How do mathematicians model infectious disease outbreaks?

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Good evening. My name is allegorically, and I would like to welcome you all to the Oxford

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Mathematics Public Lecture series. Before we start, I want to express our gratitude to

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a sponsor. Extracts market execs. Markets are leading quantitative driven

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electronic market makers with office in London, Singapore and New York.

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Obviously, this is not our usual format. We are broadcasting from home

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and I very much hope that you are also watching from the comfort of your home and that you are all in

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good health. We live extraordinary times. This

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is indicated not only by the fact that half of the world population is in one form

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or another of lockdown, but also and maybe even more extraordinary.

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We are now hearing world leaders talking about mathematics. We hear them talking

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about curves, about exponential us and about models. Mathematics

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and mathematician are playing a key role in both understanding the current crisis,

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but also in looking for possible ways out of it.

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Any form of prediction requires a mother. But what is an epidemic, mother? Where does

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one start? What data do we need? And more importantly, how do we use

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models this evening? I'm very happy that Robin Thompson has

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agreed to talk to us about these issues. Robyn is currently

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a junior research fellow in mathematical epidemiology at Christchurch College, Oxford,

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and a member of the Mathematical Institute. Robyn has been modelling infectious diseases

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for several years and since early January he has been working on important questions

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about coronavirus forecasting and control. Since then, he has been

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extremely busy contributing to the national modelling effort

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and making appearances in media about topics including social distancing and covert nineteen

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outbreak risk. If you would like to ask a question, please send it in via social media

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and we will collate them and send out answers in the next couple of days. Thank you, Robyn.

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Thanks, Allan. And hello, everybody, welcome to this Mathematical Institute public lecture, which is coming like

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from my home here in Oxford. My name is Robin Thompson. I'm a junior research fellow

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at Christchurch, and I also work in the Mathematical Institute in the Wilson Centre of

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Mathematical Biology. My specialist research area is infectious disease outbreak modelling.

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And that's a topic, of course, that's been in news a lot recently because mathematical models are being used

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in real time to inform public health measures against the Kovik 19 outbreak.

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So I'm going to do now is when I talk to you about precisely how mathematicians go about modelling infectious

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disease outbreaks. So the natural place to start then is, well, what exactly is

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a mathematical model? Well, a mathematical model is a mathematical representation

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of a system that can be used to explore that system's behaviour.

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And the goal of real time infectious disease outbreak modelling is as follows. What we do

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is we look at data from an outbreak so far. So, for example, we might have data on the numbers

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of cases per day during the outbreak. Then what we do is we construct

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a mathematical model that represents the underlying epidemiology of the system. We then use

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simulations of the model forwards to predict what might be likely to happen in future.

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So, for example, how many cases we might be expected to see per day going forwards.

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What we can then do when we've got a model look can make a sensible forecast is we can then introduce control

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interventions into the model to look at how different control interventions might affect the numbers

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of cases that we might be likely to see in future.

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So during this, let me try to focus on two main questions. The first one is how exactly do

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we build a mathematical model of an infectious disease outbreak? And then the second question

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I'm going to answer as well. Once we've got a mathematical model, how can we use it to inform public

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health measures at different stages of an outbreak? So

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these are the kind of sub questions that I'm going to address. The first thing I'm going to dress as well. Is there a characteristic

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shape of an infectious disease outbreak? And if so, that's something that's interesting because mathematicians

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are very interested in shapes. Then going to talk about how we can build a very, very basic

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infectious disease outbreak model. I'll then talk about some epidemiological concepts

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that you might have heard about during the COVA 19 outbreak so far. For example,

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policymakers have been talking a lot about the basic reproduction number or Nort,

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and the concept of herd immunity has been in the news a lot. So I'll talk about those two ideas.

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Let's talk about five ways that the very basic infectious disease outbreak model that we're going to construct

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can be extended to make them more realistic. Then after that, I going to talk about

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how mathematical models can be used to inform public health measures. Well, I'm just go to talk about three different stages

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of an outbreak. So come mathematical models be used usefully early in an outbreak

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to inform public health measures, come off. My models be used when a major outbreak is ongoing.

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So, for example, in the scenario that we're in at the moment here in the U.K. and elsewhere around the

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world. And finally, I'll talk about how mathematical models can be used usefully at the end

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of an outbreak. So the first thing

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I want to point out is that most single wave infectious disease outbreaks

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tend to have a characteristic shape. And so here's one example of this, what you

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can see here is a data set of an outbreak of influenza in a boys boarding school in the north

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of England in the 70s. And what you can see is that the virus enters the school.

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The numbers of cases gradually starts to increase and it increases until there's quite a large

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number of cases. But it doesn't keep going until everyone in the school is infected.

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Instead, it slows down again. And the outbreak peaks. And then the numbers of cases comes back down

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again to near zero. And this kind of shape is characteristic of lots and

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lots of single wave outbreaks. So here's another example. This is the foot and mouth disease outbreak in

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the U.K. in 2001. The data here is slightly different. So it's not individual cases

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of disease, but rather it's the number of infected farms. But broadly, you see the same kind

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of shape. The virus enters the population. It takes off. It goes up to a kind of high level in

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which there are lots of farms that are infected. But then it peaks and comes back down again, back down to

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near zero. So, again, broadly, you can see the same kind of shape of an outbreak. The graphs are the same

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kind of rough shape. Same thing happens if we look at third examples.

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This third example is Plake in Mumbai at the beginning of the 20th century. Again,

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this is very slightly different data. This is data on the numbers of individuals that were dying during

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that plague outbreak. But again, you see a very similar dynamic in which

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the number of deaths increases, but it increases up some high level. And when it reaches

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that high level, it turns over and falls back down again to near zero. So, again, broadly, the same

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shape. His more recent example of this is Ebola in

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West Africa, the largest Ebola outbreak in history. And again, we see a very similar

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looking outbreak shape. And here's one example

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from the ongoing Kobe 19 outbreak. This is data from China, specifically

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data on the numbers of individuals currently infected in China each day throughout

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the outbreak. And what you can see, again, is that the number of cases gradually increased.

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It increased up some high level again. Then it turned over and then the outbreak began

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to fade out. One particularly interesting feature of that specific graph

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is that you can see just here. If you look in mid-February, there's a sudden

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increase within a single day in the number of infected people.

