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?