We're back, and now we're talking a little bit more about diffusion.
And, we looked at calculating the size of a giant component in a random graph as,
as one, heuristic, approximation to the extent of a diffusion and the
probability. An now what we're going to look at is
another model, which is known as the SIS model.
And it's a simple model of diffusion. It's highly stylized.
It's not directly applicable to a lot of things.
But it's a useful model because it gives us some basic intuitions of how things
work. We can bring in degree distributions.
We can vary them. We can get comparative statics out.
So, a lot of the insights that'll come out of this model will be quite useful.
Even if the model is, is a little bit too simple and stark, to actually match a lot
of things. So, what's the, the structure of this
model. So, this model comes out of a, a, a model
that's been used and, and studied a lot in epidemiology.
There were a set of models by Bailey in the 1970s, that you know sort of defined
a lot of these things. So, what does S stand for?
Susceptible. So, basically we've got susceptible.
And then I, infected, and then susceptible again.
So, the idea here is you can recover. So you, you can catch something you
become infected so, you're, you're susceptible, you could catch it.
You get it, then you recover and this is something which you catch overtime, so it
might me something that I erase a, a virus from my computer, I'm susceptible
again. And I can catch em when new one comes, I
catch it again, I erase it and so forth. So, I go back and forth from this process
of you know, realizing that I have a virus, getting rid of it, and then
catching it again later in time. Okay, so the, the key thing is, is nodes
are going to move back and forth over time and you know, you can think of this
as, as I might be changing my mind over time.
and various things, but we'll look at the basics of it.
[COUGH] So, nodes are in these two states, infected or susceptible.
The probability that you get infected in the simplest version of this model, is
proportional to the number of infected neighbors, with some rate, let's say v
great than 0. and we'll add in a spontaneous epsilon so
that you can catch things, as in the bass model.
And then you get well in any period with, at some rate, delta.
So, this is like the bass model, except here you can actually reverse yourself
and get well, and that's going to happen with a, a rate delta greater than 0.
And let's let rho be the percent of the population that's infected any point in
time. So, what then I want to do, is make
predictions about rho as a function of the network and, and these other
parameters of the model, okay. So, what we're going to do, is start with
a simple version where all the in, individuals players agents in this
society, the node are going to mix with even probabilities.
So, you random meet one person per unit of time, and that's just going to give us
a large Markov chain, and we can do calculations on that.
And the steady state distribution is just going to be one in which the, the change
of this infection parameter rho with respect to time is 0.
So, the simplest version of this model is one where there's not actually an
explicit network structure, it's just a completely random process.
And this looks a lot like the bass model in its basic form, and then we'll bring
in network structure on top of this in just a few minutes.
So, let's start with the simplest version.
so what's the change in the infected population over time?
Well, you can only become infected if you're not infected yet, so you're
susceptible. So, this is the susceptible size of the
population. Then you catch it from a given individual
with v times rho. And epsilon is this spontaneous rate.
So, this looks lot like the bass model did.
Basically the same function form is the bass model.
But we're also going to do this. We're going to have people re, recovering
over time. And so we'll look for steady state so,
out of those who are infected. They recover at some rate delta.
And so, all put together, you're gaining new infection at this rate, and losing
infection at this rate. And in order for this to be in steady
state, these two things are going to have to balance.
The new infection rate's going to have to balance against the, the number of people
who are recovering for a period of time. And so if you solve this equation then
you get an expression for what rho looks like as a function of the rest of the
parameters. in, in this setting.
Okay. So, there we've got a simple equation and
a simple solution. And now we're looking for a steady state
and what we've done is we've enriched this bass model essentially.
To have a recovery part which then allows for a steady state distribution, which is
going to be different from everybody becoming infected.
So, if we if we let epsilon go to 0 and then we solve this basically we end up
with two solutions. One is that nobody's infected, nobody
gets infected. And then the other one, the more
interesting one is that rho is equal to 1 minus delta over v,
So, if this turns out to be greater than 0.
So, if, if delta is bigger than v, basically what does that mean?