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And the reason for that sudden increase was that precisely what constitutes an infected case

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changed the definition of a case changed in China during the coffee nineteen outbreak.

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So before that time, in order to be counted in the data, you actually had to be tested

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and found to be carrying the novel Corona virus. But obviously, when case

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numbers are very high, it's very difficult to go and test everybody. And so what happened then was there was an

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assumption that because case numbers were high, if you develop symptoms that were consistent

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with infection by the novel coronavirus, then you were included in the data. And

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so suddenly, because of this change in definition, you no longer have to be actively tested and found to

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be carrying the virus. The numbers of cases shot up very suddenly.

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But despite that, broadly, you can see the same kind of shape in this case, the virus invades,

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it goes up some high level, then the number of cases turns over and falls back down again to near

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zero. And you can see that no matter what kind of outbreak data you're looking at in these graphs, whether

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you're looking at deaths or number of infected farms, as in the case of foot and mouth or number

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of individual cases like the flu example or like in the coffee 19 example.

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Outbreaks of the characteristic shape. And what we want to do is we want to recreate that shape using a mathematical

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model. So what we could do if we wanted to is we could make the simplest possible

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assumption about how an infectious disease spreads so simple as

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possible assumption would be to assume that every single case of disease leads on

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to some fixed number of new cases every day that say. So this is one

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example. So we're going to assume that each case gives rise to three new cases every day. Well, then at the beginning

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of the outbreak. So in day zero, you might just have one infected individual.

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On day one. That one infected individual will have gone and infected three other individuals.

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So on day one, he'd have three new cases. Ban, according to this very basic

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model, we assume that on day two, each of the three eight cases from day one

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will have generated three new infections each. So in other words, on day two, you've got nine new cases.

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And then on day three, each of those nine would cause three new infections, setoff 27 new infections

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on day three and so on and so on. You can imagine this would just carry on going.

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So the assumption of this model's this model's called a geometric progression. What you'd find if you looked for

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a formula describing the numbers of cases on DADT is you'd see that the formula was just equal

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to three to the power of T. That's the number of new cases on dates. Twenty equals zero.

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The number of cases is three to the power of zero, which is one. On day one, the number of new cases is three.

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The power of one, which is three. On day two, the number of new cases is three. The power of two,

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which is nine. And so on. So this is the simplest possible

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infectious disease outbreak model. Well, does this actually look anything like real world

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data? So what I've done here on this graph is I've plotted the data from the boys boarding

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school flu epidemic. They showed you a moment ago and the data plotted in

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black. So the black line is what we actually saw in that outbreak dataset.

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What I've done then is I've overlaid and read the results of this mathematical model. So a model just

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assumes that every individual infects three new individuals every day.

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And if we overlay that, what we can see is that actually the model does a pretty good job of representing the very

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early parts of the infectious disease outbreak. The red is matching the black very well

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early on in the outbreak. But later on in the outbreak, then the

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mathematical model isn't doing a good job at all. The numbers of cases in the model is shooting off up to infinity.

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Whereas if you look at the infectious disease outbreak data, the data like we just saw

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turns over and comes back down again to near zero. It's the model prediction isn't

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doing a particularly good job of capturing the data.

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So what we have to do next, then, is a very important aspect of mathematical modelling. What we do

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next is we have to refine the model. We can't just carry on with this model. We have to make it better,

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make sure that the model matches the data more closely. We have to add in more infectious disease

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epidemiology. So clearly, one of the issues with this

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very basic model is that we've missed out a huge amount of disease biology and in particular

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we've missed out the fact that diseases gradually run out of uninfected, susceptible

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individuals to infect. That's what we can do instead is we can

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consider what's called compartmental modelling. And the idea of compartmental modelling is that you don't just keep

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track of how many individuals are infected. You keep track of individuals with all different

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infection and symptoms statuses. So the simplest possible

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compartmental model you can develop is this model here. This is called the S eye model.

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And so in the ESSI model, what you do is you divide individuals according to whether they're susceptible to the disease,

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in which case they're in the S compartment. They're in the green circle on the left.

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Or an individual can be infected and generating new infections. And if an individual

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is infected, then they're in the red circle them in the eye compartments, they're in the red circle on the right.

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And as you simulate an infectious disease outbreak using this model, what you would see is you'd see individuals

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that are susceptible becoming infected. So in other words, you'd see susceptible individuals from the green circle

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transitioning over into the red circle.

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So this is the ESSI model I just described. We can write down equations for

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this particular model so it doesn't really matter if you know whether that the key

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point of this is the idea of compartmental modelling. It's not the precise equations. But for those of you that

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are interested in the equations, I'm going to show them. So what we have is we have two separate equations.

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One equation describes the rate of change of the number of susceptible individuals,

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and the other equation describes the rate of change of the number of infected individuals.

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So we write down two equations that look a little bit like this.

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So, like I say, it doesn't matter if you haven't met equations like this before. On the left hand side

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this term here, this DST by DTT. That just means the rate of change of the

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number of susceptible individuals. And this time here is some of negative.

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That's what we've got a minus sign just here on the right of the equation. And the reason that the rate of change

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of the number of sexual individuals is negative is because the number of susceptible individuals decreases

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during the outbreak as susceptible hosts become infected.

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Similarly, what we have at the bottom is we have an equation here for the rate of change of the number of infected

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individuals. And the number of infants, individuals is something that increases during

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the outbreak as infections happen. And so the rate of change of the number

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infected individuals is something that is positive. That's why there's no negative sign on the right hand side of this equation.

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And in particular, the overall rate of infection is given by this Bita S.I. This is the rate

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at which the number of susceptible individuals decreases and the rate at which the number of infected

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individuals increases. And that rate is proportional to both S

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and I, so the infection rate is proportional both to how many susceptible individuals we've got, but also

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how many infected individuals we've got. And the reason for that is that if there's a large number of susceptible

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individuals, then there are lots of people that are targets for the disease. There are lots of people that the disease

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could, in fact. And so we'd expect the infection rate to be high. Similarly,

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if there are lots of infectious individuals, then there are lots of individuals that can do the infecting. And so, again,

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if there are lots of infectious individuals, we'd expect the overall infection rates to be high.

