1
00:00:10,940 --> 00:00:13,730
PROFESSOR: So let's build
on this concept of diffusion
2
00:00:13,730 --> 00:00:17,060
of virions in
droplets to understand
3
00:00:17,060 --> 00:00:21,890
how we would expect a
size dependent infectivity
4
00:00:21,890 --> 00:00:24,920
of virions in different
sized droplets.
5
00:00:24,920 --> 00:00:27,350
So an important
concept in epidemiology
6
00:00:27,350 --> 00:00:30,890
that we will come to
later is the infectivity,
7
00:00:30,890 --> 00:00:34,540
which is the probability
that if a virion is transferred
8
00:00:34,540 --> 00:00:37,550
that it actually causes
an infection in the host.
9
00:00:37,550 --> 00:00:40,490
That can be further
broken down into a product
10
00:00:40,490 --> 00:00:41,390
of two probabilities.
11
00:00:41,390 --> 00:00:44,810
The first is that if the virus
has escaped from the droplet,
12
00:00:44,810 --> 00:00:47,000
it actually causes an infection.
13
00:00:47,000 --> 00:00:49,300
And that's perhaps something
which is roughly constant.
14
00:00:49,300 --> 00:00:54,380
It has to do with the
physiology of the host.
15
00:00:54,380 --> 00:00:58,490
But then there is the escape
of the virion from the droplet.
16
00:00:58,490 --> 00:01:00,080
And as we've already
discussed, that's
17
00:01:00,080 --> 00:01:01,970
a strongly size
dependent quantity.
18
00:01:01,970 --> 00:01:03,680
And from very large
droplets, it's
19
00:01:03,680 --> 00:01:06,290
very difficult in a mucus
droplet, especially,
20
00:01:06,290 --> 00:01:10,280
for the virion to diffuse out
in a reasonable amount of time.
21
00:01:10,280 --> 00:01:13,970
And in fact, virions
are typically
22
00:01:13,970 --> 00:01:16,710
found to have a
period of deactivation
23
00:01:16,710 --> 00:01:18,260
where after a certain
amount of time,
24
00:01:18,260 --> 00:01:23,240
they are no longer viable
and able to basically cause
25
00:01:23,240 --> 00:01:24,420
further infection.
26
00:01:24,420 --> 00:01:26,870
And so if we assume
there's a certain time
27
00:01:26,870 --> 00:01:29,870
t, or tau v for the
virus deactivation,
28
00:01:29,870 --> 00:01:31,730
then we can ask
ourselves if the virus
29
00:01:31,730 --> 00:01:35,060
has had a chance to escape
or not as a function of size.
30
00:01:35,060 --> 00:01:37,400
So basically, to
solve this problem,
31
00:01:37,400 --> 00:01:39,500
we think of the droplet here.
32
00:01:39,500 --> 00:01:43,789
And we actually want to
solve a diffusion problem
33
00:01:43,789 --> 00:01:48,080
where C here is the
concentration of viruses
34
00:01:48,080 --> 00:01:49,560
in the domain.
35
00:01:49,560 --> 00:01:51,950
D is the diffusivity
of the viruses.
36
00:01:51,950 --> 00:01:55,670
And this is the
[INAUDIBLE] equation
37
00:01:55,670 --> 00:01:58,800
in the sphere, which is
the diffusion equation.
38
00:01:58,800 --> 00:02:02,600
And our boundary
conditions are that C of R
39
00:02:02,600 --> 00:02:05,210
and 0, the initial
condition is 0.
40
00:02:05,210 --> 00:02:11,130
And then at R and t,
it's going to be one.
41
00:02:11,130 --> 00:02:13,670
So basically what
we're imagining here
42
00:02:13,670 --> 00:02:15,230
is that we're
trying to figure out
43
00:02:15,230 --> 00:02:17,329
the C will be the
concentration viruses that
44
00:02:17,329 --> 00:02:18,940
has left the system actually.
45
00:02:18,940 --> 00:02:24,380
So what we have is if we look
as a function of the radius,
46
00:02:24,380 --> 00:02:29,570
of the radius of this
thing is R, capital R. So
47
00:02:29,570 --> 00:02:32,600
in that distance, we
have this constant--
48
00:02:32,600 --> 00:02:34,670
what I'm calling
concentration here is just
49
00:02:34,670 --> 00:02:37,550
going to jump up to one.
50
00:02:37,550 --> 00:02:41,810
And then it's going to
diffuse inward like this.
51
00:02:41,810 --> 00:02:45,290
OK, and then eventually
the final state
52
00:02:45,290 --> 00:02:49,460
is that it's entirely
basically one everywhere.