That means that the people recover so fast that this thing will never really
take root. But if delta's smaller than v, so you can
catch things faster then you can recover from them, then rho can be positive.
And basically the smaller delta is and larger v is, the larger rho is going to
be. So, rho is increasing in v and decreasing
delta. and it only has this positive, solution
as long as, delta is less than v. Right.
So, so we have this simple solution and, you know, very, very simple steady state
here. So this now hasn't brought in the network
structure at all. So, this is like the bass model, but now
with the recovery rate. And we end up with a solution here, which
makes sense as, as, as long as, delta is less than v.
Okay. So, we've, we've got, an infection at
least when, delta is less than v, where it's going to stay at some level for low
recovery rates, which can lead to large infections.
and, what we haven't brought in yet is where's the network, right?
So, this is uniformly at random interaction, we're missing the
heterogeneity degree, we're missing local patterns.
And what we're going to do, is, is we're going to start by just bringing this in.
And bringing in local patterns and explicit network structures is going to
be a lot more difficult without doing simulation.
And so what we'll start with is, is just taking a look at how we might bring in
the fact that some people are going to have more interactions per unit time than
other individuals, okay. And so exploring the, the dependence of
this one, the degree of distribution is what we're going to do is start by having
a random matching process. Where each different individual might
have a different degree, and their degree is just going to tell you how many
matches per unit of time they're going to have.
Okay? And what we're going to keep track of now
is the fraction of nodes not just overall which are infected, but also as a
function of a degree. So, it might be the people that have
three interactions per unit of time have a higher infection rate than people who
have two interactions per unit of time and so forth, okay.
And another thing we're going to keep track of is,
If I'm meeting a random person in the population.
So, I, each period, I'm meeting some number of people, my di.
So, say this is four, I'm going to meet four people per unit of time.
what's the chance that any one of those four people is infected?
And theta's going to be that fraction, okay?
Now what's going to be important, is the fraction of people over all that might
have something in their population, is not going to be the same as the fraction
of people I meet. Because I'm more likely to meet people
who are meeting lots of people. So, some people have lots of
interactions. Those are the people I'm more likely to
meet. Those are also the people who are more
likely to be infected. Okay?
So, so that's the process that's going on.
Okay, so how are we going to deal with this?
let's deal with it, again this is this random matching process.
So, let's let P of d be our degree distribution.
So, this is the fraction of nodes that have degree d.
And when I think about what's the probability that I'm going to meet
somebody, in terms of this random process, where we're all randomly
matched. I'm much more likely to meet somebody
with high degree. And in particular given that the high
degree people if somebody has ten meetings per unit of time they're
going to have to meet ten people. Somebody that has five meetings per unit
of time is only going to meet five people.
The person with ten is going to be twice as likely to be met by somebody as the
person with five meetings. So, the people with more meetings are
going to be easier to find, and the likelihood of meeting a node of degree d
is going to be directly proportional. To their degree compared to the average
degree. So we look at the fraction of those
people in the population but we have to re-weight that by what's their relative
degree compared to the average degree in a population.
Because that's going to determine how many meetings they have and how easy it
is to find them, when you're bumping into people in the population.
Okay, so that's an important thing, and that's a critical thing for understanding
contagion processes more generally. We've already seen it once earlier in the
course, and you know, this is important in, in trying to understand that fact of
the, the operation of this SIS model. Okay.
So, if we want to calculate the fraction of infected people I'm likely to meet,
well, this is the likelihood that I'm going to meet somebody of degree d.
This is how likely they are to be infected, and then we're just going to
sum across d's, and that gives us a theta.
Okay, so we have an expression now for theta, and we're going to have you solve
for this expression and see what, what it gives us.
So, this is the fraction of infected neighbors, random partners.
If we look at steady states, steady states are going to tell us for each
difference degree, we have to have the change over time of the infection rate of
different degrees all going, being 0. So, what we end up with is the infection
rate for each different type being 0. And what we know, what is those infection
rates look like for different types, and so we can then set that equal to 0.
What does that infection rate look like for different types?
