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PROFESSOR: So until now, we
based the safety guideline
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on the indoor reproductive
number, which is essentially
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the effective number
of new infections
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from a single infected
person or per infected person
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in the room.
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And in many cases, that is the
right variable to think about.
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In fact, it's essentially the
most conservative definition
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that allows us to limit
the spread of the disease
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at the level of the population.
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If every room were
doing that, we
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would control the
spread of the epidemic.
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But we also should think about
the role of the prevalence
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of infection in a given region.
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In particular, as the
number of infected people
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in the population
goes up, we should
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be increasing restrictions
to a certain point.
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And also, as the
pandemic recedes,
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we should then be decreasing
those restrictions.
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So there has to be
a role also to be
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played in using the
guideline for the prevalence
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of infection.
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So to describe this, let's
think of a random number
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of transmissions
that's going to occur.
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So this T here is going
to be the random--
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it's a random
variable, which will
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be the random number
of transmissions
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in the room with all the
usual features so in time tau
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and all the other assumptions
about this indoor space
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that we've been talking about.
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But the important thing
is that this is random
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because we don't know--
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what we're going
to focus on here
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is we don't know how many
people are in the room.
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So there are I infected people.
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And I here is the random
number of infected people.
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There are S susceptible people,
which also is a random number.
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And then there is a
transmission rate, which
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is the expected number of--
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which is the number of--
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random number of
transmissions per pair.
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So if you take an infected
person and a central person
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in this time, there's
a certain probability
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of transmission,
which is described
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by this random variable TMN.
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And so what we're going
to assume here is--
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I'll just mention some technical
assumptions, first of all,
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that this tau or TMN
is a Poisson process
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with a certain mean
rate calculated
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by the previous
model that we've been
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dealing with with a mean
rate beta times tau.
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So up to a certain
point in time,
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there's an average
transmission rate beta average.
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And that's a Poisson process.
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So what that actually means
is that the occurrence
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of transmission between
a pair of individuals
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can happen randomly
in the time sequence.
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At any infinitesimal time step,
it has no memory of the past.
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And it's an independent
random event
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with fairly low probability
in a given small time
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interval but which achieves
this certain random rate here.
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So in probability
statistics, we refer
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to that as a Poisson process.
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We also assume that
each TMN is independent
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and identically distributed
Poisson processes.
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So in other words, if I take two
different pairs of individuals,
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and I consider the transmission,
they're not correlated.
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That is an assumption.
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Because, of course, if the
infected person is sitting
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in one place, you might
expect the people nearby even,
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say, within six feet might be
more likely to be infected.
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We are leaving that out
because we are considering
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airborne transmission in
a well-mixed room where
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this should not be
any such correlations
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as a first approximation.
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Furthermore, we assume that
not only each transmission is
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independent but also that
the number of infected people
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I and all these transmissions
are also independent,
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are uncorrelated.
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So basically, the arrival
of infected people
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is uncorrelated to how
they're transmitting.
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So for example, if we have
a pack of infected people
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that arrive, we're not somehow
changing the transmission rate
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to change that.
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And that's partly we can make
the assumption because we
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are interested in the limit of
a small number infected people.
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In fact, it's
almost always going
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to be 0 or 1 because
the prevalence is not
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going to be that high in
the population generally.
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And so we can make
that assumption.
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And so if we do that, then
what we're really interested in
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is what is the expected
number of transmissions
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so the expected
value of this T here.
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So if you have a random
sum of random variables,
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then the expectation
is easy to calculate.
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If also the random number is
independent from the variables
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you're adding up so there's
no correlation between them,
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this would just be the expected
number of the total number
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of those variables,
which is IS, just
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the total number of pairs
infected and susceptible,
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times the expected value of this
tau MN, which is beta bar tau.
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And just for completeness, let
me also remind you actually
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what this beta bar is just so
that when I keep writing it,
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we're clear on it.
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It's 1 over tau integral
from 0 to tau of beta dt.
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So it's the time average beta.
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And we have further
approximated this
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by writing the beta
inverse average
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is approximately the steady
state value inverse times
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1 plus lambda C tau inverse.
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So that was a convenient
approximation.
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So in our subsequent
calculations,
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whenever you see this expression
average of beta times tau,
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you could imagine substituting
this expression where
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beta bar is given by all
the physical parameters
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that we've been discussing.
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So this is a very simple model
of the random transmission that
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can occur when you
take into account
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the randomness in the
number of infected people.
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So now let's start to write
down a model for the number
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of iffected people.
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So the simplest
thing there is that--
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I'll write here the random
number of infected persons
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is that this should be a
binomial random variable.
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And that means that
the probability
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that the infected number
is equal to some value N
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is the number of ways you can
choose little n infected people
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out of N total people in the
room times the probability
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that any one of them is
infected, which we'll call pI,
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and then qI is the probability
that the others are not
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infected.
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So the important new variable
that we have here is pI
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is the probability a
person randomly selected
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from the population and placed
in this room is infected.
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And this is also
sometimes called
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the prevalence of the
infection in the population.
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And then this qI, of
course, is just 1 minus pI.
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So it's standard notation
for binomial distribution
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is to use q is 1 minus p.
