So now we understand the diffusion model, we want to move on to something called the SIS model. Now the SIS model looks pretty much exactly the same as the diffusion model, exact for now we allow for the possibility that after someone's become infected, they can move back into the susceptible fool, pool. So they can become recovered in a sense, but then they're also susceptible again. So this would be for something like the flu, where after you're cured of the flu, the flu mutates and you can become infected again. It wouldn't work for something like measles where once you've had the measles, you're no longer gonna get the measles. So here's how our model looks. It looks exactly like our model before. There's some number of people that have it at time t+1, and that's the number they have it at time t, right, plus the people, the new people who get it, right. And this depends on somebody who has it meeting somebody who doesn't have it, right. And this is the transmission rate and this is the number of contacts. But all we do to this is we subtract off this minus aWt. Well, what is this? These are the people who become cured, right? Who no longer have the disease and they're gonna go back into this pile N. Maybe people who could get the disease anew. So I would think it's sort of like throwing in a rate of people sort of getting better. Well now here's where this model is sort of more interesting, 'cause in the diffusion model something gets spread. So it starts out slow, it goes faster, then it tails off. But here, what happens is, while people are getting sick, at some, at some other rate some rate "a", they are getting better. And if people get better faster than people get sick then what's gonna happen? The diseases ain't gonna spread. So this model's gonna produce a tipping point, so. Let's simplify things a little bit if we rearrange terms, what we get is an equation that looks like this. Now how did I get this equation? All I did was I pulled this Wt term out, right, and I crossed out some N's. Let me show you how that works, right. So up here I've got, let's just focus on this part, I've got N c tau times W over N times N minus W over N minus aW, okay? And so what I do is I say, well you know what, let's cross out this N with this N. That's fine, 'cause those N's go away. And then I've got a W here and a W here. So I'm gonna just pull out that W. And then I've got c tau. N minus W over N minus a. Alright? That's what I've got. And if I look over here, that's what I've gotta get. c tau N minus W over N minus a. So this is just a simple application. So when you run models is useful to be good at algebra. [laugh] If you can do lots of algebra, you can simplify things. Well, why do you want to simplify things? Here's why. Look, suppose we're on in the disease. So that means Wt is really small. So that means that N minus Wt over N is gonna be really close to 1, because basically a very small percentage of people in the population have the disease. So now if I look at this thing, I can say well you know, Wt plus 1 is really equal to Wt. Wt plus 1 is equal to Wt plus Wt times c tau minus a. So, this thing is gonna spread if c tau minus a is positive and it's not gonna spread if c tau minus a is negative. So this is gonna come down to: Is c tau minus a bigger than zero? Right? Or is c tau bigger than a? Or another way to write this is: Is c tau over a bigger than one? All right, and this leads to what's called the basic reproduction number. We let R0 equal c tau divided by a. If c tau divided by a is bigger than one, the disease spreads, right? Because that means that Wt plus one equals Wt plus something positive. Right? If R0's less than one, right, in other words, if c tau over a is less than one, then Wt plus one equals Wt plus something negative and the disease dies off. So this R0 is called the basic reproduction number and what it basically tells you is: Does the disease spread? But notice here, we've got at tip, right? R0 less than one, no disease spread. R0 bigger than one, disease spreads. So, let's take some real diseases. Diseases like measles, mumps, the flu. The R0s are fifteen, five and three. That's why these are real diseases. There might be a ton of diseases out there that have R0s less than one and we're never gonna hear about them. Why? Because they don't spread. Now again, for things like measles and the mumps, once you get them you don't fall back in the population. So, there's a model you use their called the SIR model, very similar to the SIS model. But for the flu, right, you fall back into the pool, so you can think of it like this SIS model. Now it's interesting here, from these basic reproduction numbers we can say, you know, that past the tipping point, the disease is gonna spread. But let's think about this figure. Why do we construct models? Well, bunch of reasons, right? But one reason is to design policies. You could say, how do we stop these things from spreading? Well, one obvious answer is vaccines. Well, then my question is, how many people do you have to vaccine [sic]? Well, turns out the model will tell us. So let's think about it this way, right? There's, let V be the percentage of people that you vaccinate. So there's this basic reproduction number R0. That's like the rate in which things spreads through the population. Well if I vaccinate some percentage of the population, that's just gonna to reduce, right, the basic reproduction number by that fraction. So if half the people were vaccinated, R0 is effectively divided by two. Right? And if 75 percent of the people are vaccinated, R0's gonna be divided in effect by four. Right? So you've only got one fourth of the population. So the question is, how many people do you have to vaccinate as a function of R0? Well, in some sense if we vaccinate like V people, it's like we've got a new r0. Right? Little r0, which is just the big R0 times the percentage of people in the population that aren't vaccinated. So what we can do is we can say, we want R0, big R0 times one minus V to equal one. Actually it's going to be less than one, right? Well we can multiply this out, we get R0 minus R0 V, right? Has gotta be less than one, right? So we can bring this over together so I'm going to get R0 minus one. We need to be less than R0 times V, so that means we need V to be bigger than one minus one over R0, right? So what we get is, we get this equation that says: This is how many people we need to vaccinate. So let's go back. Remember the measles were fifteen, the mumps were five. How many people do we need to vaccinate to prevent the measles from spreading? Well that's just one minus one over fifteen, which equals fourteen fifteenths. So we need to vaccinate fourteen fifteenths of the population against measles if we want the measles not to spread. For the mumps, right?, we need one over one-fifth. Right, we only need to vaccinate 80 percent of the population. Well here's why the model's so useful. Again, this model's isn't exact in working out all sort of things like networks, and changes in contract, contact structures, and different locations and stuff like that, but still, what this tells us is, which is really important, is depending on how variant the disease is, you have to change how many people you vaccinate in a pretty, you know, understandable way. Now the other interesting thing there's a tipping point on vaccines right? If we vaccinate 75% of the people, and we needed to vaccinate 80% of the people, but guess what? It's not gonna work. The 25% who don't get vaccinated are all gonna get the disease. But if we'd have vaccinated 81% of the people, then the 19% of the people who don't get vaccinated, they're still gonna be protected, because the disease isn't gonna spread. 'Kay, so the cool thing about this model is it's given us a tipping point, R0. It's also given us a policy, and this policy is interesting in the sense that like it's of no effect, really, vaccination has no effect really, other than the people you vaccinate, really, there's no population level effect until you get to the threshold. Once you pass the threshold, then that next person to get vaccinated, in a sense, right?, makes the whole rest of the society immune because the disease can't spread. So, step back a second. In a diffusion model, right, where we got that nice sort of S-shaped curve, there's no tip. In the SIS model, there's this R0, which is the tipping point, right? So this is value one. So there's no disease if R0's less than one. There's disease if R0's bigger than one. What we do by vaccinating people is in effect reduce R0. So that's the SIS model. In fact in the context of disease you can even think of it, though, in terms of the spread of information that we talked about. And it's an interesting model in the sense that it does also generate a tipping point. Another non-linear model, and it's a model where we get this, you know, threshold phenomena. Less than R0, no, R0 less than 1: no spread; R0 bigger than 1: everybody gets the disease. Okay. Not a happy thought. But let's move on. Thank you.
