1
00:00:01,590 --> 00:00:04,410
Okay, we're going to start week four,

2
00:00:04,410 --> 00:00:07,110
and with our first look
at continuous model

3
00:00:07,110 --> 00:00:08,280
with a very short video.

4
00:00:08,280 --> 00:00:10,570
This would be, you know, super simple

5
00:00:12,360 --> 00:00:14,460
but there are a lot of key steps.

6
00:00:14,460 --> 00:00:16,050
So simple isn't quite the right word.

7
00:00:16,050 --> 00:00:20,400
I think the idea is that
we want to do something

8
00:00:20,400 --> 00:00:22,740
that is straightforward in theory.

9
00:00:22,740 --> 00:00:24,960
We just want to take discrete models

10
00:00:24,960 --> 00:00:28,293
where we evolve a system,
one step at a time,

11
00:00:29,460 --> 00:00:32,430
using a given set of rules and mechanisms.

12
00:00:32,430 --> 00:00:34,830
And then we want to say
that those steps occur

13
00:00:34,830 --> 00:00:39,330
over progressively
smaller timescales, right?

14
00:00:39,330 --> 00:00:41,760
So we could have models,
we've discussed models,

15
00:00:41,760 --> 00:00:45,030
where we have changes of
the order of each day.

16
00:00:45,030 --> 00:00:46,320
All right, so we update the rules.

17
00:00:46,320 --> 00:00:48,390
What happens on campus today

18
00:00:48,390 --> 00:00:51,660
determines what's going
to be on campus tomorrow.

19
00:00:51,660 --> 00:00:54,120
Now let's say that our
rules are over one hour.

20
00:00:54,120 --> 00:00:57,000
So what happens on campus this hour

21
00:00:57,000 --> 00:01:01,080
is determining the state
of campus in one hour.

22
00:01:01,080 --> 00:01:04,140
We could say that our time steps
are of the order of minutes

23
00:01:04,140 --> 00:01:07,230
or seconds, and as we
shrink those time steps,

24
00:01:07,230 --> 00:01:08,640
we want to ask, "What is the limit

25
00:01:08,640 --> 00:01:11,460
of our model in continuous time?"

26
00:01:11,460 --> 00:01:12,690
The idea of continuous time

27
00:01:12,690 --> 00:01:14,850
is that continuous dynamical systems,

28
00:01:14,850 --> 00:01:16,890
calculus and differential equations,

29
00:01:16,890 --> 00:01:21,060
are going to be a incredibly
useful toolbox to unlock.

30
00:01:21,060 --> 00:01:22,830
So that's what we want
to do in those video

31
00:01:22,830 --> 00:01:25,530
is take the limit of very short timescale.

32
00:01:25,530 --> 00:01:28,590
So I'll be repeating a lot of what we saw.

33
00:01:28,590 --> 00:01:32,433
I'll use the SIS model as an example,

34
00:01:33,900 --> 00:01:37,173
so from discrete to continuous.

35
00:01:42,120 --> 00:01:45,060
I like using the SIS model as an example

36
00:01:45,060 --> 00:01:46,440
because we've discussed, you know,

37
00:01:46,440 --> 00:01:48,900
what approximations are
made to go from the reality

38
00:01:48,900 --> 00:01:52,500
of disease spread to a model like this.

39
00:01:52,500 --> 00:01:54,600
It's also a single variable model.

40
00:01:54,600 --> 00:01:56,280
So even though we have
two different states,

41
00:01:56,280 --> 00:01:59,730
two boxes, as we saw, the
fact that our population

42
00:01:59,730 --> 00:02:01,590
is conserved in the classic case

43
00:02:01,590 --> 00:02:03,930
means that we really only
have one degree of freedom.

44
00:02:03,930 --> 00:02:06,450
Give me S, I know how many N people.

45
00:02:06,450 --> 00:02:09,090
I know how many I people there are.

46
00:02:09,090 --> 00:02:12,420
So it's N minus S, right, for all times.

47
00:02:12,420 --> 00:02:14,550
So the fact that it's
only one key variable

48
00:02:14,550 --> 00:02:16,623
makes it a little easier to play with.

49
00:02:18,060 --> 00:02:20,130
Now there's two steps we're going to take.

50
00:02:20,130 --> 00:02:24,030
So in the discrete form
we add S of t plus 1

51
00:02:24,030 --> 00:02:26,343
or S of t plus delta t.

52
00:02:27,630 --> 00:02:29,070
I know I haven't been clear with that.

53
00:02:29,070 --> 00:02:30,960
Depends on whether we're counting our time

54
00:02:30,960 --> 00:02:34,140
in sort of units or in time steps.

55
00:02:34,140 --> 00:02:36,690
That doesn't matter too
much, so that's equivalent.

