1
00:00:00,720 --> 00:00:02,580
Welcome back in this short video

2
00:00:02,580 --> 00:00:03,870
for modeling complex systems.

3
00:00:03,870 --> 00:00:07,020
I'd like to share my own personal recipe

4
00:00:07,020 --> 00:00:09,840
for modeling questions
that I'm interested in.

5
00:00:09,840 --> 00:00:10,770
So this could be,

6
00:00:10,770 --> 00:00:12,870
I had a discussion with a domain expert

7
00:00:12,870 --> 00:00:15,990
or I read something and I'm
really interested in modeling

8
00:00:15,990 --> 00:00:18,123
some behavior that I'm curious about.

9
00:00:19,260 --> 00:00:20,790
And I'll always go through

10
00:00:20,790 --> 00:00:25,790
the same sort of mental
recipe to go from a question

11
00:00:26,580 --> 00:00:31,380
into a an actual model that
I can work out mathematically

12
00:00:31,380 --> 00:00:32,673
or using software.

13
00:00:35,190 --> 00:00:36,873
And this is a simple recipe.

14
00:00:37,860 --> 00:00:39,490
It's nothing fancy

15
00:00:40,740 --> 00:00:42,930
But I do hope that it is something

16
00:00:42,930 --> 00:00:45,360
that academically can be passed down

17
00:00:45,360 --> 00:00:48,000
from generation to generation.

18
00:00:48,000 --> 00:00:49,680
So the first step is just thinking about

19
00:00:49,680 --> 00:00:51,780
what are your ingredients?

20
00:00:51,780 --> 00:00:53,760
So the one thing we wanna really focus on

21
00:00:53,760 --> 00:00:55,170
is the saying that the whole is more

22
00:00:55,170 --> 00:00:56,220
than the sum of the parts.

23
00:00:56,220 --> 00:00:58,680
And we want to define all
of those words, right?

24
00:00:58,680 --> 00:01:02,850
So what is the whole, in what
space does this model occur?

25
00:01:02,850 --> 00:01:04,710
What are the parts that
we're interested in

26
00:01:04,710 --> 00:01:05,790
are we talking about?

27
00:01:05,790 --> 00:01:08,610
Neurons, molecules, people?

28
00:01:08,610 --> 00:01:09,930
And then what is the somehow,

29
00:01:09,930 --> 00:01:11,940
how do these things interact, right?

30
00:01:11,940 --> 00:01:14,730
So the first part, the whole in the parts

31
00:01:14,730 --> 00:01:15,960
are really the ingredients

32
00:01:15,960 --> 00:01:19,860
and these interactions
will be the preparation

33
00:01:19,860 --> 00:01:21,960
of the recipe, if you will.

34
00:01:21,960 --> 00:01:24,210
So when I think about this,

35
00:01:24,210 --> 00:01:26,190
I always start first as like

36
00:01:26,190 --> 00:01:29,550
in what world is my model taking place?

37
00:01:29,550 --> 00:01:33,870
And that involves both space
and time, but space first.

38
00:01:33,870 --> 00:01:37,173
So space could be discrete.

39
00:01:39,180 --> 00:01:41,553
So that could be, for example, a network,

40
00:01:44,610 --> 00:01:46,680
or the little grid that
I have on here, right?

41
00:01:46,680 --> 00:01:50,520
If I'm putting the parts of my
system in some configuration

42
00:01:50,520 --> 00:01:53,700
but they're only allowed to be
on intersection of this grid,

43
00:01:53,700 --> 00:01:54,840
well, that's a discrete space

44
00:01:54,840 --> 00:01:58,653
because they're not allowed
to take any continuous value.

45
00:02:01,410 --> 00:02:03,520
So space can also be continuous

46
00:02:04,590 --> 00:02:09,240
and that would be just our
usual Euclidean space mostly,

47
00:02:09,240 --> 00:02:11,470
it could be some other hyperbolic space

48
00:02:12,450 --> 00:02:14,250
and we would've question, for example,

49
00:02:14,250 --> 00:02:16,173
is it 2D or 3D?

