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?
