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PROFESSOR: So in the majority
of cases of indoor spreading

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that occur over the space
of hours or even a few days,

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the Wells-Riley model
of slow incubation

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where the number of
infectors is held constant

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at the initial value, is a
reasonable approximation.

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For COVID-19, the
incubation period

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is estimated to be on the
order of several days.

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For example, a
number of 5.5 days

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is often quoted as an estimated
mean incubation period.

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On the other hand,
in some cases there

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are spreading events that
involve people interacting over

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much longer periods
of time than that.

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A famous example, which
is a little bit longer

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than the time of
incubation, is the case

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of the Diamond Princess
cruise ship, which

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was quarantined in Yokohama
Port, Japan in early 2020

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when it was detected that there
were a number of cases onboard.

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The exact number wasn't
known because they

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hadn't tested the
entire population,

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but there were several cases.

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And so they decided
to quarantine

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the ship for 12 days.

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And if you look at the number
of infections versus time

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once they started
testing and felt

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they had a good
sense of the numbers,

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you can see that
it rose from value,

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which we may estimate to be on
the order of 20 possible cases

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initially.

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And it rises, but in a
very non-linear fashion.

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In fact, it starts to accelerate
and go steeper and steeper.

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Now recall, the Wells-Riley
model cannot possibly predict

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this behavior because if you
have a fixed number infectors,

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then at first you have a certain
rate of transmission but it

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always has to slow down because
if there's no new infections,

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the same factors are running
out of susceptible people

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to infect.

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The number of susceptible
is going down,

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and so you kind
of slowly saturate

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until eventually you've
infected everybody,

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when it's just one person or
a fixed number of people who

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are basically transmitting at
a constant rate of infection

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quanta to everybody else.

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But that's not what happened.

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In fact, it's a very steep rise.

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It went-- just in
the last few days,

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it jumped by several
hundred people.

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And in fact, if
they hadn't stopped

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the quarantine in 12 days--

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there were 4,000
people on that ship.

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I think 3,711.

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I'll just mention I think
that was the number.

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So way higher.

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And you see how this
is accelerating.

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If they'd gone with the
quarantine a couple more days,

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they might have had
thousands of people infected.

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So in fact, this is
an interesting warning

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for quarantines,
that just cooping up

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some people for 14 days
doesn't make everybody safe.

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If many of those people are
susceptible and are not yet

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infected, they can
become infected.

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And many interesting
things about this incident

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is that the people were not
in direct contact, obviously.

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Thousands of people were not
six feet apart from each other.

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They were often in
different rooms,

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different floors of the ship.

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And yet, very large
numbers became

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infected in a short time.

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So the important thing I'd like
to emphasize now coming back

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to our SEI model is that this
sort of non-linear increase

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can only be explained if you
have some accelerated spreading

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due to new infected people.

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And it makes sense
after about five days.

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And we also don't know
when people got infected.

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So those initially infected
people may have been

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infected five days earlier.

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So there may have been already
an increase in the number

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infected people already at time
equals zero of the quarantine.

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And so we should now account
also for the exposed people.

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Those are people that may
not be showing symptoms yet,

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but have been exposed enough
that they can then pass it on.

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And so the rate of an
exposed person becoming

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an infectious person is alpha.

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And inverse of alpha
is that 5.5 days.

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So that's the incubation time.

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And this might be something
like 5.5 days for COVID-19.

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But of course it can vary.

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But it's roughly in that order.

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And so it makes sense to
look at the Diamond Princess

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and consider what would be
the effect of accounting

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for incubation.

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So these are
non-linear equations

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that don't have a simple
solution to the full model.

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But in the same way
the Wells-Riley model

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is the limit of slow incubation,
where basically E stays zero

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and basically you only
have infected people.

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Now, we can consider
the opposite limit

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of fast incubation.

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So let's consider the opposite
limit of fast incubation.

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And this would be alpha
t much greater than one.

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So basically, we
want to make sure

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that the t is much bigger
than the incubation time.