50
00:02:17,040 --> 00:02:17,973
That might matter.

51
00:02:20,970 --> 00:02:24,093
And then we also wanna talk about time.

52
00:02:24,960 --> 00:02:29,070
So similarly, time in a
model can be discreet.

53
00:02:29,070 --> 00:02:31,950
So if we're thinking about
something like game theory,

54
00:02:31,950 --> 00:02:34,080
the prisoner dilemma, which
we're gonna talk about

55
00:02:34,080 --> 00:02:34,953
at some point.

56
00:02:36,270 --> 00:02:38,409
Well, often in game theory

57
00:02:38,409 --> 00:02:40,740
what really matters is just the rounds.

58
00:02:40,740 --> 00:02:42,480
How many rounds of this
game have you played

59
00:02:42,480 --> 00:02:43,410
with some opponent?

60
00:02:43,410 --> 00:02:45,060
So time is discreet in the sense

61
00:02:45,060 --> 00:02:46,890
that it's round after round.

62
00:02:46,890 --> 00:02:51,000
If we're thinking about genetic
mutations in a population,

63
00:02:51,000 --> 00:02:52,830
we might not care about the actual clock,

64
00:02:52,830 --> 00:02:55,050
what time is it in the day,

65
00:02:55,050 --> 00:02:57,810
but just in terms of
generations, one after the other.

66
00:02:57,810 --> 00:03:02,810
So that would be discrete
times or rounds or generations.

67
00:03:05,970 --> 00:03:07,470
And a lot of model, of course,

68
00:03:07,470 --> 00:03:10,260
will take place in continuous time,

69
00:03:10,260 --> 00:03:13,443
which is just normal clock, if you will.

70
00:03:14,340 --> 00:03:17,910
So that's part of what
I try to think about.

71
00:03:17,910 --> 00:03:19,260
Then when I think about the whole,

72
00:03:19,260 --> 00:03:20,883
there's also other questions.

73
00:03:21,780 --> 00:03:23,640
The main one that I try to think about

74
00:03:23,640 --> 00:03:28,640
is whether the system is
open or closed, right?

75
00:03:31,740 --> 00:03:34,350
So am I modeling my own little universe

76
00:03:34,350 --> 00:03:37,740
where there's nothing outside
coming into my universe

77
00:03:37,740 --> 00:03:41,400
or do I have some sort of
connection to another world

78
00:03:41,400 --> 00:03:43,590
that could be a reservoir of other parts,

79
00:03:43,590 --> 00:03:45,090
or setting a temperature,

80
00:03:45,090 --> 00:03:50,090
or forcing some behavior in
the system from the outside?

81
00:03:52,740 --> 00:03:54,360
That's gonna become a little clearer

82
00:03:54,360 --> 00:03:57,870
when we start sketching
cartoons of our models,

83
00:03:57,870 --> 00:04:00,960
meaning that that closed system

84
00:04:00,960 --> 00:04:02,430
won't have anything coming in

85
00:04:02,430 --> 00:04:03,960
where you don't know
where it's coming from

86
00:04:03,960 --> 00:04:05,250
and you're not modeling that.

87
00:04:05,250 --> 00:04:07,710
Whereas open system will
have those interactions

88
00:04:07,710 --> 00:04:08,643
from the outside.

89
00:04:10,260 --> 00:04:12,030
But these are, let me just highlight

90
00:04:12,030 --> 00:04:14,850
that these are really
question one, two, and three

91
00:04:14,850 --> 00:04:16,263
that I tried to answer.

92
00:04:18,030 --> 00:04:20,790
And this takes me to the
next important element

93
00:04:20,790 --> 00:04:21,930
of this recipe,

94
00:04:21,930 --> 00:04:25,353
all other ingredients will
involve the parts of the system.

95
00:04:28,020 --> 00:04:31,736
So we have, for different systems,

96
00:04:31,736 --> 00:04:33,960
this might look a little different,

97
00:04:33,960 --> 00:04:36,120
but roughly three or
four important questions

98
00:04:36,120 --> 00:04:37,170
that we wanna ask.