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OK, so this is the model. So, like I said, yes. By the time the green term

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is just the rate of change of the number of susceptible individuals die by details, the rate of change

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of the number of infected individuals, and then we have an overall rate of new infections,

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which is given by BITA Times. S Times I. And that depends on exactly how

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many sceptical individuals we've got and how many infected individuals we've got.

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And the promise of BITA governs the rate of infection that governs the rates at which individuals become

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infected. OK, so what happens then if we apply this very basic

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outbreak model again to the real data? Well, in black, what we have here is we

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have the data from the boys boarding school flu epidemic in the 70s.

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And what we can do is we can plot the results of this. S I model on top of the data.

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So what you see here is that he does a better job with them before. So the red line matches the dates us. The red

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line is the model prediction that matches the data pretty closely, but it only

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does so for about half the outbreak. What happens, like we saw earlier on, is that the data comes back

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down again. The numbers of cases declines back down to near zero, whereas the model output,

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the red line just stays flat. So it doesn't capture the decline in the number of cases.

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So what we found is that the epidemic, according to our model, the red line, no longer grows that bound. It doesn't go off to

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infinity anymore. So that's a good thing. But like I said, the model doesn't capture

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the decline in the number of infected hosts. So, again, we need to do what we

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did before, which is we need to refine the model further to try and get the model to match the outbreak data

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even better. So what we gonna do is going to include even

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more disease biology here. So before we were looking at the ESSI model, which just accounts professor

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sets of individuals and infected individuals, what we're gonna do now is we're

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going to include another type of individual. And they're individuals that have recovered and become immune

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to the disease. And they're in this, ah, class that you can see in the bottom model where

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you have susceptible individuals when they get infected. They become infected individuals. And when

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they recover, when the infected individual recovers, they become a recovered individual. And in this

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all class and this new model is called the asylum model. OK.

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So here's the model. We can again write down a set of equations describing the rate

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at which individuals pass from one compartment into another. So this is exactly the same

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as the Akseli model. But we have one new feature, and that new feature is that we have this

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term here. This new item, not new item, is the rates

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of which infectious individuals recover and become immune to the disease.

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And that term is only proportional to the number of infected individuals there are. And the reason for that is that if there

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is a large number of infected individuals, then we'd expect to see a large number of recoveries

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in the near future. OK, so this is the ESSI all model. So, again, we've got three equations,

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this time this size for the rate of change of the number of healthy, susceptible individuals, rate of change

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of the number of infected individuals, and then rate of change of the number of recovered individuals.

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We can write down the rates of new infections, which is the pink term that depends on and on.

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And we can write right down the rates which infected individuals recover and become immune. That's

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this new I. So now we've got two parameters, we got BITA,

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which governs the rate of infection. And we have a new parameter, MEU, which governs

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the rate of recovery. The rate at which infectious individuals recover.

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So what happens now, then? If we fit this model to the data? So, again, here

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we have the boys boarding school flu epidemic data set in black. And we can plot

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the results of this new model. And what we see in red is the results of this new model.

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What we see is that the red line, the model output, matches the infectious disease outbreak data

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pretty closely. I think you'll agree that this actually the model's doing

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a pretty incredible job. Given how simple it was, it was pretty simple to construct that model. And

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the model only included two parameters. That was a parameter governing the infection rates, which was BITA.

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And there was a parameter governing the recovery rates, which was MUE. It was a very simple

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outbreak model. And the model output matches the data very well.

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So the import conclusion is that by refining best model. So we start with one model, we refined

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it twice and eventually we included recoveries. And that allowed us to capture the overall

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shape of an infectious disease outbreak.

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This basic asylum model is kind of the prototypical infectious disease epidemic model.

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It can capture the overall shape of an outbreak. It's a very simple model and involves many

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simplifying assumptions. So, for example, there's an assumption that an individual, as soon as they

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get infected, is infectious and starts infecting other individuals. That's clearly not something

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that's true. In reality, there's an assumption that everyone mixes with everyone else. There are lots of different

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assumptions of this very basic outbreak model that may not be realistic. But what

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I'm going to come back to later is how we can extend this basic infectious disease outbreak model

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to include additional realism. Even this simple model lives, we

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even this essi all model can be used to explore different epidemiological concepts. So that's

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why I'm going to do now. It's the first concept that I'm going to talk

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about is that if the basic reproduction number or zero. This is a quantity

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that policymakers have been talking about a lot during the ongoing nineteen

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outbreak. So what is the reproduction number? Zero zero zero is the number

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of cases of disease arising from each primary case. So

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in other words, it's a measure of if I contracts an infection. It's a measure

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of the number of people that I'm likely to go on. And in fact.

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You can calculate zero by simply taking the infection rate and multiplying it by

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how long an infected individual is infectious for. That's how you go about calculating the basic

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reproduction number. And diseases that have very different

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values, if the reproduction no generate outbreaks with very different shapes.

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So, for example, if you look in the bottom, writes what you can see, as you can see, the results

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of an outbreak with very high O0 in blue. So what you can see is that the disease

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sweeps through the population very quickly. You have large numbers of infections,

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whereas if all zero is lower, like the red, then you have a much lower

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peak and the disease sweeps through the population a lot more slowly.

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If all zero is down below one, well, that means remember, the Ausra is the number of infections caused

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by each infectious individual, if I was is below one. Then on average, each

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infected individual will go on and infect fewer than one. Other individuals.

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So in other words, I'm like, it's going to infect fewer than one person. And so then what you say

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is, you see the disease doesn't spread widely in the population. Then you see somebody looks

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a bit like the purple. It's actually very difficult even to see it in this graph here. But you put in your first infected host,

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all zero is less than one. And so the outbreak simply fades out. There's a purple line running

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along the x axis just here. OK.

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So another concept you'll have heard about during this outbreak is the concept of herd immunity.

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So what is herd immunity? Well, it's the resistance to the spread of a disease. The results,

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if a sufficiently high proportion of individuals in the population are in mean.

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So in other words, maybe you've got a disease which has been around before and it's infected lots

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of people and those individuals have become immune. Well, then that creates resistance to the spread of disease

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when it reappears in the population. We can actually see how her community works

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by looking at the effects of zero swine zero, remember, was the product of an infection rate her?

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And the duration of infection term, which is this term here? Well, the infection

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rate term is equal to betore times by N. And in this equation, N represents the

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number of susceptible individuals. The number of individuals that are available to be infected.

