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PROFESSOR: So let's build
on this concept of diffusion

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of virions in
droplets to understand

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how we would expect a
size dependent infectivity

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of virions in different
sized droplets.

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So an important
concept in epidemiology

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that we will come to
later is the infectivity,

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which is the probability
that if a virion is transferred

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that it actually causes
an infection in the host.

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That can be further
broken down into a product

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of two probabilities.

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The first is that if the virus
has escaped from the droplet,

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it actually causes an infection.

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And that's perhaps something
which is roughly constant.

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It has to do with the
physiology of the host.

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But then there is the escape
of the virion from the droplet.

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And as we've already
discussed, that's

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a strongly size
dependent quantity.

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And from very large
droplets, it's

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very difficult in a mucus
droplet, especially,

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for the virion to diffuse out
in a reasonable amount of time.

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And in fact, virions
are typically

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found to have a
period of deactivation

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where after a certain
amount of time,

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they are no longer viable
and able to basically cause

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further infection.

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And so if we assume
there's a certain time

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t, or tau v for the
virus deactivation,

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then we can ask
ourselves if the virus

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has had a chance to escape
or not as a function of size.

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So basically, to
solve this problem,

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we think of the droplet here.

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And we actually want to
solve a diffusion problem

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where C here is the
concentration of viruses

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in the domain.

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D is the diffusivity
of the viruses.

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And this is the
[INAUDIBLE] equation

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in the sphere, which is
the diffusion equation.

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And our boundary
conditions are that C of R

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and 0, the initial
condition is 0.

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And then at R and t,
it's going to be one.

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So basically what
we're imagining here

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is that we're
trying to figure out

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the C will be the
concentration viruses that

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has left the system actually.

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So what we have is if we look
as a function of the radius,

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of the radius of this
thing is R, capital R. So

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in that distance, we
have this constant--

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what I'm calling
concentration here is just

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going to jump up to one.

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And then it's going to
diffuse inward like this.

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OK, and then eventually
the final state

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is that it's entirely
basically one everywhere.

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And that's when basically
the probability of removal

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has hit every part of the
drop and all of the virus

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has been removed.

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So the C is a time
dependent fraction

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of the virus, the virions
in the droplet, which

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had been removed at time t.

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So this spherical
diffusion equation

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can be solved analytically
in various ways.

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But there's not a simple closed
form solution to this problem.

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And what we're really
interested in here

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is just a rough
approximation of what

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the solution might look like.

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So let's pull out an
approximation for this.

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So I'll sketch the
droplet again here.

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Now at early times,
when there hasn't

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been a chance for the
viruses, the virions

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to diffuse very far.

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Then only those which
are close to the boundary

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actually have a
chance of leaving.

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That's this initial boundary
earlier that I sketched here,

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which is working its way in.

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So why don't we
sketch the central region

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and give that a distance delta,
which is the boundary layer

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thickness.

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So basically this outer
annulus has been--

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is really where virions
had a chance to leave.

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And that's where c
is jumping to one.

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And from-- if this
were just a plane

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with a semi-infinite diffusion
towards the center, so

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in other words, this
delta is much less than R,

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capital R, the radius, then
it's almost like diffusion

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from a planar source.

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And then we actually know
that this distance as well

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approximated by
square root of 2 DT.

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So that just comes from solving
the diffusion equation in one

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dimension leads to that
scaling of the diffusion layer

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thickness.

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So that's this thickness
of this blue region

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as it goes that way is delta.

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And it scales as
it's approximated

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by square root of 2DT.

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And now let's ask
ourselves then what

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is this concentration here?

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Well, what I'm really
interested in actually

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is this escape probability PE.

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And that's going
to be the integral

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of CDV over the volume.

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So this is the integral
over all the R's that

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are less than capital
R. So basically

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inside the drop of this
concentration field.

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So that contrary field starts
at 0 and eventually goes to 1.

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And that base is giving me
this total escape probability.

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So to calculate this integral
of the concertation field,

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I basically have a
domain at the outside

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here with the concept of
this concentration variable

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is near one and a
central region where

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it's C approximately zero.

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And here C is equal to
one on the boundary.

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This variable I've defined here.

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So therefore, I can write that
this PE is, roughly speaking,

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if we think of just
what is the volume

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of that spherical
annulus, that would be--

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and relative to
the total volume--

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that would be r-cube minus
R minus delta cube divided

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by r-cube.

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So each of the volumes has a
4/3 pi, which I've canceled off.

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So this is basically the
volume of the total sphere

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minus the volume in
the inner sphere.

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So that's just the
volume of the shell.

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Then I normalize
it properly here.

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So this is one minus delta
over r parentheses cubed.

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And if I now--

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and I have this expression here.

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So now I have at
least an approximation

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for what this might look like.

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We can also further say
that this approximation here

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was valid for the delta
B much less than r.

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And when that's the
case, then I also

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can say that this
quantity is small.

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So at early times that's small.

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And I can expand.

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All right, this is 1 minus.

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And then 1 minus
something cubed,

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where that something is
small, is 1 minus 3 times

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that something.

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That's basically a
tailor expansion.

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So that when I work this
out, the ones cancel.

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And I get three delta
over R. So what we find

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is that this PE, which
we're trying to calculate,

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has two limits that
are easy to calculate.

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One of them is this 3 delta
over R. And if that's our delta,

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then we get 3 square
root of 2 DT times R.