99
00:04:37,170 --> 00:04:39,180
The first one will sound dumb,

100
00:04:39,180 --> 00:04:41,343
but it's simply, what are the parts?

101
00:04:44,730 --> 00:04:46,350
And often it's gonna be obvious

102
00:04:46,350 --> 00:04:50,970
if we're interested in a disease model,

103
00:04:50,970 --> 00:04:54,270
of course, at the level
of a human population,

104
00:04:54,270 --> 00:04:57,090
we're gonna use people
as the individual parts.

105
00:04:57,090 --> 00:04:58,920
Sometimes it might not be obvious

106
00:04:58,920 --> 00:05:01,440
if we only have data as
their certain spatial scale,

107
00:05:01,440 --> 00:05:02,460
for example.

108
00:05:02,460 --> 00:05:05,520
Some disease models will
involve different towns

109
00:05:05,520 --> 00:05:07,110
and towns are gonna be the...

110
00:05:07,110 --> 00:05:09,660
Or communities or
neighborhoods are gonna be

111
00:05:09,660 --> 00:05:11,103
the parts of our system.

112
00:05:12,120 --> 00:05:13,260
Similarly, you could think

113
00:05:13,260 --> 00:05:17,430
about modeling individual
human cells or organs.

114
00:05:17,430 --> 00:05:19,890
If you're trying to look to
think about, for example,

115
00:05:19,890 --> 00:05:23,220
how a drug might diffuse in a human body,

116
00:05:23,220 --> 00:05:26,673
it's really a questions
of spatial scale here.

117
00:05:27,900 --> 00:05:31,050
So I'll just put that in parenthesis.

118
00:05:31,050 --> 00:05:32,950
This is what this is trying to get at.

119
00:05:36,000 --> 00:05:38,250
The most important question,

120
00:05:38,250 --> 00:05:39,240
or I'll probably say that

121
00:05:39,240 --> 00:05:40,590
for all of those questions, actually,

122
00:05:40,590 --> 00:05:42,780
they're all like defining the parts

123
00:05:42,780 --> 00:05:45,780
and doing it well is
where the art of modeling

124
00:05:45,780 --> 00:05:46,710
really comes into play.

125
00:05:46,710 --> 00:05:50,280
But one other important questions is,

126
00:05:50,280 --> 00:05:52,803
what distinguishes the part?

127
00:05:57,180 --> 00:06:02,180
And basically here I want
you to think about the parts

128
00:06:02,250 --> 00:06:04,500
as items we wanna put away, right?

129
00:06:04,500 --> 00:06:09,500
Like balls in a kid's room
and/or toys in a kid's room.

130
00:06:09,930 --> 00:06:12,870
And then we wanna put the
toys in relevant boxes

131
00:06:12,870 --> 00:06:15,020
so that they're easy
to find in the future.

132
00:06:15,960 --> 00:06:17,913
So basically the question is,

133
00:06:18,840 --> 00:06:20,940
how do I distinguish the different parts

134
00:06:20,940 --> 00:06:23,100
so that once they're in their boxes,

135
00:06:23,100 --> 00:06:25,770
everyone in the boxes sort
of has the same behavior

136
00:06:25,770 --> 00:06:27,510
or serve the same purpose?

137
00:06:27,510 --> 00:06:29,880
For example, again, in a disease model

138
00:06:29,880 --> 00:06:31,560
we might wanna have two boxes,

139
00:06:31,560 --> 00:06:34,440
healthy and sick people, right?

140
00:06:34,440 --> 00:06:35,760
In a forest fire model,

141
00:06:35,760 --> 00:06:39,900
we might wanna have healthy
trees and burning trees.

142
00:06:39,900 --> 00:06:44,160
If we're interested in something
about how forests regrow,

143
00:06:44,160 --> 00:06:47,640
we might wanna have old healthy
trees, young healthy trees,

144
00:06:47,640 --> 00:06:49,530
and burning trees.

145
00:06:49,530 --> 00:06:52,050
So that's the kind of
questions we wanna ask.

146
00:06:52,050 --> 00:06:54,750
How do I distinguish the
different parts of my system?