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So if there's a high amount of immunity in the population, what that does is it reduces

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the number of individuals that are available to be infected. So in other words, it reduces and.

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And so if N is reduced, well, then that means that the basic reproduction number on north is also reduced because

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of this formula here. That's what we can do then, is we can look at those graphs

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again in the context of immunity. Well, if there's no immunity at all,

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then O0 is high. And so you get an outbreak that looks a little

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bit like the blue curve. If there's some immunity in the system, though, if N reduces

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on wards effectively, then there's some immunity in the system and you instead see an outright that looks a

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little bit like the red. And if lots of individuals are immune, then you instead see an

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outbreak that looks a little bit like the purple. So, again, immunity being

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in the system significantly changes the dynamics of an infectious disease outbreak.

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OK, so we talk now about a very basic epidemiological model, the ESSI Amahl, and we

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talked about a few concepts relating to infectious disease outbreaks. So concepts like the basic

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reproduction number and the concept of herd immunity. I'm going to go on and do now is talk about how

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we can extend the basic asylum model. And these extensions are ones

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that had uncommonly by infectious disease outbreak models. So the first extension

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is to note that infectious disease outbreaks are inherently random.

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So in other words, all of the curves that we've seen so far for the number of infected individuals to

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a time have been these very smooth curves. Right. The numbers of cases went up. It peaks

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and then it comes back down again. And it does so in a very smooth fashion. But real infectious

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disease outbreak dates just doesn't look like that. So here are a couple of examples. So

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what we have at the top is we have the number of new coronavirus cases every

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day in South Korea since February 15th. And what you can see is

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that we get the general overall shape of the outbreak. The number of cases goes up, it peaks, and then it comes

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back down again in this particular wave that the outbreak.

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But it doesn't do is it doesn't do that in a really smooth fashion. The numbers of cases kind of jacks around a little bit.

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Similarly, if we look at the bottoms, this is data from Italy. We see not a

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complete wave of the outbreak, but we see, again, a similar looking pattern where the numbers

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of cases goes up. It then looks to peak and it looks like it's starting to come back down again.

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But again, it does it by sort of jagging around a little bit. Not like the smooth curves that we've seen so far

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for the essi all model. And we can include this randomness in simulations of an infectious

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disease outbreak by using what's called stochastic models.

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So the idea of a sarcastic model is that an infectious disease outbreak

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isn't simply a deterministic process. Instead, what happens? So if you want

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to simulate a stochastic S.A. model, you can simply flip a coin lots and lots of times.

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And then according to the results of each coin flip, you can either generate a new infection. So

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a susceptible individual becomes infected or you can make one of your infected individuals

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recover and become immune. And just to be clear, when we're flipping this coin lots and lots

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of times, this isn't a fair coin. It's not 50/50 whether or not there's an infection events

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or or recovery events. Instead, the coin is waited and it's weighted according to

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the number of sceptical individuals and the number of infectious individuals that are in the population

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at that time. So as an example, if you imagine there are lots of healthy individuals

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in the population at moment. Well, that means that there are lots of targets for infection. So you'd

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expect the chance that the next event is an infection event to be high because there are lots of potential

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individuals that could be infected. And so you'd wipe the coin when the number of susceptible

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individuals is high so that the chance that the next event is an infection event is high compared

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to the chance that the next event is a recovery event.

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So what we do is we flip a coin lots and lots of times, and according to the results of each coin flip, we change

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the state of the system. So this is one example. Again, this is nothing to do with Kovar 19, but it's just to show

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the idea. So we flip a coin lots and lots of times and we see this graph, the number of infected

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individuals, three time. We can then repeat that if we

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want to. We can generate a new simulation of an infectious disease outbreak by

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simply flipping a coin lots and lots of times again. And if we do that in a second

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simulation on drop suit, identical conditions, we might see an outbreak that instead

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looks like the blue curve there. So qualitatively, the blue curve and

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the black curve. Pretty much identical. The only difference. So the reason they don't look exactly the same

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is that we've got a slightly different sequence of coin flips. We've got a slightly different sequence

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of infection and recovery events. So what we can do if we want to

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is we can actually buy another simulation again under absolutely identical conditions. So again, we generate

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a sequence of coin flips. And if we do that, we might instead see something that looks a little

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bit like the red line there. So what's happened this time is we started with one infected individual.

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We've generated a sequence of coin flips. But the first coin flip, as indicated,

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is the one infected individual has recovered without infecting anyone else. And because they've

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been because they've recovered without insects getting anyone else, the disease isn't in the population anymore.

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And so the outbreak simply fights out. So by including this randomness in the model,

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then you can get very different qualitative to behaviour each time you want a simulation. You can either get

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a big outbreak like the blue or the black, or you can get a very small outbreak like the

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red. So another

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thing you can do to extend the basic asylum model is you can include all sorts of different infectious

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disease epidemiology. So I'm going to talk about one particular example in the context

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of some of the current government guidance for Kovar 19. It's the current government guidance,

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Povey 19 is that if you live with others and you're the first in your household to have symptoms

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of a coronavirus infection, then you should stay at home for seven days. Well,

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the other members of your household, on the other hand, must stay at home for 14 days.

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So what you can see there is that the other members of your household, if you will, the infected one initially,

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the other members of your household may not even be carrying the virus yet. They are expected to stay at home

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for longer than you are. But this rule, in fact, makes complete sense. And the

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reason for that is that there's a delay between an individual becoming infected and an individual

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showing symptoms and starting to infect other individuals. So what that means

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is that we're supposed to. I get the infection. Well, let's suppose that I'm symptomatic,

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let's say for seven days, within seven days, then I might go in, infect someone else in my household

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and they may only show symptoms and start infecting other individuals in the second set of seven days.

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So that's why it's very important that other household members stay at home for a longer period, for a period

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of 14 days. So like I say, the key thing here is that there's a delay between

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an individual becoming infected and that individual starting to show symptoms or infects other individuals.

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And we can include that in the basic CSI all model by simply including another class

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in between being susceptible and being infectious.