56
00:02:39,870 --> 00:02:42,453
And I wrote, in a previous video,

57
00:02:43,440 --> 00:02:46,140
that it's proportional to everyone

58
00:02:46,140 --> 00:02:48,840
that's currently susceptible,

59
00:02:48,840 --> 00:02:52,263
are going to stay susceptible
if they don't get infected,

60
00:02:53,190 --> 00:02:57,120
and we add 1 minus beta times delta t

61
00:02:57,120 --> 00:02:58,410
or beta times delta t.

62
00:02:58,410 --> 00:03:01,590
Remember, if we're simulating
this process, let's say,

63
00:03:01,590 --> 00:03:04,170
for every contact with
an infectious individual,

64
00:03:04,170 --> 00:03:06,360
beta times delta t is my transmission rate

65
00:03:06,360 --> 00:03:07,530
times the duration.

66
00:03:07,530 --> 00:03:09,510
So my transmission rate
might be, you know,

67
00:03:09,510 --> 00:03:12,510
0.01 transmission per hour.

68
00:03:12,510 --> 00:03:14,636
If my delta t is one day,

69
00:03:14,636 --> 00:03:19,636
If my delta T is one day

70
00:03:20,070 --> 00:03:23,010
And I applied this as
a discrete probability

71
00:03:23,010 --> 00:03:25,590
for every one of my infectious contact.

72
00:03:25,590 --> 00:03:26,977
Everyone that can infect me, we ask,

73
00:03:26,977 --> 00:03:28,950
"Do you infect me? Yes or no?"

74
00:03:28,950 --> 00:03:30,900
And I'm only going to remain susceptible

75
00:03:30,900 --> 00:03:32,130
if they all answer no.

76
00:03:32,130 --> 00:03:32,963
So they answer no

77
00:03:32,963 --> 00:03:37,230
with probability 1 minus the
transmission probability,

78
00:03:37,230 --> 00:03:42,230
and I have I of t potential
infectious people infecting me.

79
00:03:42,750 --> 00:03:47,250
Then we had an average term
here I of t for recoveries,

80
00:03:47,250 --> 00:03:50,310
so flow of individuals back from I to S.

81
00:03:50,310 --> 00:03:52,920
And then I don't need
to write I of t plus 1

82
00:03:52,920 --> 00:03:54,540
or I of t plus delta t

83
00:03:54,540 --> 00:03:56,520
'cause I know it's N, my total population,

84
00:03:56,520 --> 00:03:58,170
minus whoever's in susceptible.

85
00:03:58,170 --> 00:04:00,753
So the system is fully fixed by this.

86
00:04:02,280 --> 00:04:07,280
We saw in the bonus
video for Taylor series

87
00:04:07,380 --> 00:04:12,380
that one thing you can do
is develop this probability

88
00:04:13,110 --> 00:04:14,430
as a Taylor series,

89
00:04:14,430 --> 00:04:16,350
meaning we're going to
make the approximation

90
00:04:16,350 --> 00:04:19,173
that beta times delta t
is really, really small.

91
00:04:20,160 --> 00:04:22,830
And that's going to give us something

92
00:04:22,830 --> 00:04:27,830
like 1 minus beta delta t times I of t,

93
00:04:29,370 --> 00:04:31,443
and then plus some terms,

94
00:04:36,000 --> 00:04:38,550
whoops, that go as beta times delta t,

95
00:04:38,550 --> 00:04:39,600
the whole thing squared,

96
00:04:39,600 --> 00:04:42,270
so beta squared times delta t squared.

97
00:04:42,270 --> 00:04:44,550
The key part is that, if that probability,

98
00:04:44,550 --> 00:04:46,170
beta times delta t, is very small,

99
00:04:46,170 --> 00:04:48,483
we can ignore those other terms.

100
00:04:49,740 --> 00:04:51,600
So that's one thing we can do when delta t

101
00:04:51,600 --> 00:04:53,403
becomes really, really small.

102
00:04:55,080 --> 00:04:56,460
And that's going to be super useful.

103
00:04:56,460 --> 00:04:59,733
So if this is a little confusing
right now, what's in red,

104
00:05:00,570 --> 00:05:03,540
I invite you to go and
rewatch the bonus video

105
00:05:03,540 --> 00:05:06,060
for a formal definition of Taylor series

106
00:05:06,060 --> 00:05:07,983
in this particular example.

107
00:05:09,090 --> 00:05:11,910
But now what we're going
to want to do is remember

108
00:05:11,910 --> 00:05:15,840
that that simple result
for small probabilities,

109
00:05:15,840 --> 00:05:19,620
and we're going to ask,
"As delta t goes to zero,"

110
00:05:19,620 --> 00:05:22,173
so as my windows get smaller and smaller,

111
00:05:23,167 --> 00:05:25,870
"what is the instantaneous change

112
00:05:29,160 --> 00:05:33,237
between a time step t and a
time step t plus delta t?"

113
00:05:37,980 --> 00:05:39,280
That's what we care about.

114
00:05:44,700 --> 00:05:48,603
So let's assume that we now
know the result that was in red.