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And as I said, the
incubation time

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doesn't necessarily
start at time zero.

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The infected people in
this case may have already

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been infected five days earlier.

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In fact, the cruise
ship had been going

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for actually weeks before that.

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So no one knows exactly
when the infection began.

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So it may be even likely
that that was happening.

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So let's consider the
fast incubation limit.

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So what this then tells us is
that now the exposed portion

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is roughly zero.

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So if alpha is very quick,
then you pretty much quickly

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go through the
exposed, and you end up

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immediately being infected.

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And so this is actually
a much simpler model

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where the number of
susceptibles is just n minus I.

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So there's no exposed
compartment anymore.

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So the Wells-Riley
model in some sense

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is the SE model, where there
are only exposed people

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and susceptible, but the
number infected doesn't change.

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This is really the
SI model, where

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we don't worry about the
exposed compartment, OK?

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So the equation we
want to solve then,

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if we realize that
S is n minus I

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is that dS dt is
minus beta of t SI.

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So that's the same equation
I wrote down earlier.

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But now let's substitute.

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Let's derive an equation
for the number of infected.

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So that will be from here, dI
dt will be beta of t times I.

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And then S is n minus
I. Because again,

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if we go straight through
the exposed fraction,

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then basically this rate
of losing susceptibles

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is equal to the rate of
creating new infected people.

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Again, because n is fixed.

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The number of people is fixed.

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So dS dt is minus the dI dt.

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So this is the equation
now that we can solve

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for this limit of the model.

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And fortunately, this is a
simple equation to solve.

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It's a first order separable
differential equation.

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So we can write
this as dI over I

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times n minus I is
equal to beta of t dt.

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On this side, I
can write this as--

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I can factor out a one over n,
and write this as one over I

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plus one over n minus I. So
when I combine these two,

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I get In minus nine
in the denominator.

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And the numerator, I get n
minus I plus I. So just get n.

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But then it divides by
that n, so I do come back

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to what I started with.

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And this times dI.

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And so now I can integrate
both sides this equation.

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Take into account
the initial condition

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is that at t equals zero, the
number of infectors is I0.

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So basically, I can integrate
by going from zero to time, t.

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And on this side, in
terms of the infectors,

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I'm going from I0 to the
current number of infectors, I.

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OK, so now we're ready to
integrate this equation.

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So let's multiply the
n to the other side

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and do the integrals.

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So basically, the integral
of one over I is log I.

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So we have log I minus
log of n minus I.

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And that's because
there's a minus I there,

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so that leads to a
minus sign out front.

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On the other side, if I
multiply through by n,

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I have n integral
from zero to t of beta

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of t dt plus a constant
of integration.

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The boundary condition that I
need is that I of zero is I0.

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So a t equals zero,
this term vanishes,

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and this expression must be
evaluated with I equal to I0.

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So therefore, there
must be on this side

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of the equation,
some constants log

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of I0 minus log of n minus I0.

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So now we satisfy the boundary
condition or initial condition

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of t equals zero.

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Now, I can also write this in
terms of the quanta emission

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rate for the initial infectors,
which we defined earlier.

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So for the Wells-Riley model,
we talked about writing q of t

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is the number of quanta emitted
by the initial infector.

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So if we just define it as
I zero times the integral

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over time of beta
of t dt, this is

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the sort of infection
quanta emitted

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by the I0 initial infectors.

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So if we take that here,
we can express the solution

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a somewhat different way.

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If I take an exponential
on both sides,

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the difference of two logs
is the log of the ratio.

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And when I exponentiate,
I get rid of the log.

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So this side turns into I over
n minus I. And the other side,

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we have from the similar
expression, I0 over n minus I0.

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But times, now, the
exponential of--

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well, if we want to express it
in terms of q0, it would be n.

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And then this d beta dt
has a one over I0 q of t.

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So this is the
solution in this case.

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Now, if we look at early times--

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and so then that would be
less than incubation time.