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So in other words, we can build a model that looks like this one on the screen now where you have two separate individuals

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and after infection, they end this E class and each class represents individuals that are infected,

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but they're not yet generating new infections. And then only some time later to infect

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individuals, become infectious and enter the AI class. And then eventually, after they've been infectious,

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they recover and enter the all class. So this novel here includes

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this sort of additional bit of epidemiology, the fact that there's a delay between being infected

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and then starting to generate new infections. And you could include other types of epidemiology

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in compartmental models in a similar way. You can simply introduce new compartments into the

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model to represent different aspects of the underlying epidemiology.

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So another thing that you might like to include in the infectious disease outbreak model is the idea

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that individuals of different ages contacts other individuals at different rates.

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So here's an example. This is a graph that shows this quite clearly, I think. This is data

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from a paper by Premiss Al and plus computational biology. And this graph was created by one of my Ph.D.

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students. And what you can see is so on the x axis, you've got

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the age of an individual. And then on the Y axis, you've got ages of

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the individuals contacted by that first individual. And then the colour

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in the graph represents the number of contacts per day between individuals of those ages.

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So what you can see, as you can see, this diagonal here, which is quite dog. So this diagonal

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represents individuals on the x axis, contacting individuals on

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the y axis. And what you can say is that because it's this diagonal that's quite dark. That means that it's reasonably likely

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that an individual of a certain age will contact other individuals of the same age.

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You can also see some other kind of dark areas as well on this on this particular graph. So here are some dark

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areas up here and there were also some dark areas down here. This dark area down

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here represents an adult that's roughly of an age of parents

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contacting someone that's the age of a child. So this particular streak represents

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parents contacting their children. This kind of dark

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area up here, which this represents individuals that are at the age of the children of a child.

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Contacting individuals that are the age of an adult. So this particular area up here represents

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children contacting their parents.

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She can really clearly see that there are some very strict age structure in who contacts whom within a

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population. You can also look at contacts in different settings. So that's what we have

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here. So this middle graph here represents home contacts and you can see very similar

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patterns to the ones that I just described in the left graph. So in particular, what you can see

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here is that at home it's very likely that individuals contacts individuals of the same age.

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And it's also very likely that individuals so parents contact children and children

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contact parents. But if you look in another setting, then contacts will look different.

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So if you look, for example, at the right graph, well, the right graph represents school age contacts.

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And what you see in schools is that the majority of contacts, children contacting individuals

406
00:36:26,580 --> 00:36:31,710
of the same age. So this kind of

407
00:36:31,710 --> 00:36:36,850
age structure wasn't represented in the basic essi all model that we looked at earlier. The assumption

408
00:36:36,850 --> 00:36:41,850
in the I all model was that everyone of every age contacts other individuals that a kind

409
00:36:41,850 --> 00:36:46,950
of constant rates, whereas the data not shown on this particular slide here

410
00:36:46,950 --> 00:36:51,990
shows that that clearly isn't the case. But age structure can very straightforwardly

411
00:36:51,990 --> 00:36:57,540
be inclusion and model like the asylum model. So here's one way to do it. So in this

412
00:36:57,540 --> 00:37:02,850
particular model I'm showing here, we just have individuals of two ages. So we just have children

413
00:37:02,850 --> 00:37:07,920
and we have adults. This approach, though, is the approach I'm showing you here can easily be extended to

414
00:37:07,920 --> 00:37:13,830
any number of age groups you want. So in particular, the data on the previous slide can all be incorporated

415
00:37:13,830 --> 00:37:18,880
in an essay type model. If we want to do the. But here in

416
00:37:18,880 --> 00:37:24,220
this very basic extension, we have individuals that are children and adults. And what we essentially have is we have an essay

417
00:37:24,220 --> 00:37:29,220
all model for children. So that's this one on the top where you have susceptible children, infected

418
00:37:29,220 --> 00:37:34,220
children and recovered and immune children. And then we also have an

419
00:37:34,220 --> 00:37:39,500
essay model on the bottom for adults. So, again, we got susceptible adults, infected adults

420
00:37:39,500 --> 00:37:44,510
and recovered and immune adults. Something to notice there is if we

421
00:37:44,510 --> 00:37:49,820
look at the infection rates term, which of these terms in here, the infection rates depend

422
00:37:49,820 --> 00:37:55,130
on. So this is the rate at which children become infected. That depends on the rate at which children

423
00:37:55,130 --> 00:38:00,650
infects other children. This beta CCE term. And it also depends

424
00:38:00,650 --> 00:38:07,020
on the rate at which adults infects children. So it depends on this beta AC term.

425
00:38:07,020 --> 00:38:12,130
So if children infect others. If children have lots of contact with other children, then you might

426
00:38:12,130 --> 00:38:17,170
expect the B to C, C to the infection rate between children would be high. And

427
00:38:17,170 --> 00:38:22,420
so then the rate at which children become infected will also be high. Similarly, if adults have a lot of contact

428
00:38:22,420 --> 00:38:27,490
with children, well, then the beta AC term would be high because the number of contacts is

429
00:38:27,490 --> 00:38:32,500
high and the infection rate is also high. And then again, children would be infected.

430
00:38:32,500 --> 00:38:38,380
That's a very high rate. So in other words, the data from the previous slide,

431
00:38:38,380 --> 00:38:43,930
the data on the numbers of contacts between individuals of different ages, is included in this model

432
00:38:43,930 --> 00:38:52,940
by the various different beta terms. So via the various different infection rates.

433
00:38:52,940 --> 00:38:58,790
Another thing that we can include in other extensions, the basic S.A.M. is to include asymptomatic

434
00:38:58,790 --> 00:39:04,130
transmission or transmission from individuals with very few symptoms. This has been something

435
00:39:04,130 --> 00:39:09,590
that's been talked about a lot in the context of kov. At Nineteens of Cauvin, 19

436
00:39:09,590 --> 00:39:14,810
infected individuals could have any of a wide spectrum of symptoms.

437
00:39:14,810 --> 00:39:20,420
So an individual could have very clear symptoms. But it is also possible that they have very few symptoms.

438
00:39:20,420 --> 00:39:25,880
And if an infected individual has very few symptoms, then that makes the outbreak difficult to control

439
00:39:25,880 --> 00:39:32,270
because an individual can be spreading the virus without even knowing that they're doing so.

440
00:39:32,270 --> 00:39:37,560
With that in mind, I want to show you this figure here in the. On the left. So this is the figure from

441
00:39:37,560 --> 00:39:43,140
a paper by Christophe Fraser in PNAS in 2004.