53
00:02:49,460 --> 00:02:51,920
And that's when basically
the probability of removal
54
00:02:51,920 --> 00:02:55,270
has hit every part of the
drop and all of the virus
55
00:02:55,270 --> 00:02:55,980
has been removed.
56
00:02:55,980 --> 00:03:00,450
So the C is a time
dependent fraction
57
00:03:00,450 --> 00:03:04,580
of the virus, the virions
in the droplet, which
58
00:03:04,580 --> 00:03:07,430
had been removed at time t.
59
00:03:07,430 --> 00:03:09,470
So this spherical
diffusion equation
60
00:03:09,470 --> 00:03:12,170
can be solved analytically
in various ways.
61
00:03:12,170 --> 00:03:15,270
But there's not a simple closed
form solution to this problem.
62
00:03:15,270 --> 00:03:16,940
And what we're really
interested in here
63
00:03:16,940 --> 00:03:19,130
is just a rough
approximation of what
64
00:03:19,130 --> 00:03:21,270
the solution might look like.
65
00:03:21,270 --> 00:03:23,960
So let's pull out an
approximation for this.
66
00:03:23,960 --> 00:03:28,430
So I'll sketch the
droplet again here.
67
00:03:28,430 --> 00:03:31,760
Now at early times,
when there hasn't
68
00:03:31,760 --> 00:03:34,100
been a chance for the
viruses, the virions
69
00:03:34,100 --> 00:03:35,630
to diffuse very far.
70
00:03:35,630 --> 00:03:37,640
Then only those which
are close to the boundary
71
00:03:37,640 --> 00:03:38,780
actually have a
chance of leaving.
72
00:03:38,780 --> 00:03:41,400
That's this initial boundary
earlier that I sketched here,
73
00:03:41,400 --> 00:03:43,280
which is working its way in.
74
00:03:43,280 --> 00:03:47,079
So why don't we
sketch the central region
75
00:03:47,079 --> 00:03:51,130
and give that a distance delta,
which is the boundary layer
76
00:03:51,130 --> 00:03:51,740
thickness.
77
00:03:51,740 --> 00:03:54,980
So basically this outer
annulus has been--
78
00:03:54,980 --> 00:03:57,760
is really where virions
had a chance to leave.
79
00:03:57,760 --> 00:04:01,870
And that's where c
is jumping to one.
80
00:04:01,870 --> 00:04:05,620
And from-- if this
were just a plane
81
00:04:05,620 --> 00:04:08,460
with a semi-infinite diffusion
towards the center, so
82
00:04:08,460 --> 00:04:12,160
in other words, this
delta is much less than R,
83
00:04:12,160 --> 00:04:15,010
capital R, the radius, then
it's almost like diffusion
84
00:04:15,010 --> 00:04:16,120
from a planar source.
85
00:04:16,120 --> 00:04:21,190
And then we actually know
that this distance as well
86
00:04:21,190 --> 00:04:24,730
approximated by
square root of 2 DT.
87
00:04:24,730 --> 00:04:27,490
So that just comes from solving
the diffusion equation in one
88
00:04:27,490 --> 00:04:30,250
dimension leads to that
scaling of the diffusion layer
89
00:04:30,250 --> 00:04:30,760
thickness.
90
00:04:30,760 --> 00:04:34,190
So that's this thickness
of this blue region
91
00:04:34,190 --> 00:04:36,400
as it goes that way is delta.
92
00:04:36,400 --> 00:04:39,340
And it scales as
it's approximated
93
00:04:39,340 --> 00:04:41,360
by square root of 2DT.
94
00:04:41,360 --> 00:04:43,180
And now let's ask
ourselves then what
95
00:04:43,180 --> 00:04:47,480
is this concentration here?
96
00:04:47,480 --> 00:04:49,870
Well, what I'm really
interested in actually
97
00:04:49,870 --> 00:04:53,440
is this escape probability PE.
98
00:04:53,440 --> 00:04:55,600
And that's going
to be the integral
99
00:04:55,600 --> 00:04:58,670
of CDV over the volume.
100
00:04:58,670 --> 00:05:01,600
So this is the integral
over all the R's that
101
00:05:01,600 --> 00:05:03,310
are less than capital
R. So basically
102
00:05:03,310 --> 00:05:06,320
inside the drop of this
concentration field.
103
00:05:06,320 --> 00:05:10,390
So that contrary field starts
at 0 and eventually goes to 1.