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And so let's make some
further assumptions
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about this random number
of infected people.
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So first of all, by assuming
this binomial distribution,
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we are assuming that
at any moment in time,
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the number of iffected people is
somehow refreshed continuously
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to reflect the same
kind of distribution
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that you find in the population.
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So the variable I is
refreshed continuously in time
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to reflect the
population prevalence.
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So people are coming
and going from the room.
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But there's always a certain
number of inffected people
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that reflects the chance of
running into an infected person
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in the population.
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So that's a
reasonable assumption.
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A more sophisticated
model for a given space
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might take into account the
actual probability or rate
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of arrival and the rate
of removal of people
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and, similar to models
from queuing theory,
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might derive a distribution
which is more complicated
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and depends on those
other parameters
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for the fluctuations
in the number
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of infected people in a room.
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But this is the
simplest way to start
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is that the room essentially
reflects the population
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statistically.
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Another assumption, though,
which we're going to make,
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which gives us some
more simplicity
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and also allows us to
be more conservative,
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is that we neglect
exposure and essentially
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allow infection to happen at
the same rate more than once
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even and allow transmission
to proceed at the same rate
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where we essentially are
assuming that S is N minus I.
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So in other words,
the susceptibles
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never get converted into
an exposed group that can
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no longer be infected again.
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We'll just say that the
rate is still always going
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to be I times S where
S is just N minus I.
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So we're letting the number
of infected people fluctuate.
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But everyone else in the room
is considered susceptible.
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That's a conservative
approximation.
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Because in reality, the
number of susceptible people
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would go down as they
converted to the exposed Group.
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Over very long times,
eventually there
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be new infected
people generated.
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But that's really relevant for
situations like the Diamond
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Princess quarantine
where same people
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are confined to the same space
for a period, a longer period.
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Here, we're really thinking
of just this indoor space that
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is reflecting the statistics of
prevalence in the population.
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Let me now do a couple
quick calculations based
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on this model of some quantities
that we're going to need.
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So the first is that given
a binomial distribution,
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the expected value of
this random variable
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I is just the number in the
room times pI the prevalence.
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And furthermore,
the variance of I,
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which is the expected
value of I squared
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minus the expected value of I
quantity squared, is NpIqI--
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basic result for a
binomial random variable.
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00:11:21,510 --> 00:11:24,300
From there, if you then take
these two relationships,
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you can solve for the
expected value of I squared
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00:11:28,530 --> 00:11:36,000
and find that that is equal
to this Npq plus the Np
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00:11:36,000 --> 00:11:37,240
quantity squared--
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00:11:37,240 --> 00:11:38,160
I'll just write that--
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00:11:38,160 --> 00:11:43,290
Npq plus Np quantity squared.
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That's wrong.
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00:11:45,190 --> 00:11:52,040
And so that's can be written
as I expected value times--
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well, let's see here.
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00:11:52,920 --> 00:12:01,750
We have a qI plus
expected value of I.
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And now using this
relationship, if we
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00:12:04,870 --> 00:12:06,790
look at the number
of susceptibles,
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00:12:06,790 --> 00:12:12,250
S, the expected value, is just
N minus the expected value of I,
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00:12:12,250 --> 00:12:14,600
which is Np.
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00:12:14,600 --> 00:12:16,270
And finally, to
calculate transmissions,
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00:12:16,270 --> 00:12:21,220
what we're really interested in,
the expected value of I times
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00:12:21,220 --> 00:12:22,630
s--
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00:12:22,630 --> 00:12:26,980
so that would be,
if we substitute,
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00:12:26,980 --> 00:12:30,280
that would be N times
the expected value of I
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00:12:30,280 --> 00:12:33,680
minus the expected
value of I squared,
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00:12:33,680 --> 00:12:36,040
which we have right here.
218
00:12:36,040 --> 00:12:40,300
And if we substitute
this expression here,
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00:12:40,300 --> 00:12:42,550
you see we have a factor of I--
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00:12:42,550 --> 00:12:44,200
expected value that
we can factor out,
221
00:12:44,200 --> 00:12:50,480
and then we get N minus the
expected value I minus q, qI.
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00:12:50,480 --> 00:12:53,450
And then, finally,
substituting the expected value
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00:12:53,450 --> 00:12:56,740
of Is and pI, and
1 minus pI is qI,
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00:12:56,740 --> 00:13:01,930
you can find that
this is pIqI N N
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00:13:01,930 --> 00:13:07,030
minus 1, which can
also be written
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00:13:07,030 --> 00:13:12,540
sigma I squared N minus 1.
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00:13:12,540 --> 00:13:15,900
So now we basically
have an expression
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00:13:15,900 --> 00:13:20,430
for the transmission rate in
terms of the number of people
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00:13:20,430 --> 00:13:21,130
in the room.
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00:13:21,130 --> 00:13:23,860
So essentially, it's N minus 1,
which, in fact, you may recall,
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was the transmission rate when
there's one infected person
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and N minus 1 susceptibles.
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00:13:28,980 --> 00:13:31,720
But time is factor
sigma I squared,
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which is actually
the fluctuation
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in the infected number.