Welcome back in this short video
for modeling complex systems.
I'd like to share my own personal recipe
for modeling questions that I'm interested in.
So this could be,
I had a discussion with a domain expert
or I read something and I'm really interested in modeling
some behavior that I'm curious about.
And I'll always go through
the same sort of mental recipe to go from a question
into a an actual model that I can work out mathematically
or using software.
And this is a simple recipe.
It's nothing fancy
But I do hope that it is something
that academically can be passed down
from generation to generation.
So the first step is just thinking about
what are your ingredients?
So the one thing we wanna really focus on
is the saying that the whole is more
than the sum of the parts.
And we want to define all of those words, right?
So what is the whole, in what space does this model occur?
What are the parts that we're interested in
are we talking about?
Neurons, molecules, people?
And then what is the somehow,
how do these things interact, right?
So the first part, the whole in the parts
are really the ingredients
and these interactions will be the preparation
of the recipe, if you will.
So when I think about this,
I always start first as like
in what world is my model taking place?
And that involves both space and time, but space first.
So space could be discrete.
So that could be, for example, a network,
or the little grid that I have on here, right?
If I'm putting the parts of my system in some configuration
but they're only allowed to be on intersection of this grid,
well, that's a discrete space
because they're not allowed to take any continuous value.
So space can also be continuous
and that would be just our usual Euclidean space mostly,
it could be some other hyperbolic space
and we would've question, for example,
is it 2D or 3D?
That might matter.
And then we also wanna talk about time.
So similarly, time in a model can be discreet.
So if we're thinking about something like game theory,
the prisoner dilemma, which we're gonna talk about
at some point.
Well, often in game theory
what really matters is just the rounds.
How many rounds of this game have you played
with some opponent?
So time is discreet in the sense
that it's round after round.
If we're thinking about genetic mutations in a population,
we might not care about the actual clock,
what time is it in the day,
but just in terms of generations, one after the other.
So that would be discrete times or rounds or generations.
And a lot of model, of course,
will take place in continuous time,
which is just normal clock, if you will.
So that's part of what I try to think about.
Then when I think about the whole,
there's also other questions.
The main one that I try to think about
is whether the system is open or closed, right?
So am I modeling my own little universe
where there's nothing outside coming into my universe
or do I have some sort of connection to another world
that could be a reservoir of other parts,
or setting a temperature,
or forcing some behavior in the system from the outside?
That's gonna become a little clearer
when we start sketching cartoons of our models,
meaning that that closed system
won't have anything coming in
where you don't know where it's coming from
and you're not modeling that.
Whereas open system will have those interactions
from the outside.
But these are, let me just highlight
that these are really question one, two, and three
that I tried to answer.
And this takes me to the next important element
of this recipe,
all other ingredients will involve the parts of the system.
So we have, for different systems,
this might look a little different,
but roughly three or four important questions
that we wanna ask.
The first one will sound dumb,
but it's simply, what are the parts?
And often it's gonna be obvious
if we're interested in a disease model,
of course, at the level of a human population,
we're gonna use people as the individual parts.
Sometimes it might not be obvious
if we only have data as their certain spatial scale,
for example.
Some disease models will involve different towns
and towns are gonna be the...
Or communities or neighborhoods are gonna be
the parts of our system.
Similarly, you could think
about modeling individual human cells or organs.
If you're trying to look to think about, for example,
how a drug might diffuse in a human body,
it's really a questions of spatial scale here.
So I'll just put that in parenthesis.
This is what this is trying to get at.
The most important question,
or I'll probably say that
for all of those questions, actually,
they're all like defining the parts
and doing it well is where the art of modeling
really comes into play.
But one other important questions is,
what distinguishes the part?
And basically here I want you to think about the parts
as items we wanna put away, right?
Like balls in a kid's room and/or toys in a kid's room.
And then we wanna put the toys in relevant boxes
so that they're easy to find in the future.
So basically the question is,
how do I distinguish the different parts
so that once they're in their boxes,
everyone in the boxes sort of has the same behavior
or serve the same purpose?
For example, again, in a disease model
we might wanna have two boxes,
healthy and sick people, right?
In a forest fire model,
we might wanna have healthy trees and burning trees.
If we're interested in something about how forests regrow,
we might wanna have old healthy trees, young healthy trees,
and burning trees.
So that's the kind of questions we wanna ask.
How do I distinguish the different parts of my system?
I'm gonna do a little 2.5 here
because we're gonna ignore debt for a while.
2.5 would be, what do the parts know?
Do the parts have memory about their past states?
Do they know what else is occurring in the system?
So in really simple models,
often the answer to that is gonna be nothing.
They know nothing, they have no memory,
and we're still gonna be able to answer great questions.
Eventually, after six weeks or so,
maybe starting with the six week,
we're gonna start including some knowledge about the system
and then parts will be able to react accordingly.
I'm putting it as a 2.5
because what they know in their memory
is also one way to distinguish different parts
of your system.
And then the final question,
which I'm not gonna say a lot about here is,
how do they interact?
There's not much to say here
just because it's fairly system specific, right?
But once you said,
I have a disease model,
it takes place in a discreet space
where everything's interconnected,
a fully connected network,
and it occurs in continuous time
and it's a closed population.
My parts are different people that I distinguish
based on whether they're healthy or sick,
how do they interact?
Sick people infect the healthy people,
sick people recover sometimes.
That's my model.
I just like walk you through the most classic disease model
that is still used
in a lot of media right now to try and forecast COVID-19.
So this recipe is fairly simple
but that's how I think about these systems
in a way that helps me formalize my intuition.
In the next video, we'll do some basic examples
and I'll introduce one type of models
that we're gonna use throughout the semesters
which are called compartmental modeling.
And it's really based under this question two,
what distinguishes the parts.
So, I'll see you in the next one.