We're now going to look at our second model of tipping points, and this is a model from epidemiology and it's known as the SIS model, for "susceptible, infected and then susceptible". That is, you're susceptible to some disease, then you get infected, and then after you get infected you're cured, but then you can become susceptible again if the disease is mutated in some way, like a flu virus. There's also something called the SIR model, where after you become infected then you're recovered, then there's no chance of getting the disease again. Alright, What we want to do with this model is show that it produces a tipping point. It's going to be a variable that comes out of our model, called the basic reproduction number. And the basic reproduction number, if it is bigger than one means that everybody is going to get this disease. If it's less than one means that no one will. So it's going to be a lot like, like our percolation model where we get this, you know, tipping point. R's are less than one, no spread of the disease; R's are bigger than one, somebody gets the disease. Now this model is pretty intuitive, but it's got a lot of notation. So to sort of build this up to, I'm gonna start with something simpler known as the pure diffusion model. So in the diffusion model, everybody just gets it. There's no, you know, sort of getting cured. So this thing of this is diffusion of information through a system or disease that everybody's just gonna get. Alright? So the diffusion method sorta works as follows. Let's suppose that there's some new disease called the wobblies. And we're gonna let W sub T be the number of people who got the wobblies at time T. Now, if there's N people total in our population, so this could be a community, or this could be an entire society, N minus WT is gonna be the number who don't have the wobblies. And so we can imagine that, like, tau, right, this variable tau, is just the transmission rate. So it's the likelihood that someone with the wobblies gives the wobblies to someone who doesn't have the wobblies, right? So it could be that you meet, you don't get it. And it could be that if two people meet, or one has it, and one doesn't, but they do get it. Tau is just the rate at which that occurs. So, if two people meet, what's the likelihood that one person would give it to the other? Well, remember, W is the number that have the wobblies, and N minus W is the number that don't. So, what you need is, you need one person to have the wobblies, and one person not to. So what's the probability that someone has the wobblies? Well, that's just W over N. That's the probability of someone having the wobblies. What's the probability of the other person not having the wobblies? Well, that's just N minus W over N. So if you want to think about, what's the probability of two people meeting where it could get transmitted? This is it. And then you just have to multiply that by tau. Right? Because that's the probability that if those two people do meet, then in fact, the disease moves from one person to the other. Now, instead of a disease, you can think of this as a new technology, or as a piece of information. It's the same model. Tau's just the probability that I tell you the piece of information. Right? Or that I tell you about the technology and you adopt it. Okay, so now we get this much fancier formula. This is the probability it's gonna move from one person to someone else, if two people happen to meet, and again tau is just the transmission rate. And the only thing that's different here is now I get these fancy t's here to represent that this is the rate at which it's going to go at time period t. Right, because the number of people that have the wobblies is going to change from period to period. Well, now I'm going to add one more thing, which is the contact rate. Because it depends on how often do people actually meet. So you can imagine a situation where people don't meet very often. Or you can imagine a situation where people meet a lot. Now often what counts as a meeting, right, could differ, right? So, if it's a disease, then meeting would have to be a physical meeting. If we're talking about a piece of information, that meeting could be on the Internet, or over the phone. So if there's a contact rate c, what you can imagine is, that I've got that formula, right? Tau, which is the transmission rate, and then we've got the people who have it, which is W over N. And the people who don't have it, which is N minus W of N. So this is the probability someone has. Right? Someone not, someone doesn't have. And this is, sort of the rates, if they meet. Well, then I'm gonna multiply this by the probability that two people would meet, right? So, c's the contact rate. And if there's N people, I've got to multiply this whole thing times N, times c. 'Cause c's the rate, so N times c is gonna be, sort of, the number of meetings. Right? So, this is gonna be the number of meetings. This is gonna be the rate at which it transfers. And this is gonna be the probability the meeting is between one person who has it, and one person who doesn't. And now, I get this incredibly complicated formula, [laugh] looks like this. The number that have it, at time t+1, is the number at time t, plus the number of new people. Right? That's what I mean by, a lot of notations. Even though there's are a lot of notation here, nothing's complicated. So, I said to you, how many people have the wobblies at time t+1? You'd say, well, the number that have it at time t, plus the number of people who get it in the next period. That's it. It's get-, gets a little bit complicated to write down. So, what epidemiologists do, is then they use this equation trying to say, okay, what does this tell us about the spread of the disease, or the spread of diffusion, in this case, of disease, right? Well, it's gonna look like this, it's gonna start out really low. And then it's going to go really fast. And then it's gonna get really slow again, but why is that? Well, let's look at this equation a little bit. And I want you to focus on this part right here, this W over N, times N minus W over N. When W's small, then what you get is something that is just supposed to be one person. You get 1 over N times N-1 over N, right? So that's not very big, right cause that's just gonna be basically 1 over N, right? That's gonna be approximately equal to 1 over N. But when I get in the middle, and W equals say N over 2, like half the population have it, then I get N over 2 divided by N, times N over 2 divided by N, and now those N's cancel and I get 1/4. So what's gonna happen is that early on, since not many people have it, there's not many people who can spread it. In the middle, half the people have it, half the people don't. So, it's going to spread really fast. Later on, when W is almost equal to N, a lot of people have it, but there's very few people who don't have it. So, there's not that many people who it can spread to. So that's why early on, there's few people who have it. And so it can't spread very fast. Later on, there's few people who don't have it, and so it can't spread very fast. So you get an S-shaped curve. So, again, this is nonlinear. Here's the point though, no tipping point. [laugh] Yeah, right. This isn't a tipping point, this is just diffusion. Now, you might look at this graph and say, oh, boy, here's a tip right here, where it suddenly speeds up. And here's another tip. Nothing tips. All this is, is just the natural diffusion of a process. Diffusion starts out slow, it then goes fast, it accelerates. So this is an acceleration, but it's not a tip. And then it decelerates because there's very few people who don't have it. So just because you see a kink, doesn't mean there's a tip. So kink, right, does not equal tip. It could just be an acceleration. So if we look at something like that Facebook graph, right, the number of Facebook users, ooh, we see this kink here. We see, boy, it really accelerated here, this acceleration. That does not mean there's a tip. It just means that Facebook was diffusing. And the thing is, you, you could say, well yeah, but Facebook is still going up, up, up, up, up. Well, at some point, there's gonna be no more people to get Facebook, right? It's gonna diffuse to the whole society, and it's gonna flatten out. Right? So it's a pure diffusion process. All right, so now we've got the diffusion model down, let's move on to what I call the SIS model.