115
00:05:50,610 --> 00:05:52,350
We have that stored in memory forever.

116
00:05:52,350 --> 00:05:55,650
That's going to be useful often.

117
00:05:55,650 --> 00:05:57,453
So let's just store that in memory.

118
00:06:03,780 --> 00:06:07,263
And now I'm going to expand
my terms a little bit.

119
00:06:10,590 --> 00:06:12,300
I'm going to have S of t.

120
00:06:12,300 --> 00:06:15,063
I'm using the result that I just erased,

121
00:06:16,410 --> 00:06:19,950
and that's going to give
me this delta t times beta

122
00:06:19,950 --> 00:06:23,883
times I of t times S of t.

123
00:06:28,260 --> 00:06:31,830
Let me do something to help
me write here more clearly.

124
00:06:31,830 --> 00:06:32,910
'cause I have a lot of terms.

125
00:06:32,910 --> 00:06:37,910
I'm just going to omit the
explicit dependency on times.

126
00:06:38,850 --> 00:06:40,920
I'm just going to use I times S

127
00:06:40,920 --> 00:06:43,140
instead of I of t for everything.

128
00:06:43,140 --> 00:06:45,153
That's delta t gamma I.

129
00:06:50,580 --> 00:06:53,550
I was missing a delta t here
in my previous equation.

130
00:06:53,550 --> 00:06:57,663
The probability of recovery,
again, was gamma terms delta t.

131
00:06:59,160 --> 00:07:01,080
Good thing I have notes.

132
00:07:01,080 --> 00:07:04,353
Okay, so that's this
whole thing over delta t.

133
00:07:05,370 --> 00:07:09,390
You know, bonus point for those
who corrected my term here,

134
00:07:09,390 --> 00:07:13,257
initially, the same way I
add beta times delta t here,

135
00:07:15,510 --> 00:07:18,933
I needed gamma times delta t times I of t.

136
00:07:25,530 --> 00:07:27,360
Well now I have some
terms that cancel out.

137
00:07:27,360 --> 00:07:29,340
So this limit is actually going to be

138
00:07:29,340 --> 00:07:30,390
pretty straightforward to do.

139
00:07:30,390 --> 00:07:32,403
I have a minus S and a plus S,

140
00:07:33,360 --> 00:07:36,663
and then everything that's left
is proportional to delta t.

141
00:07:37,530 --> 00:07:41,610
So really what I have
is delta t over delta t,

142
00:07:41,610 --> 00:07:46,610
and then gamma I minus beta IS.

143
00:07:51,570 --> 00:07:54,150
So then I can easily take the limit.

144
00:07:54,150 --> 00:07:57,750
The two delta t's cancel
out and it doesn't matter.

145
00:07:57,750 --> 00:08:01,530
And then I simply have gamma I of t.

146
00:08:01,530 --> 00:08:02,760
This is my final result,

147
00:08:02,760 --> 00:08:05,583
so I'll put the time
dependency back on there,

148
00:08:06,810 --> 00:08:08,553
I of t, S of t.

149
00:08:09,420 --> 00:08:13,890
And then this, we'll say, is
equal to the time derivative

150
00:08:13,890 --> 00:08:15,033
of S of t.

151
00:08:17,250 --> 00:08:19,620
Because if you remember
from your calculus courses,

152
00:08:19,620 --> 00:08:22,620
this instantaneous change
that we just calculated,

153
00:08:22,620 --> 00:08:26,673
that's the, you know, purest
definition of a derivative.

154
00:08:27,630 --> 00:08:31,410
So this last thing here is
essentially just by definition,

155
00:08:31,410 --> 00:08:34,770
which we've calculated is
the differential equation

156
00:08:34,770 --> 00:08:37,590
governing the continuous time evolution

157
00:08:37,590 --> 00:08:40,320
of this S of t variable.

158
00:08:40,320 --> 00:08:42,030
And it looks a little funny here,

159
00:08:42,030 --> 00:08:45,123
and if you wanted to make it
a little more explicit, again,

160
00:08:46,530 --> 00:08:49,200
everything here we have
to remember that I of t

161
00:08:49,200 --> 00:08:51,600
is equal to N which is just a constant,

162
00:08:51,600 --> 00:08:55,050
number of people in the
model, minus S of t.

163
00:08:55,050 --> 00:08:57,720
So I could replace both my I of t's here

164
00:08:57,720 --> 00:09:01,110
by this N minus S of t form,

165
00:09:01,110 --> 00:09:04,770
and I would have a complete
differential equation

166
00:09:04,770 --> 00:09:08,910
or continuous time model
for the SIS process.

167
00:09:08,910 --> 00:09:11,820
So we're going to look into
how to solve this model

168
00:09:11,820 --> 00:09:13,860
and analyze this model,
simulate this model,

169
00:09:13,860 --> 00:09:15,870
a little further in the subsequent video.

170
00:09:15,870 --> 00:09:17,673
So I'll see you all in the next one.