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And specifically, the p is
defined up to a certain time

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tau v. So I'll now
replace t with tau v

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because that is my timescale
for virus deactivation.

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And so this would be in the
case where this quantity is--

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basically, this ratio here
is much less than one.

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And then in the
opposite limit where

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this diffusion has completely
spanned the particle

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and is getting
much bigger than R,

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then this obviously
has to tend to 1.

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OK, now, I can write
down a function

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that makes this transition
right about when

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this thing is of order
one in a variety of ways.

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One way we could
do that would be

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to write that PE
is approximately

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given by 1 minus the exponential
of minus this quantity.

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So minus 3 square root of
2 d tau v divided by R.

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And you can see there we
have a-- there's sort of--

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you could either write
this in terms of a time

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where the critical time is--

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so we could write this--

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just to get more
insight into it,

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we could write PE is
approximately 1 minus e

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to the minus tau v
over some timescale--

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I'll call it tau
d for diffusion--

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where we see here that
tau d is R squared over--

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and then it's--

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To bring inside the square
root, this 3 becomes 9.

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And then times 2 is 18.

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So 18D.

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Now, you may recall from
our last calculation,

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the average first passage
time in the sphere calculated

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exactly was R squared over 15D.

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So this very simple
calculation is clearly

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giving us roughly the
right order of magnitude

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for that time.

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But we're actually
not interested so much

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in writing this
in terms of time.

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We'd actually like to
write in terms of radius.

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So I can also write PE is
1 minus E to the minus R.

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I'll call it maybe Rd
for diffusion over R

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where Rd is basically all this
stuff, 3 root 2 d tau v. OK,

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so this is maybe another
useful way to write that.

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And what does this function
look like as a function of R,

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this one right here?

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So maybe if I sketch
that out, I'll

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look at this a little
bit more carefully.

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Let's plot this.

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So as a function of R, here is
this Rd, this critical size.

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When we are smaller than that
critical size, then basically

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we have that PE.

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The escape probability,
essentially,

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is very close to 1, OK,
because then we have--

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that's basically just
what we were just arguing.

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It's this limit right here.

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But then it's a
function that when

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it gets much larger
than our d, then it

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decays as we suggested
here as sort of 1 over R.

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So it's actually a fairly
slow decay in the long run.

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So basically, there's
this limit here.

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And I just wanted to
get to this picture.

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Just to point out that even
though there are obviously

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physiological characteristics
having to do with the way

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the a virion would actually
get into a host cell

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and whether they
would get infected,

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but a lot of those
properties should

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be independent of the delivery
of the virion in a droplet.

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It's really more once the virion
gets out, there's some process.

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But what this
calculation shows is

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00:11:04,860 --> 00:11:08,070
that we would expect a
fairly strong dependence

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of the infectivity on
the size of the drop.

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Well, in particular, if
we calculate this Rd.

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We'll have some
idea that droplets

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00:11:14,280 --> 00:11:16,410
that are smaller
than that are highly

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infectious because every
virion in those droplets

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can get out and
infect the host cell.

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00:11:22,290 --> 00:11:24,500
Whereas if the virus is--

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00:11:24,500 --> 00:11:26,100
the droplet is much
bigger than that,

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00:11:26,100 --> 00:11:29,220
then you have this problem where
this dead region in the middle.

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And those virions
are not going to be

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able to get out in a
reasonable amount of time,

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00:11:33,550 --> 00:11:35,340
which is set by this tau v.

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00:11:35,340 --> 00:11:41,160
So for example, a tau v for
SARS COV-2, the coronavirus,

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is estimated to be
anywhere from one hour.

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There was one study
in aerosol droplets

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finding that kind of decay.

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But another study found
that after 16 hours,

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it was still viable.

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00:11:51,030 --> 00:11:53,680
So there's not quite a
consensus forming yet.

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But it may be a time on the
order of hours, certainly

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days, over which the virion
needs to get out of the droplet

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00:12:02,250 --> 00:12:05,550
in order to be able
to cause infection.

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00:12:05,550 --> 00:12:07,690
And this calculation shows
you that as a result,

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you would expect a size
dependent diffusivity--

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or excuse me, a size
dependent infectivity.

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And roughly speaking, if
we plug-in the numbers,

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00:12:17,860 --> 00:12:19,830
these are the aerosol droplets.

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00:12:19,830 --> 00:12:22,900
And these are the large drops.

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We've already done that.

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00:12:23,900 --> 00:12:27,140
That was our previous
calculation based on this time

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00:12:27,140 --> 00:12:29,540
here, which is what
I'm calling tau

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00:12:29,540 --> 00:12:31,310
d here, which
corresponds to this,

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00:12:31,310 --> 00:12:34,400
is also pretty close
to what we call tau 0.

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00:12:34,400 --> 00:12:39,720
That was the-- well, that
was the longest escape time,

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00:12:39,720 --> 00:12:41,720
actually what I call, I
think, tau bar actually,

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which was R squared over 15d.

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00:12:46,520 --> 00:12:48,240
That was the
average escape time.

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So basically, we've already
shown that that average escape

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time starts to become
days or even months when

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we get to large drops.

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But for the aerosol
droplets in mucus

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anyway, this timescale
is of order minutes

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00:13:02,960 --> 00:13:04,370
to hours, which is reasonable.

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And you would expect those to
be very infectious droplets.