147
00:06:56,370 --> 00:06:59,760
I'm gonna do a little 2.5 here

148
00:06:59,760 --> 00:07:03,723
because we're gonna
ignore debt for a while.

149
00:07:06,330 --> 00:07:10,073
2.5 would be, what do the parts know?

150
00:07:18,300 --> 00:07:21,330
Do the parts have memory
about their past states?

151
00:07:21,330 --> 00:07:24,237
Do they know what else is
occurring in the system?

152
00:07:24,237 --> 00:07:26,490
So in really simple models,

153
00:07:26,490 --> 00:07:28,500
often the answer to that
is gonna be nothing.

154
00:07:28,500 --> 00:07:30,570
They know nothing, they have no memory,

155
00:07:30,570 --> 00:07:32,970
and we're still gonna be able
to answer great questions.

156
00:07:32,970 --> 00:07:37,470
Eventually, after six weeks or so,

157
00:07:37,470 --> 00:07:39,450
maybe starting with the six week,

158
00:07:39,450 --> 00:07:42,300
we're gonna start including
some knowledge about the system

159
00:07:42,300 --> 00:07:45,000
and then parts will be
able to react accordingly.

160
00:07:45,000 --> 00:07:46,586
I'm putting it as a 2.5

161
00:07:46,586 --> 00:07:49,110
because what they know in their memory

162
00:07:49,110 --> 00:07:51,900
is also one way to
distinguish different parts

163
00:07:51,900 --> 00:07:53,400
of your system.

164
00:07:53,400 --> 00:07:55,950
And then the final question,

165
00:07:55,950 --> 00:07:59,340
which I'm not gonna say
a lot about here is,

166
00:07:59,340 --> 00:08:00,543
how do they interact?

167
00:08:05,640 --> 00:08:07,350
There's not much to say here

168
00:08:07,350 --> 00:08:10,740
just because it's fairly
system specific, right?

169
00:08:10,740 --> 00:08:12,750
But once you said,

170
00:08:12,750 --> 00:08:14,220
I have a disease model,

171
00:08:14,220 --> 00:08:18,120
it takes place in a discreet space

172
00:08:18,120 --> 00:08:19,440
where everything's interconnected,

173
00:08:19,440 --> 00:08:20,730
a fully connected network,

174
00:08:20,730 --> 00:08:22,380
and it occurs in continuous time

175
00:08:22,380 --> 00:08:24,150
and it's a closed population.

176
00:08:24,150 --> 00:08:27,750
My parts are different
people that I distinguish

177
00:08:27,750 --> 00:08:29,640
based on whether they're healthy or sick,

178
00:08:29,640 --> 00:08:30,630
how do they interact?

179
00:08:30,630 --> 00:08:33,360
Sick people infect the healthy people,

180
00:08:33,360 --> 00:08:35,490
sick people recover sometimes.

181
00:08:35,490 --> 00:08:36,360
That's my model.

182
00:08:36,360 --> 00:08:41,360
I just like walk you through
the most classic disease model

183
00:08:41,520 --> 00:08:43,020
that is still used

184
00:08:43,020 --> 00:08:46,740
in a lot of media right now
to try and forecast COVID-19.

185
00:08:46,740 --> 00:08:49,170
So this recipe is fairly simple

186
00:08:49,170 --> 00:08:51,540
but that's how I think about these systems

187
00:08:51,540 --> 00:08:56,540
in a way that helps me
formalize my intuition.

188
00:08:56,610 --> 00:08:59,790
In the next video, we'll
do some basic examples

189
00:08:59,790 --> 00:09:03,450
and I'll introduce one type of models

190
00:09:03,450 --> 00:09:05,460
that we're gonna use
throughout the semesters

191
00:09:05,460 --> 00:09:07,440
which are called compartmental modeling.

192
00:09:07,440 --> 00:09:10,140
And it's really based
under this question two,

193
00:09:10,140 --> 00:09:11,700
what distinguishes the parts.

194
00:09:11,700 --> 00:09:13,350
So, I'll see you in the next one.