Well, so, the, the fraction of people of type d that are currently susceptible is
1 minus rho of d. The chance that they meet infected
individuals. Number of infected individuals per unit
of time time they're likely to meet, is theta times d.
And the chance they get affected by one of them is, is v.
So, this is the, the rate at which there going to gain infections, and then they
get better at a rate delta. So they recover.
They, they get rid of their computer virus.
they recover from whatever cold or they had.
so, so here we've got a situation where we've got an expression now that involved
our theta and we can solve this for each rho of d.
This has to be equal to 0. So, a steady state sets us equal to 0.
And basically that tells us that the rho of d for any given d, is going to be
proportional to this expression over here, lambda theta d over lambda theta d
plus 1. Where lambda is v over delta.
So, whats the relative rate at which you get infected compared to the rate at
which you get better? so generally for this infection to take
hold, this expression is going to be something which is bigger than 1.
And so here we've got something which sort of captures the, well, it doesn't
necessarily have to be bigger than 1 in instances where we have two different
degree distributions. But so this, this is going to be a very
useful parameter, and this happens because when we solve through its only
the relative rates of v and d that matter, and not the absolute rate.
So, if we double both of these, and the expression is already solved we'll get
the same solution, so, These are scaling together and it's, it's
relative rates at which you get infected and recover that matter, not the absolute
rates. Okay.
So, solving this equation we've got this expression now, for rho of d.
We can plug that back in to our expression for theta.
And then we end up with theta equals a function of theta, which now depends only
on the primitives of what's the degree distribution.
what's the infection relative infection of compared to recovery rate in this.
and those are the only expressions that n are in.
So, we've got expected degrees and so forth, and now we've gotta solve this for
theta. Okay, so we've got theta as a function of
h of theta which is where h of theta is, this big expression over here.
so we've got some function of theta. and basically we're, we're going to look
for a fixed point of this expression. Okay?
So, now we've boiled everything down, this model to a simple equation that one
can solve. Okay.
So, solving this equation we, we know that theta has, is some expression which
is a function of the Ps and so forth. generally this isn't going to be easy to
solve. We can solve it by simulation, it's a
nonlinear equation. It depends on the expected degrees and
the full degree distribution and the lambdas.
but what we can do, is do some fairly easy comparative statics.
And say, let's suppose that we increase the probability of higher degree nodes,
what's that going to do to theta? what happens as we change the expected
degree, just what happens as we change lambda.
So, we can do different comparative statics and begin to understand how these
things work. Okay, so in particular we want to ask
what H of theta looks like, and how it depends on these different.
Parameters, the degree distribution the expected degree and so forth, so solving
this would depend on making sense out of these.
So, let me just go through how this looks in terms of the solutions.
So, basically when we go back, to this we're trying to, to look for thetas,
which solve this equation. We want theta equal to H of theta.
H of theta is going to be an increasing in concave function, and we'll talk about
that in little bit. So, H of theta is increasing in theta and
its concave function. And in particular when we look at looking
for thetas and H of theta, that intersect.
One possibility is is theta 0, so if nobody's infected, then nobody gets
infected in this model. And so one steady state is no infection,
and once you eradicate something, it just doesn't come back.
Another possible steady state is a positive one, but it's going to depend on
what this H function looks like. So, it could be that this H function is
so shallow, that there is no positive solution.
It could be the steeper function and that concavity will give us a positive
solution to it. So, understanding whether this works, in
terms of having a, a nonzero steady state in this, is going to depend on the
properties of this H function. Which depend on the dis, degree
distribution the relative infection rate and so forth.
So, even though there's a little bit of technicality here, the intuition's are
fairly simple. basically more degree higher degree
nodes, more interactions, higher infection rates in terms of the v
compared to delta. Are going to lead to higher H's, which
are going to lead to higher steady states, and more infection in a
population, okay? So, we're going to take a look at that in
some detail next. that'll be our next look in, in, in more
detail at the diffusion process. So, we'll solve out the SIS model for,
for explicit expressions in different settings.
And then look at what we can say about comparative statics.