Hi folks. So, let's talk a little bit more about

getting solutions in the SIS model. So, as we had talked about, in terms of

finding the steady state, we have an equation which relates the fraction of

people that you're going to randomly meet who are infected.

To an expression which involves the degree distribution and parts of that

degree distribution, as well as parameters indicating the relative

frequency of infection compared to recovery.

So what we need to do, is see what this H looks like.

And then figure out what the steady state solution looks like.

So again, as we pointed out, H is going to be a function where, if we're

finding we want to find thetas that equal H of theta then basically if we've got a

situation where H prime at 0 is less than 1 and it's a concave function.

There's going to be no steady state and otherwise, we'll find a positive steady

state. So let's take a look at this function in

a little more detail. So we want to take derivatives of this

with respect to theta to figure out what its shape looks like.

And if we take the first derivative of this with respect to theta that's quite

easy. And in fact we'll find that this

expression is greater than 0 and therefore we we end up with a, an

increasing function. So you can take the derivative here.

And verify that this is the expression you get, and indeed, what we end up with

is a positive function. and then if we take the second derivative

of this function then we get a, an expression, which is less then 0, so here

I'm not going to explicitly solve for the derivative.

You can go through and, and take those derivatives.

it's relatively straight forward, these are just polynomial functions.

Take the derivatives, the first derivative is positive, second derivative

is negative. and so here, we've got a function which

is going to be increasing and strictly concave, and that's going to tell us that

the picture that I was showing you before is an accurate picture.

And then the question is, where is this going to hit?

and have an intersection. So are we in a situation where if H prime

at 0 is bigger than 1, then initially, it's going above the 45 degree line

again, it's given as strictly concave. Either we're going to have a steady state

all the way up at 1, so we'll end up hitting up here, or it'll cross somewhere

and have a, a non-zero steady state. Or if H prime 0 is less than 1, then

basically, there's no possibility of sustaining a, an infection, the only

theta solution's going to be 0. And that's going to be in a situation

basically where the degree distribution is putting weight on very low degrees,

and the lambda's fairly low, so you don't have much infection that's going to be a

situation where H prime has a fairly low derivative.

So if we look at H prime at 0, we can plug in 0 for theta and then see what

this thing looks like. And if we calculate H prime at 0, it

looks out to be lambda e of d squared over e of d, so it's looking at the

relative expectation of the square of the degree compared to the expectation of the

degree, and weighting that by lambda, where we recall lambda's looking at the

relative infection rate compared to the recovery rate.

So we have those expressions and so the theorem then we get that the conditions

for a steady state process of of the SIS model to have a non-zero steady state.

Is going to be if and only if, this land is large enough, and what it needs to be

larger than is e of d compared to e of d squared.

Okay, so you need the infection recovery rate to be high enough relative to the

average degree divided by a second moment of, roughly think of variance there, so

large enough lambda is, is going to give us a steady state that's non-zero.

Now the interesting thing here is increasing the variance, if you do a

mean-preserving spread. So you increase the variance.

That makes it easier to satisfy this equation, right?

So in a situation where we keep E-d constant and increase E of d squared,

then we're going to be in a situation where basically, what we're doing is

spreading out the degrees. But that makes higher degree nodes, which

are going to have. serve as hubs and be conduits for

infection and that aids in, in the spread of, of the infection.

And allows us to have a steady state that's non-zero.

So, if we think about this condition, then we can plug in what we know for

various different models. And for regular network, what does this

turn out to be? Well in a regular network, everybody has

the same degree. So the expected degree is just whatever

that degree is. the expected degree squared is just the

expected degree of squared. So, in this case for regular network, E

of d squared, is just equal to E of d. Everybody has the same squared.

So, in that case then you just need lambda to be bigger than 1 over the

expectation. So, in that case, the larger the expected

degree, the easier it is to satisfy this, so that, that makes sense, and, and also

the larger lambda obviously the easier it is to sustain a positive steady state.

For an Erdős–Rényi random network if you work through what E of d is and E of d

squared are for, a Posone random network, then you end up with a situation where

the e of d squared is just equal to e of d times 1 plus e of d, so in that, in

that model, then we end up with in this case, lambda with 1 over 1 plus the

expanded degree. If you work in a power-law network.

so if we have, say for instance we work with one where the density function looks

like c times d to the minus gamma. Then what we end up with is if you do a

calculation say, integrate that and, and look for the variance e of d squared

actually becomes infinite. And if that becomes infinite, then this

whole expression becomes 0, and so we end up with lambda greater than 0.

And so, it's you, you basically always have a non-zero steady state.

And what's happening here is, in a power-law network, at least in the, in

sort of the limit, if you have a very large network, you're going to have very,

very large degree nodes. They're going to interact and, and always

become infected, and carry the infection through the society.

So, in that setting you end up putting weight on the tails.

And sufficient weight on the tails that you always sustained an, an infection.

Now, if you, you, you , if you do a power-law where you actually truncate the

distribution and have some maximum degree, then you won't quite find this.

But you'll find that that that expression converges to 0 as you let that, whatever

the maximum degree you allow in the society, to go to infinity.

So, so in the limit you always have a non-zero steady state in that model.

And so, basically what we find is the, the presence of hub kinds of nodes helps

substantially in sustaining non-zero steady states.

So the idea here, is these high degree nodes are more prone to infection.

They serve as conduits. Higher variance allows more such nodes,

and that enables infection, and we see that directly in the theorem.

So this is one of the kinds of insights that comes out of the SIS model.

Which is a useful insight and, and made explicit in this particular model.

And it also then allows us to compare degree distributions, showing that if you

have the same mean but you're increasing the e of d squared, then it's easier to

satisfy these conditions. Okay, so, so that shows some insights

that we get out of the SIS model. next will take a little more, a close

look. So what this did, is allow us to know

when it is that we get a non-zero steady-state.

We can also ask questions about how large that steady state is and, and whether we

can do comparative statics in that. And that's not going to be exactly the

same kind of answer as just when there exists one, which was just looking at

that derivative at 0. And more generally we can, we can go

through and try and solve this model, and say something about, what's the average

infection rate in the society?

5.6: Solving the SIS Model

Social and Economic Networks: Models and Analysis

Meet the Instructors