Welcome back.
Welcome back for a third week
of Modeling in Complex Systems.
Last week when I left you in the videos,
I gave a very vague definition
of the first type of modeling
that we started doing using compartmental models.
The definition I gave was something like,
you know, compartmental models
are reduction of complex systems
in schematics of boxes and arrows
where boxes represent subpopulations of your system,
of the parts of your system,
where approximation of homogeneity holds.
And this approximation of homogeneity, really,
I didn't really define.
So I thought it would be nice to revisit in this video
in light of all the good discussions
that we have this week, so.
Lemme share a little bit of slides here.
So I really wanna revisit
this idea of compartmental modeling.
And maybe a clearer definition
would be really that compartments are boxes
where we put indistinguishable parts of the system.
So it's not just that we can approximate
that their homogeneous,
really these approximation of homogeneity
is the idealization that we make to get to the model.
But within the model,
parts that are put in the same boxes
are literally indistinguishable.
So if we are curious about an ecosystem,
for example, with wolves eating rabbits
and rabbits reproducing,
we might have two boxes, one for wolves, one for rabbits.
And we're saying that all wolves are exactly the same.
They behave in the same way.
They have no individual features,
they have no memory of their past dates.
And same with rabbits.
So once we do this compression, really,
what we're trying to do is to think about
the different parts of our system, which is heterogeneous,
sort of by nature of complex system.
And then we try to reduce this heterogeneity into boxes
where there's no heterogeneity within the boxes,
only across, so discrete types of parts.
And then once we have that,
what we try and do is reduce all the interactions,
all the flow of energy or information
within that system as arrows between the boxes.
We can have different types of arrows.
But really they're just a way to formalize what we mean
by interaction and flow of parts across the system.
The example that we discussed at great length
was inspired by, you know, current events in the world.
And we thought about outbreaks of COVID
on a university campus, right?
And then we saw the classic disease models which say,
well, we have this complex human population,
different types of people and all that,
but at the very minimum,
what we care about in terms of building a disease model
is their state regarding this infectious disease.
So in the classic case, we would've three boxes,
so Box S, I, and R to distinguish our population
in terms of whether people are susceptible, meaning healthy,
and potentially could be infected in the future.
Currently infected and infectious.
So this I box.
And then recovered.
And we simply have two arrows,
one from S to I, and one from I to R.
So those two arrows, really the way to think about this is,
I mean, the way I like to think about it
is that boxes are simply backed up
that get filled with water representing a certain density
or fraction of the total system.
And then mechanisms, the arrows are simply pipes
through which water can flow, in this case.
So you have a lot of water initially
in the susceptible bathtub or compartment,
which flows at some rate that might change in time.
But this pipe has a certain width
through which water flows from the S to I bathtub
and that models infections in the system.
And then the I, we get a little bit of water
that grows in the I bathtub
and another pipe that flows with a different width
into the R bathtub, representing recovered individual.
The reason why I use this idea of water
will become clearer in the next videos for this week
when we start thinking about the mathematical description
that we can get from these systems of box and arrows.
And then we'll see that they're not necessarily ideal
at dealing with discrete quantities.
So it's often better to think of it
as a density of water.
Now the one thing I'll say at this point
is that we have three boxes that we've labeled,
you know, susceptible, infected,
or infectious and recovered.
And we have pipes that we label infection events
or recovery events.
And I discussed this
with the entire class throughout the week
but I wanted to put it on the record in one video.
My number one modeling tip is to be very careful
with the labels that we put on boxes and on arrows.
Putting label is super useful to remind ourselves
of, okay, this compartment,
this box represent infected individual.
And then we have an idea
of what this means in terms of mechanisms
and it help us think about the mathematical description.
But when comes the time to discuss the result of the model,
we have to be careful about not putting too much weight
on the words that we chose
to use to label these boxes and arrows, right?
So if I take, you know, some other model,
we might think of some arrows as,
you know, brain complexity,
or some boxes as individuals
with more technological savviness, right?
And then if our conclusions read something like,
imagine headlines read something like,
with increases in brain complexity
or increases in technological savviness comes X, Y and Z.
Well that's not quite right, right?