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So if basically our alpha
t is much less than one--

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so not much incubation
is occurred.

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And basically during that time
I is approximately still equal

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to I0.

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So that would be kind of in
the early, early stages here

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of this dynamics--

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then we could write that
I of t is, well, from this

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we could write it as n minus I0.

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And then I0 times the time
interval of beta, or n minus I0

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times q, or it's the number
of susceptibles times q of t.

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So this is a result
that can come directly

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from analyzing this expression.

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We can also see it here,
that if I is not changing,

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then we have I, n minus
I. And we can also

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see this expression
here where we just

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integrate both sides in time
to get to this equation here.

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So it's basically the
same as the Wells-Riley

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in an early time.

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And that's an
interesting observation,

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which is that there
is a universal sort

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of small transmission limit
in both limits of this model.

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And what that is,
if we take this--

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at least for the
SEI model, if we

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write how many exposed
plus infected relative

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to the initial number
of infectors, OK?

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So that is telling us how
many transmissions there

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are, either to make
someone exposed

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or to make them infectious
from the initial number

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of infectors.

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That at early times,
we get this same result

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that we had before
because we got

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the same thing for the
Wells-Riley model for E,

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here it's for I. And
we find that this

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is this what we call Rn, the
indoor transmission number.

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Which is the initial number of
susceptibles times the number

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of quanta transferred.

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And in the case where if S0 were
equal to n minus one, and I0

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were equal to one, this
would be n minus one integral

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over t of beta dt.

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That would be that case
that we talked about before

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for the indoor
transmission number.

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But more generally, it
would look like this.

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So that is at early times,
before many transmissions

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have happened.

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It really doesn't matter
what's the details

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of the model in terms of
the non-linear response.

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So even if after a
longer amount of time

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00:13:34,860 --> 00:13:36,930
more and more
people get infected,

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the initial moments are always
universal and are really

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just governed by this
transmission rate

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beta and the number of
susceptible people and number

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00:13:44,100 --> 00:13:46,210
of infectious people
initially in the room.

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So it's kind of independent
of all these details here.

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And so then to kind of
summarize that picture,

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we could plot versus time here
what happens in terms of the--

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so we have a rate
of transmission

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where if I look at
the total number of

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exposed plus
infected people, OK.

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And then here's the total
number of people in the room, n.

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So in the Wells-Riley
model, everyone's exposed,

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but nobody becomes infectious.

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And we know that we get this
kind of exponential relaxation

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as we eventually run out
of susceptible people.

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And the timescale for
that is beta inverse, OK.

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And that gives you
the transmission time

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for just a fixed
number of infectors

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to slowly infect everybody else.

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But in the case like
we described here

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where we have some
non-linear acceleration,

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that has to start out the same.

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That's what I'm
trying to say here,

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is the initial transmission
rate doesn't matter

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if there is an incubation rate
until you reach the incubation

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time.

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So there's kind of
an alpha inverse

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here, which is the
incubation time.

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This one here is the
transmission time.

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And at this time
scale, you start

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to see an exponential increase
until it saturates basically

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once all the transmission has
occurred because now there's

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more and more infectious people.

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And so this is the fast
incubation and slow incubation.

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And this one is the Wells-Riley
model, which is widely used.

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But if you're fitting spreading
data where there may actually

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be some incubation going on,
and also potentially removal--

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we could add another equation
for the removal of people.

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00:15:53,240 --> 00:15:55,290
We need to be careful on
how we fit data in order

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00:15:55,290 --> 00:15:57,670
to extract information about
the transmission rate, which

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00:15:57,670 --> 00:15:59,290
is what we're
interested in what we're

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00:15:59,290 --> 00:16:01,300
trying to interpret the data.

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00:16:01,300 --> 00:16:06,310
So maybe an important
conclusion from this

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is that infection
quanta, this notion--

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00:16:11,870 --> 00:16:14,860
or the infection quantum, I
guess you can say, one of them.

