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# R nought and vaccine coverage

## Video transcript

- I have the definition of R nought here and I'm just going to take some quick notes. It says the number of new cases that an existing case generates on average and it's over the infectious period in a susceptible population. In here, what they're kind of referring to is non-vaccinated population. The way to think about this is for me anyway, is to kind of put it into math terms. If you have let's say, I'm just going to write these down here. Let's say existing cases here. Existing cases. And you have new cases. You literally can just take the two and just divide them by each other. You could say, new divided by existing is going to be my R nought, equals R nought. Right? Let's just go through a few scenarios. Let's say R nought is less than one. If it's less than one, let's think of how that might play out. Let's see if two existing cases and you come back and you say, Well, how many new ones are there in my population? There's only one new person even though you used to have two and so, you say, okay, that's one divided by two equals .5 and that's less than one. And what does that mean exactly? It means that if you have less than one, overtime, you can see how basically this infection is going to die out. It's not going to spread. It's just going to slowly die out because fewer and fewer people are going to get it and in this case, maybe even just one generation down the road, there will be no one to potentially get this infection. These were always the nice ones because they're going to die out by themselves and that's always wonderful for humanity when an infection dies out. Another scenario could be an R nought of one. In this case, you might have two people with an infection and you follow them overtime and you say, well, how many new people got infected with this infection? and they say, actually we found two new people that got infected. So, we have two old people, two new people and so, two divided by two equals one and just in terms of keeping track of this infection overtime that means that basically, this is a stable infection. It's not spreading. The numbers are basically always two and it's also not dying out. What's the third scenario? The third one, of course, is going to be greater than one. So, it's less than one and equal to one. The third scenario is greater than one and you start out with two again, let's just say and you say, okay, how many new people got this infection? And this one, let's say, I'm just going to make this really, really contagious. Let's say a lot of people got this infection. Let's say eight new people and how many existing? Well, there were only two just like all the other examples. We have an R nought of four, which is greater than one. And so, what is really happening here? This is an infection that's spreading. This infection is the one that we're worried about. It's spreading and we want to either allow it to stabilize. One goal might be, hey, let's just get this at least stable, right? So that it's not infecting more and more people. Another goal might be, let's actually eradicate this disease. Let's go all the way down to this theoretical R nought of zero and what would R nought of zero, really, what does that mean? It means that this number new is zero. The numerator is zero and that means that there are no new cases and that would mean eradication. So, if you want to eradicate the disease like smallpox, for example, this is what you have to do. You have to figure out how to get all the way down to zero but for most diseases, you just want to stop the spread. You want to stop the spread. If you want to stop the spread, this is the goal that you have to think about. This is goal number one. Let's put this in the context of a real infection. Let's take the mumps virus. Mumps causes fevers and headaches and general lousy feelings but also, it swells up your cheeks because it affects your saliva gland and it makes you look a little bit like a chipmunk. Mumps virus is not a pleasant virus and this infection when studied has an R nought of about four. It's perfectly modeled with this picture right here with two going to eight. And fortunate for us, we have an MMR vaccine and that stands for Measles, Mumps and Rubella and the measles, mumps, rubella vaccine is given to a lot of folks and it's going to help us kind of think through how to stop the spread of mumps, for example. Thinking through this, if the goal is to stop the spread, what does that mean? It means that our goal R nought is going to be one in this case and what is our current R nought? What are we starting from? In this case, it's going to be four. And so, the proportion of people that are still going to get sick. I mean even with this wonderful goal, the proportion that are still going to get sick are one quarter and actually, I can maybe show them right here. Maybe these two will still get sick with mumps even if we're able to stop the spread because that's exactly what happened here, right? But the wonderful news and this is the good part about this is that you're able to protect this many people with this goal at least. You're able to protect this many people and that basically is reflective as one minus one over four. One minus the goal R nought over current R nought. This is the math way to represent this fraction. This represents this fraction right here and this represents this fraction right there. What proportion need vaccination? What proportion of our community, let's say, what proportion needs vaccine to at least stop the spread? That's the goal that we said we have, right? What proportion needs vaccine? We'd say well, it's just what we laid out. It would be one minus the goal R nought, which is one divided by the current R nought, which is four. We'd say, well, this is three quarters or 75%. In other words, if we are able to vaccinate 75% of people, meaning these folks. Six out of eight is 75%. Then we basically have changed the way that mumps looks. It no longer looks like an R nought of four. It looks like our goal R nought, which is one. Now, I'm going to grab another example just to make sure we have a clear handle on how this works. Measles has an R nought of 18. Put into words, that means for every one existing case, we see 18 new cases over the infectious period usually a few weeks in a non-vaccinated population. If that's the case, how many people, what proportion needs vaccine to stop this spread? We say, okay, well that means one minus, our goal is to stop the spread, so one divided by, we said the R nought is 18. This is the proportion that we need to vaccinate and that works out to about 94%. Holy cow, that's much higher than what we saw with mumps. Let's do one more example. Let's do diphtheria. Kind of hard to spell but D-I-P-T-H, diphtheria has an R nought of let's say, about six and these values, six, 18, I said mumps was four, really depends on which study you're looking up but most of these are presented as ranges but in case, if we accept the number six then we say, okay, it's one minus one because that's the goal R nought is one divided by six and that equals 83%. Basically, the higher the R nought, the better your vaccine coverage has to be in order to prevent an outbreak from happening and of course, we're only talking about one way to protect folks and that's through vaccine. The other way to protect folks, of course, is to try to limit the number of new cases by saying, well, let's just physically surround people and maybe put people in quarantine and having people separated so they can't actually catch infection like I'm doing here. Maybe doing that can also obviously, prevent new cases from happening. I don't want to imply that vaccine is the only way but of course, it's a very, very effective way of preventing diseases from spreading.