104
00:05:10,390 --> 00:05:15,800
And that base is giving me
this total escape probability.
105
00:05:15,800 --> 00:05:18,160
So to calculate this integral
of the concertation field,
106
00:05:18,160 --> 00:05:20,170
I basically have a
domain at the outside
107
00:05:20,170 --> 00:05:22,420
here with the concept of
this concentration variable
108
00:05:22,420 --> 00:05:25,180
is near one and a
central region where
109
00:05:25,180 --> 00:05:27,820
it's C approximately zero.
110
00:05:27,820 --> 00:05:30,370
And here C is equal to
one on the boundary.
111
00:05:30,370 --> 00:05:32,600
This variable I've defined here.
112
00:05:32,600 --> 00:05:39,920
So therefore, I can write that
this PE is, roughly speaking,
113
00:05:39,920 --> 00:05:41,680
if we think of just
what is the volume
114
00:05:41,680 --> 00:05:46,030
of that spherical
annulus, that would be--
115
00:05:46,030 --> 00:05:48,130
and relative to
the total volume--
116
00:05:48,130 --> 00:05:54,820
that would be r-cube minus
R minus delta cube divided
117
00:05:54,820 --> 00:05:55,570
by r-cube.
118
00:05:55,570 --> 00:05:59,320
So each of the volumes has a
4/3 pi, which I've canceled off.
119
00:05:59,320 --> 00:06:01,450
So this is basically the
volume of the total sphere
120
00:06:01,450 --> 00:06:03,200
minus the volume in
the inner sphere.
121
00:06:03,200 --> 00:06:04,970
So that's just the
volume of the shell.
122
00:06:04,970 --> 00:06:07,070
Then I normalize
it properly here.
123
00:06:07,070 --> 00:06:16,690
So this is one minus delta
over r parentheses cubed.
124
00:06:16,690 --> 00:06:18,290
And if I now--
125
00:06:18,290 --> 00:06:19,840
and I have this expression here.
126
00:06:19,840 --> 00:06:21,810
So now I have at
least an approximation
127
00:06:21,810 --> 00:06:24,690
for what this might look like.
128
00:06:24,690 --> 00:06:28,080
We can also further say
that this approximation here
129
00:06:28,080 --> 00:06:31,800
was valid for the delta
B much less than r.
130
00:06:31,800 --> 00:06:34,950
And when that's the
case, then I also
131
00:06:34,950 --> 00:06:36,600
can say that this
quantity is small.
132
00:06:36,600 --> 00:06:38,640
So at early times that's small.
133
00:06:38,640 --> 00:06:39,870
And I can expand.
134
00:06:39,870 --> 00:06:41,760
All right, this is 1 minus.
135
00:06:41,760 --> 00:06:43,590
And then 1 minus
something cubed,
136
00:06:43,590 --> 00:06:46,920
where that something is
small, is 1 minus 3 times
137
00:06:46,920 --> 00:06:50,100
that something.
138
00:06:50,100 --> 00:06:52,060
That's basically a
tailor expansion.
139
00:06:52,060 --> 00:06:55,020
So that when I work this
out, the ones cancel.
140
00:06:55,020 --> 00:07:01,490
And I get three delta
over R. So what we find
141
00:07:01,490 --> 00:07:08,300
is that this PE, which
we're trying to calculate,
142
00:07:08,300 --> 00:07:12,080
has two limits that
are easy to calculate.
143
00:07:12,080 --> 00:07:15,380
One of them is this 3 delta
over R. And if that's our delta,
144
00:07:15,380 --> 00:07:20,940
then we get 3 square
root of 2 DT times R.
145
00:07:20,940 --> 00:07:24,060
And specifically, the p is
defined up to a certain time
146
00:07:24,060 --> 00:07:27,840
tau v. So I'll now
replace t with tau v
147
00:07:27,840 --> 00:07:32,190
because that is my timescale
for virus deactivation.
148
00:07:32,190 --> 00:07:38,730
And so this would be in the
case where this quantity is--
149
00:07:38,730 --> 00:07:43,030
basically, this ratio here
is much less than one.
150
00:07:43,030 --> 00:07:45,480
And then in the
opposite limit where
151
00:07:45,480 --> 00:07:47,870
this diffusion has completely
spanned the particle
152
00:07:47,870 --> 00:07:49,290
and is getting
much bigger than R,
153
00:07:49,290 --> 00:07:52,120
then this obviously
has to tend to 1.
154
00:07:52,120 --> 00:07:57,300
OK, now, I can write
down a function
155
00:07:57,300 --> 00:08:00,090
that makes this transition
right about when
156
00:08:00,090 --> 00:08:03,820
this thing is of order
one in a variety of ways.