442
00:39:43,140 --> 00:39:48,180
Christophe Fraser is now based in Oxford on what you can see as. If so, the value

443
00:39:48,180 --> 00:39:53,280
on the x axis is the proportion of new infections that arise from

444
00:39:53,280 --> 00:39:58,440
individuals that have very few symptoms. So it occurs from individuals either prior to them developing

445
00:39:58,440 --> 00:40:03,540
symptoms or from individuals that never develop symptoms. And what you can see

446
00:40:03,540 --> 00:40:09,120
is that this source cluster here, they source cluster is on the left hand side of this graph.

447
00:40:09,120 --> 00:40:14,610
So what that means is that almost all Saar's infections were from individuals

448
00:40:14,610 --> 00:40:20,450
that were displaying clear symptoms. So that's very important because

449
00:40:20,450 --> 00:40:25,490
what it means is so infections are arising from individuals who are displaying clear symptoms. So

450
00:40:25,490 --> 00:40:30,830
in order to control the outbreak of Saar's in 2003, what could be done is

451
00:40:30,830 --> 00:40:35,960
we could go out and actually find individuals that are displaying clear symptoms. We can make sure

452
00:40:35,960 --> 00:40:41,210
that we isolate those individuals to make sure that those individuals with clear symptoms don't transmit it to anyone

453
00:40:41,210 --> 00:40:46,760
else. And so by isolating individuals with clear symptoms, you can actually bring the outbreak

454
00:40:46,760 --> 00:40:52,200
under control. That's something is unlikely to be possible for Kovik 19, because

455
00:40:52,200 --> 00:40:57,540
it isn't the case that the only infection's out there are driven by individuals with clear symptoms.

456
00:40:57,540 --> 00:41:02,580
So, in fact, a large number of infections are driven by individuals who don't have clear

457
00:41:02,580 --> 00:41:07,860
symptoms. And so that means that we have to introduce control strategies that don't only target

458
00:41:07,860 --> 00:41:12,990
symptomatic hosts, but also target all of us because we might be spreading

459
00:41:12,990 --> 00:41:18,390
the virus without knowing that we're doing so. That's why we need to implement measures like social distancing

460
00:41:18,390 --> 00:41:23,510
that have a huge impact on everybody. So this idea

461
00:41:23,510 --> 00:41:29,670
that about asymptomatic transmission, well, that can be included in a model like the ESSI all model.

462
00:41:29,670 --> 00:41:34,890
So here's an example. So basically, this is exactly the same as the S.A.M. The only thing I want to point

463
00:41:34,890 --> 00:41:39,900
out is that we have this additional class in the model, this a class, and this represents individuals

464
00:41:39,900 --> 00:41:44,950
that are infected and transmits it, transmitting the underlying disease, but

465
00:41:44,950 --> 00:41:53,710
they're not showing any clear symptoms. These are the asymptomatic carriers of the disease.

466
00:41:53,710 --> 00:41:58,750
Something else that can be included in the basic compartmental modelling

467
00:41:58,750 --> 00:42:03,880
framework is the idea of spatial structure. So here on the left, there's

468
00:42:03,880 --> 00:42:10,190
a graph which represents the population density throughout England and Wales.

469
00:42:10,190 --> 00:42:15,460
And there are two features that I want to point out. The first feature is that the population is very wide

470
00:42:15,460 --> 00:42:20,590
spread. And actually, it's pretty unlikely that if you've got an infected individual on the south coast,

471
00:42:20,590 --> 00:42:25,810
let's say that they go on and directly infect someone living in the very north of England. That

472
00:42:25,810 --> 00:42:30,820
is very unlikely. Similarly, so another thing you can notice from this graph

473
00:42:30,820 --> 00:42:36,090
is that the population density vary substantially throughout the country.

474
00:42:36,090 --> 00:42:41,210
And these things can all be included in infectious disease outbreak models like the Asylum

475
00:42:41,210 --> 00:42:46,440
Armidale, on the right hand side is a graph of air traffic routes over

476
00:42:46,440 --> 00:42:51,660
Eurasia. And what you can see here is that some areas are very well connected

477
00:42:51,660 --> 00:42:57,150
and other areas are a lot less well connected. And again, that's something that we can include in an exile

478
00:42:57,150 --> 00:43:02,250
type model if we want to. And this is how we do it. So what I'm considering here

479
00:43:02,250 --> 00:43:07,690
is I'm considering a model which only has two regions and it's a two spatially distinct areas.

480
00:43:07,690 --> 00:43:12,840
You can extend this idea to any number of regions you want. And so what we have in

481
00:43:12,840 --> 00:43:17,970
the first region is we have an essay, although in the second region we also

482
00:43:17,970 --> 00:43:23,430
have an essay or model. And we can include individuals moving between these two regions.

483
00:43:23,430 --> 00:43:28,650
There's some kind of coupling between these two regions, a particular right, which in this model is represented

484
00:43:28,650 --> 00:43:34,140
by Lambda. And if you have regions, some of which are better connected than others,

485
00:43:34,140 --> 00:43:39,750
then you would simply change the value of LAMDA between two different regions. So a higher value of lambda

486
00:43:39,750 --> 00:43:44,970
would represent a better connexion between two particular regions.

487
00:43:44,970 --> 00:43:50,010
You can include different population densities in these models by simply having different numbers of

488
00:43:50,010 --> 00:43:55,160
individuals in total within each region. And in regions

489
00:43:55,160 --> 00:44:00,320
that are a long way apart. So you could include that if you want to in the model by again, assuming a small value

490
00:44:00,320 --> 00:44:06,280
of lambda. Because transmission is perhaps less likely if regions are alone apart from each other.

491
00:44:06,280 --> 00:44:11,330
So that's how one of the lines, the spatial structure can be included in a model.

492
00:44:11,330 --> 00:44:16,570
Yes, I will. So we talked about so far

493
00:44:16,570 --> 00:44:21,870
as we talked about how a very basic infectious disease outbreak model can be developed. That outbreak

494
00:44:21,870 --> 00:44:26,910
model matches the kind of shape of most infectious disease outbreaks.

