such
as that from social distancing.

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So you could take the
area of the room, divide

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by a little six foot
radius around each person--

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or a three foot
radius or a six foot

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separation-- and
get at an occupancy.

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So you fix n.

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The problem with that is
that time has to play a role.

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If a certain number
of people are

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in a room for a very short
time, it's really unlikely

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to have any transmission.

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But if the same number
stays for a very long time,

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eventually some transmission
must take place.

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So time has to be important.

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You will also find
limits on time.

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The CDC defines a contact
where transmission

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is possible as being within six
feet of an infectious person

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for more than 15 minutes.

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So there is a time
constraint there.

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So it says that
after 15 minutes,

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you should expect to be
potentially infected.

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But the problem
is that that's not

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accounting for transmission rate
factors, such as ventilation

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and all the things
that we've discussed,

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and also the occupancy.

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So how many people could
you transmit it to?

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If you're by yourself
in a room or there's

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only two or three people
in a very large space,

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are you really going to
transmit in 15 minutes?

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Not necessarily.

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And anyway, so
that is the problem

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with those kinds of criteria.

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And it can be easily seen by
plotting this relationship

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here.

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So by the way,
this quantity here,

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which is the product of n
and t, or n minus one and t,

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is what I like to call the
cumulative you exposure time.

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So the time that you're exposed
to the infected individual,

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tau, is multiplied by how many
other people are in the room

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because that's how many
people could get infected.

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So the more people that there
are, the more chance that one

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of them could be infected
in a well-mixed room.

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And so you really don't have
a bound just on time.

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It's really this product
that's important.

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And so let's plot what
this looks like now.

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So if we have here the time
and here the occupancy,

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then this guideline is a
bound that looks like this.

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It saturates at one.

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So basically, if you're
below this guideline here,

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you're considered safe.

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And up here is
potentially unsafe.

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Meaning that given the
tolerance that you've chosen,

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you would expect
that there could

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be a transmission with
greater than that probability.

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OK.

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It saturates at one because
it will never go below one.

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Because only one
person, you're not

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going to transmit to anybody.

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So it's OK.

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So an occupancy of one is
OK, except in a building

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where perhaps other
people have come and gone

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from that room or ventilation
is bringing particles

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from other rooms.

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And we'll come back to that.

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But just at the simple
level of analyzing one room,

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obviously one is fine.

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And here you see a very
fundamental problem

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that I was just
alluding to, which

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is that if I put the standard
guidelines on-- for example,

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a limit on number of people.

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So this could be
fixed occupancy.

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And in fact, in Massachusetts,
for example, right now

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there is a guideline which
says n is less than 25.

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No more than 25 persons
can congregate in a room.

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In fact, I'm teaching
class right now which

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we have 51 people, and I was
able to split into two rooms.

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And then this rule
came along, and then we

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had to start doing three,
remotely broadcasting

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between those rooms because
we have a 25 person rule.

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The problem with that rule is it
doesn't take into account time.

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What if my class is
only five minutes long?

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Or let's say it's
one-hour long, OK.

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In that time, if I don't have
an expectation transmission,

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I should be OK.

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On the other hand,
if those 20 people

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sit in the classroom
for several weeks,

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it's pretty likely that if an
infected person is among them,

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there will be a transmission.

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So time has to come in.

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And that you see
very clearly here

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from the crossing of the fixed
occupancy with the safety

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guideline.

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So for a short amount of
time, the fixed occupancy,

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which is telling you
should be under this--

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so basically you should
only have a lower occupancy.

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So this is a fixed
occupancy bound.

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For example, the one
in Massachusetts.

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At first, this is
too conservative.

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So you are telling people
they cannot be in the space,

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but until this amount
of time has passed,

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it's very unlikely that
anybody would transmit.

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Imagine this time as one minute.

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It's pretty unlikely that you're
going to have a transmission.

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On the other hand,
if you keep waiting,

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you always cross
the yellow line.

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And over here it's too risky
and too dangerous, basically.

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Because you're allowing
people to think that

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they're safe because
there's only 25.

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But let's say this crossover
happens after one hour,

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and the people are in
the room for five hours,

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there's a very high
risk of transmission.

