Hi. So we came up with a simplified model of flu transmission that breaks the population into three classes, susceptibles, infecteds and recovers. There are two parameters that govern how people are moving from these different compartments or classes into the others. In the optional video, we showed how we could construct an algorithm to simulate this model in a computer. So let's just recap what that algorithm is, and of course in order for us to put these into a computer, we need to pick parameter values and we're going to need to pick parameter values. For those parameters are two parameters. B which regulates what is the rate or probability that infections will occur or all the potential contacts that Ss could have with Is given by the multiplication and R governs the rate at which infected individuals are going to recover. So if we pick those parameters then we'll need to also pick initial conditions. So we start with an N susceptible individuals. Maybe one person infected and zero recovered, then we can start simulating. We of course need at least someone to be infected to begin with otherwise there's not going to be anything happening and we can look at two and see how the actually it's true. So then each day what we're going to do it, we're going to simulate one day at a time. We're going to start with the number of susceptible, infected and recovers at the beginning of the day. At the end of the day we should update the numbers based on how many people we expect to get infected. So we argued that the number of people that would get infected should be a function of the potential contacts between Ss and Is and then times the rate at which those contexts lead to a successful infection. So it would be B times S times I. So then we would need to update the number of Is and the number of Ss, increase Is by B times S times I and decrease S by V times S times I. It's zero sum game. If someone becomes infected then they're not susceptible anymore. In terms of the recovery we expect the number of recoveries to occur every day to be R times I. If R is measuring the daily rate of recovery per infected individual, so then we should update the number of Rs to what we started with the capital R plus lowercase r times I. Then also if someone recovers they're not infectious anymore. So we subtract that number from the I box. This is an algorithm that we can easily implement in a computer and that's what we have done. In the tool that we've shown, there's slightly different version but basically the same principles are governing what you're seeing in the web tool. To finalize then now when we have this model, we said we went to create a model to look at the impact of vaccination. Then we need, we're missing that box or a class or a group of people who were vaccinated. Those who were vaccinated before the beginning of the outbreak. So what do we need to do? So we need to add a new class and you who that accounts for those who were vaccinated. Then what happens to vaccinated? The what and how. So in an ideal situation the vaccine works and it's perfect. So that means if you're vaccinated you are not at risk of getting infected anymore. So in this case the vaccinated people would be pretty much like recovered in the model and basically the role that they're doing is simply decrease the pool of people who could actually get infected at the beginning of the simulation. So really there isn't much that we need to add to what we already have other than account for those vaccinated people and make sure that then we take them out from the people who can get infected. So in terms of the diagram our new diagram really is something where we would think about susceptibles before the beginning of the outbreak maybe going into being vaccinated in a different class, the now that class can not actually lead to infections. In this case this is how we would for instance we could update our model. Then using that new diagram we can update our algorithm that you're going to see is not that different from what we had before pick values for b and r and then now start with a population of n minus M people's susceptible where now capital M represents the vaccinated. So now rather than starting with everyone being susceptible, now I start with n minus m susceptible, M been vaccinated. Let's start with one infected and zero recovered. Then everything else will remain the same. The infected or the number of infections every day will be governed by how many potential successful contacts, which mean Ss and Is occur that actually transmit the virus. Then we'll update the Is and the Ss and then also taking into account how many people recover from the infection every day. So mark embodies the only places where the vaccinated people affected our algorithm, it's just in this case was basically only in the initial conditions. Everything else remains the same. So it was in some sense a nice effects because of the way that we set things up. Of course, we could make things more complicated, people getting vaccinated not only before the flu shows up but during the outbreak and many many other things. But for now I think that's enough for today. So one activity that we're going to ask you to do is, well now that you seen this, now that you have some better idea of what the different components of the tool that we showed you at the beginning are then why don't you go ahead and play with it and maybe pick some parameters for b and r and then some initial conditions, and then ask the question of how many people you wouldn't need to vaccinate in order for an epidemic to be prevented? Then think about what does that mean that no epidemic occurred. Then maybe play with different values of r to see how does that answer that you come up with might vary as a function of r as well as better as well. Great hope you were able to get some answers and more importantly to see how do in this case the number of vaccinated might affect the outcomes that you get in terms of propagation of the flu. It turns out that that's one really important question in public health. We've had a lot of people think about this for many many years and that's something that we think about not only for flu but for other diseases, how many people we need to vaccinate to prevent an epidemic or to prevent the propagation of certain diseases? For things that maybe you've recently been hearing about in the media like measles, that measles had gone away but now it's back. It's back because we're actually vaccinating slightly less than what we should be vaccinated to prevent measles outbreak. So we'd gone below that threshold of how many people should be vaccinated to prevent the transmission of measles. So that's actually an important question and an important problem. We've developed a very simple model you can play with, you can actually start getting some sense of why this is actually irrelevant, that's for some diseases that very transmissible. Pretty much we need almost everyone to be vaccinated or close to a 100 percent for those instances not propagate just because there's so transmissible and contagious.