495
00:44:26,910 --> 00:44:32,100
We talked about various concepts like the reproduction number and like herd immunity. And we talked about

496
00:44:32,100 --> 00:44:37,110
how models can be extended from the very basic model I showed you near the beginning to much more complex

497
00:44:37,110 --> 00:44:42,270
settings that are much more realistic. One to talk about now is how

498
00:44:42,270 --> 00:44:47,550
models can be used to inform control during outbreaks. So clearly

499
00:44:47,550 --> 00:44:52,740
one example of this is the ongoing KOVA 19 pandemic. I first heard about this

500
00:44:52,740 --> 00:44:57,780
pandemic right back in early January. So I think it was something like the April the 9th January

501
00:44:57,780 --> 00:45:02,910
when I saw the note that's on the right hand side of this figure. So I saw

502
00:45:02,910 --> 00:45:08,580
this note. It was it was posted on Twitter. And this is a public health notice that was posted

503
00:45:08,580 --> 00:45:13,740
in the city of Wuhan in China. And this public health notice says

504
00:45:13,740 --> 00:45:18,860
that there have been a number of cases of atypical pneumonia in the city of Wuhan

505
00:45:18,860 --> 00:45:24,140
and thing that these cases appear to all have in common is travel to the Juan Island seafood

506
00:45:24,140 --> 00:45:29,260
market in that city. And so one of the first things that I did

507
00:45:29,260 --> 00:45:34,450
was I started to go about collecting information about Cauvin 19. And I started to go about

508
00:45:34,450 --> 00:45:39,520
developing mathematical models for this particular outbreak. What we

509
00:45:39,520 --> 00:45:45,220
saw as January went on so as January this year went on, is that the numbers of cases starts to accumulate

510
00:45:45,220 --> 00:45:51,010
within China. So on the 20th of January, there were 291 reported cases,

511
00:45:51,010 --> 00:45:56,050
only a couple of days later. There were already four hundred and forty six cases. And by the time we got to the

512
00:45:56,050 --> 00:46:01,730
twenty sixth of January, there were over two thousand reported cases.

513
00:46:01,730 --> 00:46:06,850
And so the question that I want to answer as the outbreak was spreading within China is, well, what

514
00:46:06,850 --> 00:46:12,130
is the risk of getting outbreaks in other countries? How likely is it we see an outbreak

515
00:46:12,130 --> 00:46:17,250
like the outbreak that we're seeing in China, but in the U.K.?

516
00:46:17,250 --> 00:46:22,300
And again, models can be very useful to explore questions like that. So again, this particular graph is

517
00:46:22,300 --> 00:46:27,720
not specifically for Cauvin 19, but it demonstrates a key epidemiological principle.

518
00:46:27,720 --> 00:46:33,550
And that key epidemiological principle is that of the epidemic risk.

519
00:46:33,550 --> 00:46:39,200
So the epidemic risk. So what is that? That means every time you get an imported case, it's simply

520
00:46:39,200 --> 00:46:44,420
the risk that that imported case in a new location generates a large outbreak

521
00:46:44,420 --> 00:46:50,880
there. So do they start chains of transmission that lead on to a large outbreak?

522
00:46:50,880 --> 00:46:56,250
The graph that I'm showing here is the graph. I showed you a bit earlier in that graph.

523
00:46:56,250 --> 00:47:01,290
What I showed you was that when you run up to Capstick epidemic model. You

524
00:47:01,290 --> 00:47:06,780
can either see a large outbreak like the blue or the black, and you could or you can see a small outbreak

525
00:47:06,780 --> 00:47:11,790
like the red. So if the epidemic risk is zero,

526
00:47:11,790 --> 00:47:17,160
what that means is that every time you get an imported case, you're not going to get a large outbreak.

527
00:47:17,160 --> 00:47:23,340
You're always going to see something looks like the looks like the red and not something that looks like the blue or the black.

528
00:47:23,340 --> 00:47:28,710
If, on the other hand, the epidemic risk is one, well, then a major epidemic is definitely going to occur.

529
00:47:28,710 --> 00:47:33,720
So in other words, every time you get an imported case, they're going to start chains of transmission that lead to a large

530
00:47:33,720 --> 00:47:39,390
outbreak like the blue or the black, rather than a small outbreak like the red.

531
00:47:39,390 --> 00:47:44,880
Usually the epidemic risk isn't zero one. Instead, the epidemic risk takes a value between

532
00:47:44,880 --> 00:47:50,160
zero and one, and that value represents the chance that any single imported

533
00:47:50,160 --> 00:47:55,350
case will lead onto a large outbreak.

534
00:47:55,350 --> 00:48:00,440
So the economic risk is the probability that an imported case leads onto a major epidemic.

535
00:48:00,440 --> 00:48:05,730
And if you have a higher value of the basic wheat production number like we saw before, that means that the

536
00:48:05,730 --> 00:48:11,160
disease is more transmissible. You have a higher value of the basically production number. Then you have a higher

537
00:48:11,160 --> 00:48:16,170
epidemic risk. In other words, it's more likely that you're going to go on and see an outbreak. It looks like the

538
00:48:16,170 --> 00:48:22,510
blue or black as opposed to an outbreak that simply fades out like the red.

539
00:48:22,510 --> 00:48:27,600
And remember the Arnaut? So the basic reproduction number could be calculated by taking

540
00:48:27,600 --> 00:48:33,400
the infection rate and multiplying it by the duration of infection.

541
00:48:33,400 --> 00:48:38,500
So if you can reduce Donalds, then you can reduce the epidemic risk. And

542
00:48:38,500 --> 00:48:44,110
so right back at the beginning of the Cove, it 19 outbreak when there were very few cases in the UK.

543
00:48:44,110 --> 00:48:49,300
Only cases that were imported from outside. The question was, well, how can we reduce our Nords

544
00:48:49,300 --> 00:48:55,530
and therefore reduce the chance of getting a large outbreak in this country?

545
00:48:55,530 --> 00:49:00,530
And the answer to that was, well, either you could reduce the infection rate in some way or you

546
00:49:00,530 --> 00:49:06,030
can reduce the length of time that individuals are infectious. Four.

547
00:49:06,030 --> 00:49:11,090
And so the initial policy in the U.K. aimed at controlling the Kovar 19 outbreak was to

548
00:49:11,090 --> 00:49:16,160
reduce the length of time that individuals are infectious four. So, in other words, to go out

549
00:49:16,160 --> 00:49:21,290
there and find imported cases and find all of their contacts

550
00:49:21,290 --> 00:49:27,080
and make sure that those individuals were isolated quickly by selecting those individuals quickly.