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So just putting out
a number like 25

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doesn't really protect you
because these lines always

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cross.

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You will always cross the safety
guideline at a certain time.

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That's the safe time.

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OK.

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Similarly, so basically-- so
let me just kind of stress that

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here.

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So this is basically, what
we're really concerned is this.

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We're very concerned
about situations

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where a guideline is giving
people the sense of protection,

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and in fact they're
at high-risk.

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We're also concerned
about this case here.

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Because for example, as
in the case of my class,

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we might be causing some
damage to people's education

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or to their businesses
or to the economy

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by shutting down a certain
space or imposing a limit that

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doesn't have really a strong
scientific basis because we

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don't expect
transmission to happen

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under those circumstances.

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So we're concerned about both.

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But I'm particularly concerned
about the overly risky case

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because that is contributing
to transmission of the disease,

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and potentially loss of life.

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Now, if we look in
the other direction,

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we have the same kind of issue.

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So if we have a time limit--
for example, let's say here.

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You know, this
could be like this.

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So this would be a fixed time.

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For example, 15
minutes from the CDC

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is the time they
recommend for contact.

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So that's a pretty short time.

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OK, and what a
guideline like that says

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is that if you're
in the presence

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of an infected person--

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sufficiently close-- for 15
minutes, then here you're safe,

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but over here you're unsafe.

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But again, you have
the same phenomena.

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Now, 15 minutes is
a pretty short time,

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so that's typically
going to be safe.

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But again, in this region
it's too conservative.

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So the blue here is
still too conservative.

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But now, when you get up
here, even the 15-minute rule

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eventually becomes unsafe.

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Because if there's a very
large amount people in a room

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and they also can interact with
each other through well-mixed

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space--

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so I'm not talking about
a really, really big room.

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I'm talking about
a small enough room

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that you could expect air to be
transmitted between the people

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in that space--

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then if I keep
increasing occupancy,

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again I cross that line.

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There's always a crossing.

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So you cannot have a guideline
only based on time or only

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based on occupancy.

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If you think of this
limit-- for example,

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how about we take for
the 15-minute rule,

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what if I put 15 people or maybe
20 people into a small tent--

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just let's say a
little bit bigger

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in the size of this board--

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and everybody's
standing close together.

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We might pass the infection in
much less time than 15 minutes

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because we have a chance
to infect each other,

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and if I keep going up, I'm
going to cross that line.

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And the tent also has a very
small volume, poor ventilation,

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and this yellow
curve is very low.

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OK, so this concept
of the cumulative exposure time

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is really important
to understand

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because it's very general.

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It really isn't so tied to
the details of the model.

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And it just shows
you that any bound

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on one parameter such
as occupancy, time,

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and I should mention
also social distance.

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Because that's the big one
which is happening right now.

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Social distance
guidelines-- so this

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would be that the distance,
d, is greater than six feet.

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00:13:35,860 --> 00:13:38,540
And that's a CDC guideline
in the United States.

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It can be greater than one
meter, which is 3 feet,

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and that's from the World
Health Organization.

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So about half the distance.

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00:13:47,780 --> 00:13:49,820
That leads to a guideline
where there's still

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a maximum occupancy, which is
the area of the room divided

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00:13:57,680 --> 00:14:00,920
by d squared, or by some
other factor depending

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00:14:00,920 --> 00:14:02,840
how you think people are
going to be arranged.

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00:14:02,840 --> 00:14:07,550
But basically, you go into the
room, you map out that spacing,

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and you arrive at
a fixed occupancy.

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00:14:09,320 --> 00:14:11,090
This is being done
everywhere, including

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00:14:11,090 --> 00:14:12,890
here at MIT right now.

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00:14:12,890 --> 00:14:15,860
And that still leads
to fixed occupancy.

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00:14:15,860 --> 00:14:17,420
So regardless of
how it was derived,

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00:14:17,420 --> 00:14:18,810
you still have a
fixed occupancy,

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00:14:18,810 --> 00:14:20,690
which is too
conservative at first,

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and eventually is too risky.

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00:14:23,160 --> 00:14:25,070
And you must know
where is that crossover

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point because occupancy
and time are linked.