Technological sevens or brain complexities
are what our model is trying to describe,
which really it's with this decomposition
and all the idealization that we've made
that comes X, Y, and Z.
And so sometimes putting things on,
putting things with realistic label
can be a little misleading.
'Cause we forget that we're really talking about
our model at this point and not reality, right?
So seeing it by the dip,
the representation is not the real world
and sometimes labels can lead us to forget that.
And that's my number one tip
about modeling to be careful about the labels.
Oka, but for getting to this now,
we have, you know, three boxes, S, I, and R,
and two arrows labeled infection and recoveries.
And this little description
is the simplest possible disease model
for something like COVID on campus
where R would be, you know, people that have recovered,
that are either dead or immune.
In any case, they're not gonna participate,
they're not gonna get infected anymore.
So they're not gonna spread the disease further.
And then we discussed ways to generalize this model.
Really, the way we did that was to try and list
all the possible idealization that had to be made
by the modelers who came up with the S I R model
to go from a complex system, a human population,
and an infectious pathogen spreading therein
to this simple, like three boxes, two arrow system.
One idealization is that infectious individuals
are all the same.
In terms of COVID, we know some people are asymptomatic,
some are not.
So one improvement that we made
was to introduce two boxes for two different types
of infectious individuals,
whether they are aware, IA, or unaware, IU,
of their infectious state, right?
So untested, asymptomatic individual would be an IU
and you can imagine that those individuals contribute more
to the flow of individual from S to IA or IU
than IA individuals.
Meaning that IU individual might be more contagious
'cause they're not taking any precautions,
they don't even know they're sick.
Another important generalizations that we made
was to not only consider recovered individuals,
that all individuals previously infectious.
One reason why we might wanna model disease
is to be able to test different interventions.
While in which case, you know, it really matters
whether people recovered and are now immune,
recovered and might have long-term consequences from COVID,
or if they're dead, right?
So disease individuals,
what we wanna be able to evaluate
is the cost of this disease burden.
So then we split all of that up into a disease compartment,
a recovered compartment,
and a long-lasting consequence compartment.
In terms of dynamics of the spreading,
they're all the same,
meaning they're not gonna get sick anymore,
they're not gonna infected anyone else.
But in terms of evaluating different outcomes,
it really makes a big difference.
So again, here we have a more complex model,
a lot more arrows.
And mathematically, that can be annoying.
It's gonna be harder to analyze.
But if the purpose of our model is to evaluate the quality
of different outcomes with different interventions,
well this is essential, right?
The final generalization that I wanna talk about
that was made during the discussion group
is also to consider
more different types of susceptible individuals.
So we're not all as susceptible on campus, right?
So if you're a remote student,
so you're in town,
you might get sick through some other venue
but you're not physically on campus,
you're a low risk individual.
And we might call that SL.
If you're on campus,
but you take all your precautions,
you might be medium risk, SM.
If you're on campus and you have pre-existing conditions
and a lot of contact,
then you might be a high risk individual, SH.
And as you can see, like it's easy, if you want.
There are so many idealizations that we made
to go from the real world
to a system of three boxes and two arrows.
It's easy to think of generalizations to make.
The key part is in making the right generalization
that don't introduce too much more,
too much additional complexity in the model,
but are required for the purpose of the model, right?
So if we wanna consider the role of having remote student,
then we have to distinguish on these three different types
or at least two types of susceptible individual.
It's all about the balance here.
So to wrap it up,
before we start going from boxes and arrows
to mathematical description,
I'll give yet another definition of compartmental model.
One way to think about it
is that it's a decomposition representation
or compression of a real system
into different types of homogeneous parts.
Really, the key thing is we wanna acknowledge
that per nature of complex system,
we have a collection of parts,
multiple interconnected parts, and they're not all the same.
They're often heterogeneous.
But we wanna ask what is the minimum,
minimum number of of different types that I can distinguish.
And then within those types, then I'll start ignoring
all individual heterogeneity for convenience.
And then if you can get to this minimum description,
then you have a good compartmental model
that's worth going through a mathematical description
and solving.
And then that's what we will be up to
over the next few videos.
So hopefully that clears it up.
Really, this is the type of of modeling tool
that only becomes obvious the more you do it.