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Which is a quantity that
was introduced by Wells,

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00:16:17,680 --> 00:16:19,830
really based on this
Wells-Riley model,

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00:16:19,830 --> 00:16:22,000
is basically-- think of
this exponential relaxation,

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00:16:22,000 --> 00:16:27,250
and saying when let's say 63%
of people become infected,

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00:16:27,250 --> 00:16:29,650
that's what you say is
one infection quantum

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00:16:29,650 --> 00:16:33,910
has been transmitted to
each of those people.

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00:16:33,910 --> 00:16:37,300
Then really, it's
better defined--

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00:16:37,300 --> 00:16:40,930
I'll just write here it's
defined by the transmission

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00:16:40,930 --> 00:16:47,180
rate and not by the
number of people

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00:16:47,180 --> 00:16:48,310
that actually get infected.

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00:16:48,310 --> 00:16:49,730
So if you look at some
data like the Diamond

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00:16:49,730 --> 00:16:52,420
Princess or other data that
we're going to look at later,

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00:16:52,420 --> 00:16:54,650
you are seeing
spreading happening

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00:16:54,650 --> 00:16:56,330
and there could be
lots of contributions

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00:16:56,330 --> 00:16:57,950
to the number of people
that actually get sick.

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00:16:57,950 --> 00:17:00,270
For example, there could
be incubation going on.

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00:17:00,270 --> 00:17:02,630
So what's sometimes called
the secondary attack

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00:17:02,630 --> 00:17:09,470
rate is E plus I divided by S0.

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00:17:09,470 --> 00:17:11,770
So the secondary
attack rate is sort

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00:17:11,770 --> 00:17:13,190
of the fraction
of people that are

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00:17:13,190 --> 00:17:15,170
susceptible that got infected.

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00:17:15,170 --> 00:17:18,500
And the Wells-Riley would
say, when that 63% then you've

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00:17:18,500 --> 00:17:21,530
transmitted a quantum
to each of those people.

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00:17:21,530 --> 00:17:24,599
Whereas, as we see in the case
of the fast incubation model,

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00:17:24,599 --> 00:17:26,480
that's not how you would
interpret that data.

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00:17:26,480 --> 00:17:28,490
But on the other hand,
beta is well-defined.

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00:17:28,490 --> 00:17:30,980
It's just each person
is transferring quanta

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00:17:30,980 --> 00:17:32,690
at a certain rate
and has the potential

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00:17:32,690 --> 00:17:34,280
to infect other people.

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00:17:34,280 --> 00:17:37,040
Now, I don't want to
overstate the relevance

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00:17:37,040 --> 00:17:39,680
of this model for a particular
case like the Diamond Princess

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00:17:39,680 --> 00:17:40,220
cruise ship.

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00:17:40,220 --> 00:17:41,940
We'll come back to this later.

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00:17:41,940 --> 00:17:43,490
But just simply to
illustrate that it

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00:17:43,490 --> 00:17:45,470
has this kind of
non-linear feature

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00:17:45,470 --> 00:17:47,570
which is suggestive
of incubation

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00:17:47,570 --> 00:17:50,810
occurring and an increase in
the number of infected people.

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00:17:50,810 --> 00:17:54,620
And just to point out that
this sort of simple modeling

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00:17:54,620 --> 00:17:56,990
leads us to kind of a
universal expression

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00:17:56,990 --> 00:18:00,860
for the initial transmission
from the initial in factors

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00:18:00,860 --> 00:18:04,190
for each of them to transfer
to sort of one other,

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00:18:04,190 --> 00:18:07,700
which is this indoor
reproductive number.

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00:18:07,700 --> 00:18:11,580
And that's really what's
valid at the early times here.

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00:18:11,580 --> 00:18:15,620
And that's where we have
kind of a universal behavior.

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00:18:15,620 --> 00:18:18,290
And that's useful in
formulating safety guidelines,

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00:18:18,290 --> 00:18:20,380
which we will do next.