157
00:08:03,820 --> 00:08:05,310
One way we could
do that would be
158
00:08:05,310 --> 00:08:07,050
to write that PE
is approximately
159
00:08:07,050 --> 00:08:11,850
given by 1 minus the exponential
of minus this quantity.
160
00:08:11,850 --> 00:08:21,220
So minus 3 square root of
2 d tau v divided by R.
161
00:08:21,220 --> 00:08:26,150
And you can see there we
have a-- there's sort of--
162
00:08:26,150 --> 00:08:29,630
you could either write
this in terms of a time
163
00:08:29,630 --> 00:08:32,840
where the critical time is--
164
00:08:32,840 --> 00:08:34,640
so we could write this--
165
00:08:34,640 --> 00:08:36,169
just to get more
insight into it,
166
00:08:36,169 --> 00:08:40,190
we could write PE is
approximately 1 minus e
167
00:08:40,190 --> 00:08:46,340
to the minus tau v
over some timescale--
168
00:08:46,340 --> 00:08:48,650
I'll call it tau
d for diffusion--
169
00:08:48,650 --> 00:08:57,170
where we see here that
tau d is R squared over--
170
00:08:57,170 --> 00:08:58,950
and then it's--
171
00:08:58,950 --> 00:09:01,430
To bring inside the square
root, this 3 becomes 9.
172
00:09:01,430 --> 00:09:02,660
And then times 2 is 18.
173
00:09:02,660 --> 00:09:05,240
So 18D.
174
00:09:05,240 --> 00:09:07,280
Now, you may recall from
our last calculation,
175
00:09:07,280 --> 00:09:10,310
the average first passage
time in the sphere calculated
176
00:09:10,310 --> 00:09:13,910
exactly was R squared over 15D.
177
00:09:13,910 --> 00:09:16,100
So this very simple
calculation is clearly
178
00:09:16,100 --> 00:09:18,110
giving us roughly the
right order of magnitude
179
00:09:18,110 --> 00:09:19,430
for that time.
180
00:09:19,430 --> 00:09:21,140
But we're actually
not interested so much
181
00:09:21,140 --> 00:09:22,510
in writing this
in terms of time.
182
00:09:22,510 --> 00:09:24,630
We'd actually like to
write in terms of radius.
183
00:09:24,630 --> 00:09:30,500
So I can also write PE is
1 minus E to the minus R.
184
00:09:30,500 --> 00:09:34,250
I'll call it maybe Rd
for diffusion over R
185
00:09:34,250 --> 00:09:43,970
where Rd is basically all this
stuff, 3 root 2 d tau v. OK,
186
00:09:43,970 --> 00:09:47,590
so this is maybe another
useful way to write that.
187
00:09:47,590 --> 00:09:51,200
And what does this function
look like as a function of R,
188
00:09:51,200 --> 00:09:52,040
this one right here?
189
00:09:52,040 --> 00:09:55,090
So maybe if I sketch
that out, I'll
190
00:09:55,090 --> 00:09:57,390
look at this a little
bit more carefully.
191
00:09:57,390 --> 00:09:58,920
Let's plot this.
192
00:09:58,920 --> 00:10:07,660
So as a function of R, here is
this Rd, this critical size.
193
00:10:07,660 --> 00:10:12,850
When we are smaller than that
critical size, then basically
194
00:10:12,850 --> 00:10:14,650
we have that PE.
195
00:10:14,650 --> 00:10:16,840
The escape probability,
essentially,
196
00:10:16,840 --> 00:10:19,940
is very close to 1, OK,
because then we have--
197
00:10:19,940 --> 00:10:21,940
that's basically just
what we were just arguing.
198
00:10:21,940 --> 00:10:24,200
It's this limit right here.
199
00:10:24,200 --> 00:10:25,620
But then it's a
function that when
200
00:10:25,620 --> 00:10:27,990
it gets much larger
than our d, then it
201
00:10:27,990 --> 00:10:30,780
decays as we suggested
here as sort of 1 over R.
202
00:10:30,780 --> 00:10:37,060
So it's actually a fairly
slow decay in the long run.
203
00:10:37,060 --> 00:10:39,300
So basically, there's
this limit here.
204
00:10:39,300 --> 00:10:41,790
And I just wanted to
get to this picture.
