[MUSIC] So far, we've been studying a simple SIR model where people get infected, they stay immune forever, and there's no replacement of individuals in the population. Under these conditions the amount of susceptibility in a population only as it goes in one direction, downwards. As a result, the effect of reproduction number also decreases over time. As I've mentioned before, this simple picture is a good approximation for infections that give long lasting immunity like measles. And where the timescale of the epidemic is sufficiently short, that we can ignore changes in the population over time. But in reality, these conditions may be violated by a range of factors. If we're modeling over longer time periods, then people who've gained immunity can grow old and die, only to be replaced by newborns who are fully susceptible. Also there are certain diseases to which immunity can wane over several years. Well, both of these processes can increase the susceptibility in the population. We can also counteract these trends by vaccination. This week we'll extend the SRL model to capture each of these processes in term. And you'll be doing more of your own modeling this week too. So have a look at the help sheet on our coding to make sure that you feel comfortable with the basic commands that you will use if you need. But don't worry if you run into difficulties, as these are an important part of learning how to write a model and to code in general. And at the end of each exercise, we will issue correct code to show you how the code should look. The important thing is that you attempt to do the coding yourself first, so that you can better understand and interpret the correct code when you receive it. So let's begin with population turnover. Starting from our basic SIR model, we want to adapt it to include births and deaths in the population. First, we'll include background mortality as a constant hazard coming out of each model compartment as everyone is subjected to this background mortality rate. Naturally, everyone who dies is lost from the population. But in special cases like a builder, we may want to construct a separate compartment for people who have died since these individuals may still contribute to transmission. But that's a fairly special case. Here we're going to assume that people who have died are no longer contributing to the infection, so there's no need to count them in any separate compartment. Note that just for simplicity, we're ignoring disease-induced mortality, and instead focusing here on deaths from all other causes. Now, you can guess that in this model, the population will decline over time and eventually vanish because people are being lost through death, but we don't have any births. We model births in a very simple way by including an influx into the S compartment. And note here that we're assuming that all newborns are susceptible. But if, for example, you wanted to model some maternal immunity or neonatal vaccination, you would have a certain proportion of births going into the R compartment as well. And we'll come back to that a little later on. Now, over to you. In this next activity, you will extend your basic SIR model to include births and deaths, according to these equations. You'll then use this model to explore the dynamics of an SIR system when you have a slowly changing population and when you have a rapidly changing one. As you'll see, the same disease can show very different dynamics in these two scenarios. [MUSIC]