Hi. I'm Justin Lessler. In this module, I'll be talking about Basic Epidemic Dynamics in the Terrible Law that drives it. The idea of a terrible law comes from a quote from Robert Lowe to the British House of Commons when he was talking about a rinderpest outbreak or cattle plague outbreak in 1866. He said, "If we do not get the disease under by the middle of April, and my under he means under control, prepare yourself for a calamity beyond all calculation. Wait, and you will see the averages, which had been thousands, grow to tens of thousands, for there is no reason why the same terrible law of increase which have prevailed hitherto should not prevail henceforth". So he had this idea that since the epidemic was growing exponentially, it would keep on growing exponentially for the foreseeable future. But, was the law of epidemic grief really that terrible? William Farr, who was the general statistician for the UK at the time, felt differently. He had made a prediction based on observing many epidemics about how epidemics moved and how they came up gradually and then eventually came down. He predicted that the rinderpest outbreak would begin the declines shortly after the time that Robert Lowe was making his statement. In fact, while his prediction wasn't perfect, Farr was correct. So the purpose of this module is to give you a basic familiarity with the types of ideas that allow us to understand how epidemics move through populations and how diseases move through individuals. In particular, our objectives in this module are to introduce basic measures of disease transmissibility, to find the reproductive number and how it varies over the course of an epidemic, explain the dynamic quantities that characterize the natural history of an infectious disease, and relate measures of transmissibility and disease progression to epidemic dynamics, how epidemic curves look, and control. For some recommended readings and resources, you might consider consulting Chapter 6 of Infectious Disease Epidemiology Theory and Practice, the third edition by Conrad Nelson. It gives an overview of many of the basic concepts of epidemic dynamics. There's a nice discussion of R-naught an NPR called "No Seriously, How contagious is Ebola". It's also quite good. There are a couple of papers; one on opportunities and challenging and modeling emerging infectious diseases, another on understanding what factors make infectious disease outbreaks controllable, and finally a rather old paper on infectiousness of communicable diseases in households. These give a deeper discussion of many of the concepts that will be introduced in this module, and also a glossary of terms for infectious disease modeling is available online, that can help you understand and look up some of the terms I'll be using and give you a different perspective in their definition. So let's start out by talking about measures of transmissibility. In particular, we'll start talking about attack rates. We'll talk about a few measures of transmission. First we'll talk about attack rates and three flavors of the attack rate: the basic attack rate, a household secondary attack rate, and the susceptible exposed attack rate. As we go through it, it should become clear why we might have different measures of attack rates. Next we'll talk about the reproductive number which is a measure of disease transmission, and we'll talk about the basic reproductive number and the general reproductive number and it will become clear to you what those terms mean as we go on. The raw attack rate or just the attack rate, is simply a measure of those at risk versus those effective. It takes no account to differences in the who's exposed, who is susceptible to infection, that is not immune or any other factors. So the attack rate is simply the number of cases that occur in the epidemic divided by the population size. In the case here at the right, three out of five people were infected, so the attack rate was equal to 60 percent. So the thing about the attack rate is we don't really know who is exposed to the disease, but it's a good measure of what the impact of the disease was on a population. It's not necessarily a great measure of how transmissible the disease was. A better measure of that would be the household secondary attack rate. This is the proportion of cases infected in a household due to household exposure, and it's usually calculated assuming a single index case. So what we mean by that is that, it's calculated based on the idea. On the right, we show an illustration of this. The infection has been introduced in the household infected a single individual shown as the central individual in this diagram. There are four arrows showing potential possible exposure to the four other household members. Two of those arrows point to people shown in red and are shaded in, those people were actually infected. Two of those arrows are not shaded pointing to people shown in green, those people were not infected by the disease. So this is showing of the potential exposures are the people who were exposed and could be potentially infected, but proportion of those people were actually infected, and this is a more direct measure of infectiousness because all of the people involved in the denominator were at risk. As I said, we usually calculate this assuming a single index case or a single introduction into the household. More sophisticated methods do exist and it serves as a proxy for infectiousness dependent on exposure. So the household secondary attack rate or SAR, is the number of people who were infected by the end of the epidemic and the household minus one for that index case, divided by the total number of people in that household, once again minus one for that index case. In this case, that's two out of four or 50 percent. So the household secondary attack rate assumes that the index case had one single exposure to everyone in the household and that a certain proportion were infected and a certain proportion weren't infected. But we know that that's not the only way that a disease can move from a household. For instance, I could be infected and I could infect my brother, who could then infect my sister. So two other people were infected in the household, but I didn't directly infect both of them. So an even more nuanced measure of transmissibility is the susceptible exposed attack rate. This is the proportion of a potential infection events that actually occur, and the denominator here is those at risk for infection in each generation of transmission. So I illustrate this on the bottom and I'm going to walk you through this. So at the first time after the disease is introduced in the household, there's one person infected, that's the index case, and the four other people in the household are still at risk for infection. So then at the next time and this next row of people time equals two, one of those four people has been infected. So there were four potential transmission events, are shown by the arrows, and one that occurred, shown by the red arrow. So now on this next generation, we have one infected person shown in red, three people at risk, and then that original case shown in gray who is no longer at risk for being infected. Now, of those three people at risk or those three potential infection events, once again shown by the arrows, one occurs. So one out of those three people is infected, shown by this red arrow. So now we have one infected individual in the household, two people at risk, and there's two people who are immune because they were previously infected. Then in the final generation, that last infected person who has two other people shown in green they might effect, doesn't infect anyone. That is zero out of two people at risk who are infected. So overall, we have two infection events that actually occurred, one in that first generation and one in that second generation, and nine potential infection events, four in that first generation, third in that second generation, and two in that third generation. So that gives us a susceptible exposed attack rate of two out of nine or 22 percent. So these attack rates are the most basic measures of transmission and transmissibility. They're simply a measure of how many people who could have been infected and how many people were infected. More precise definitions of the risk set can make attack rates more directly capture transmissibility. So if we have a full population as to risk set, we get a raw attack rate. It just gives us a measure of how things happened across the entire population. If we'd look at non-indexed household members and households where the disease is spread, that gives us a household secondary attack rate, which is a better measure of people who might actually have been exposed to disease whether or not they got sick. Then if we have a denominator of possible infection events in each generation, we can calculate the susceptible exposed attack rate which is a very direct measure of how likely a person is to infect someone when that infection had a possibility of occurring. What should be clear to you is that, as we go down this hierarchy from raw attack rate to susceptible exposed attack rate, we're needing more and more information in each set. So not all of these attack rates can be realistically estimated in all circumstances. For instance, if we only have data on an entire population, we can only calculate the raw attack rate. If we don't have any temporal information about when people were infected in households, we can only calculate the household secondary attack rate. For the susceptible exposed attack rate, we need to have household information and know when people actually got sick. For an exercise, consider a population of 20 people living in five equally sized households, you observe eight cases spread across three of these households. So calculate the overall attack rate. Calculate the household secondary attack rate. What assumptions did you need to calculate this household secondary attack rate? What additional information would you need to calculate the susceptible exposed attack rate?