205
00:10:41,790 --> 00:10:44,160
Just to point out that even
though there are obviously
206
00:10:44,160 --> 00:10:48,630
physiological characteristics
having to do with the way
207
00:10:48,630 --> 00:10:51,090
the a virion would actually
get into a host cell
208
00:10:51,090 --> 00:10:52,910
and whether they
would get infected,
209
00:10:52,910 --> 00:10:55,380
but a lot of those
properties should
210
00:10:55,380 --> 00:10:59,160
be independent of the delivery
of the virion in a droplet.
211
00:10:59,160 --> 00:11:02,880
It's really more once the virion
gets out, there's some process.
212
00:11:02,880 --> 00:11:04,860
But what this
calculation shows is
213
00:11:04,860 --> 00:11:08,070
that we would expect a
fairly strong dependence
214
00:11:08,070 --> 00:11:10,200
of the infectivity on
the size of the drop.
215
00:11:10,200 --> 00:11:12,070
Well, in particular, if
we calculate this Rd.
216
00:11:12,070 --> 00:11:14,280
We'll have some
idea that droplets
217
00:11:14,280 --> 00:11:16,410
that are smaller
than that are highly
218
00:11:16,410 --> 00:11:19,440
infectious because every
virion in those droplets
219
00:11:19,440 --> 00:11:22,290
can get out and
infect the host cell.
220
00:11:22,290 --> 00:11:24,500
Whereas if the virus is--
221
00:11:24,500 --> 00:11:26,100
the droplet is much
bigger than that,
222
00:11:26,100 --> 00:11:29,220
then you have this problem where
this dead region in the middle.
223
00:11:29,220 --> 00:11:30,990
And those virions
are not going to be
224
00:11:30,990 --> 00:11:33,550
able to get out in a
reasonable amount of time,
225
00:11:33,550 --> 00:11:35,340
which is set by this tau v.
226
00:11:35,340 --> 00:11:41,160
So for example, a tau v for
SARS COV-2, the coronavirus,
227
00:11:41,160 --> 00:11:44,100
is estimated to be
anywhere from one hour.
228
00:11:44,100 --> 00:11:46,350
There was one study
in aerosol droplets
229
00:11:46,350 --> 00:11:47,850
finding that kind of decay.
230
00:11:47,850 --> 00:11:50,130
But another study found
that after 16 hours,
231
00:11:50,130 --> 00:11:51,030
it was still viable.
232
00:11:51,030 --> 00:11:53,680
So there's not quite a
consensus forming yet.
233
00:11:53,680 --> 00:11:57,780
But it may be a time on the
order of hours, certainly
234
00:11:57,780 --> 00:12:02,250
days, over which the virion
needs to get out of the droplet
235
00:12:02,250 --> 00:12:05,550
in order to be able
to cause infection.
236
00:12:05,550 --> 00:12:07,690
And this calculation shows
you that as a result,
237
00:12:07,690 --> 00:12:12,180
you would expect a size
dependent diffusivity--
238
00:12:12,180 --> 00:12:14,220
or excuse me, a size
dependent infectivity.
239
00:12:14,220 --> 00:12:17,860
And roughly speaking, if
we plug-in the numbers,
240
00:12:17,860 --> 00:12:19,830
these are the aerosol droplets.
241
00:12:19,830 --> 00:12:22,900
And these are the large drops.
242
00:12:22,900 --> 00:12:23,900
We've already done that.
243
00:12:23,900 --> 00:12:27,140
That was our previous
calculation based on this time
244
00:12:27,140 --> 00:12:29,540
here, which is what
I'm calling tau
245
00:12:29,540 --> 00:12:31,310
d here, which
corresponds to this,
246
00:12:31,310 --> 00:12:34,400
is also pretty close
to what we call tau 0.
247
00:12:34,400 --> 00:12:39,720
That was the-- well, that
was the longest escape time,
248
00:12:39,720 --> 00:12:41,720
actually what I call, I
think, tau bar actually,
249
00:12:41,720 --> 00:12:46,520
which was R squared over 15d.
250
00:12:46,520 --> 00:12:48,240
That was the
average escape time.
251
00:12:48,240 --> 00:12:50,660
So basically, we've already
shown that that average escape
252
00:12:50,660 --> 00:12:54,350
time starts to become
days or even months when
253
00:12:54,350 --> 00:12:55,320
we get to large drops.
254
00:12:55,320 --> 00:12:58,500
But for the aerosol
droplets in mucus
255
00:12:58,500 --> 00:13:02,960
anyway, this timescale
is of order minutes
256
00:13:02,960 --> 00:13:04,370
to hours, which is reasonable.
257
00:13:04,370 --> 00:13:07,690
And you would expect those to
be very infectious droplets.