551
00:49:27,080 --> 00:49:32,270
You're reducing the duration of infection for those individuals who will therefore reducing

552
00:49:32,270 --> 00:49:39,010
are zero. And in turn, your reducing the chance of a large epidemic in the U.K.

553
00:49:39,010 --> 00:49:45,520
So that was the initial policy. This is a paper that I wrote on this topic right back in January.

554
00:49:45,520 --> 00:49:50,890
And the key conclusion is fast isolation of imported cases can reduce the epidemic

555
00:49:50,890 --> 00:49:56,650
risk in countries other than China. So what you need to do is you really need to rapidly

556
00:49:56,650 --> 00:50:01,840
isolate any imported case and their contacts if you want to reduce

557
00:50:01,840 --> 00:50:07,470
the chance of having a large outbreak in other countries. Unfortunately,

558
00:50:07,470 --> 00:50:12,800
what then went on to happen is we did see large outbreaks elsewhere. The epidemic risk wasn't reduced

559
00:50:12,800 --> 00:50:20,100
sufficiently. And so what then happens is we enter a different phase of the outbreak.

560
00:50:20,100 --> 00:50:25,110
So then the question for modellers is, well, what can we do when a major outbreak is ongoing? And so this is the

561
00:50:25,110 --> 00:50:31,390
kind of procedure that we go through. What we do is we observe data from the ongoing outbreak.

562
00:50:31,390 --> 00:50:36,430
We then develop an epidemiological model, so a sort of compartmental model of the type that I

563
00:50:36,430 --> 00:50:41,860
showed you earlier. The model I've shown here in step number two is a model that I developed for Ebola

564
00:50:41,860 --> 00:50:46,900
virus disease. What we then do is we estimate the parameters of the model.

565
00:50:46,900 --> 00:50:52,060
So we estimate the values of parameters like the infection rate that we looked to earlier or like

566
00:50:52,060 --> 00:50:57,070
the recovery rates. So we choose those parameters so that the output of the model

567
00:50:57,070 --> 00:51:02,110
is consistent with the data that we've observed. And then once we've got a mathematical

568
00:51:02,110 --> 00:51:07,140
model, we can run simulations of it forwards to make a forecast

569
00:51:07,140 --> 00:51:12,400
and then we can take those simulations and introduce different control interventions in the model. And look at what happens

570
00:51:12,400 --> 00:51:17,860
in the model under different possible control interventions. And the aim there isn't to predict

571
00:51:17,860 --> 00:51:22,900
precisely how many cases there are going to be or precisely when the outbreak is going to peak.

572
00:51:22,900 --> 00:51:27,940
Instead, the aim is to look at different possible controlled interventions and look at which are most

573
00:51:27,940 --> 00:51:33,040
likely to have beneficial effects and which are most likely to have the largest beneficial

574
00:51:33,040 --> 00:51:38,170
effects in terms of reducing case numbers. So these

575
00:51:38,170 --> 00:51:43,360
are the kinds of things that one might find. So, again, this is not for Cauvin 19. This is just general

576
00:51:43,360 --> 00:51:48,540
infectious disease outbreak model. Something that you might have heard policymakers,

577
00:51:48,540 --> 00:51:53,610
the prime minister's talked about this a lot, is the idea of flattening the curve. One of the things that we want to

578
00:51:53,610 --> 00:51:58,740
do is we want to flatten the curve. What does that actually mean? Well, we can think about that

579
00:51:58,740 --> 00:52:04,020
in terms of social distancing. So social distancing involves reducing

580
00:52:04,020 --> 00:52:09,720
the rates that we contact other individuals. If we reduce the rate at which we contact individuals,

581
00:52:09,720 --> 00:52:15,270
we reduce the overall infection rates because everyone's having a lot fewer contacts. And so we're likely

582
00:52:15,270 --> 00:52:20,370
to pass the disease on to fewer people. And so if we reduce the infection rate,

583
00:52:20,370 --> 00:52:25,470
what we do is we reduce this parameter BCO. And if we reduce this parameter BITA, the knock

584
00:52:25,470 --> 00:52:30,630
on effects of that is to reduce are zero because our zero depends on this parameter.

585
00:52:30,630 --> 00:52:36,310
Bita. And if we reduce our zero, we'll epidemiological

586
00:52:36,310 --> 00:52:41,530
models tell us. That we can go from something like the blue curve in this graph

587
00:52:41,530 --> 00:52:47,140
to something that's a bit more like the red curve. And this is highly desirable

588
00:52:47,140 --> 00:52:52,330
during an infectious disease outbreak. And the reason for that is not. Well,

589
00:52:52,330 --> 00:52:57,400
one of the key things that you might want to do during an infectious disease outbreak is make sure that the number of

590
00:52:57,400 --> 00:53:02,440
infected individuals at any single time remains below the capacity for

591
00:53:02,440 --> 00:53:07,660
treatment. So in other words, try and manage the number of infected individuals in such

592
00:53:07,660 --> 00:53:12,870
a way that health sector health care services can cope.

593
00:53:12,870 --> 00:53:18,820
So what you can see here is that in the blue case. So when honour is high. So without something like a social distancing

594
00:53:18,820 --> 00:53:23,920
intervention, the disease sweeps through the population very fast. The peak

595
00:53:23,920 --> 00:53:28,960
of this outbreak is very high. What that means is that there's a large number of individuals infected at any

596
00:53:28,960 --> 00:53:34,550
one time. And because of that, it's very difficult for healthcare services to cope

597
00:53:34,550 --> 00:53:40,290
because they have to deal with a huge amount of cases all at a single time.

598
00:53:40,290 --> 00:53:45,870
In contrast, if you introduce an intervention that reduces our Nords, that has two effects.

599
00:53:45,870 --> 00:53:51,540
Firstly, it reduces the size of the peak. So if you look at the red curve, then the peak is much lower

600
00:53:51,540 --> 00:53:56,610
than for the blue curve. What that means is that at any one time, health

601
00:53:56,610 --> 00:54:01,770
care services aren't having to treat as many infected individuals. And that's

602
00:54:01,770 --> 00:54:06,970
clearly a beneficial thing. But the other thing, the other benefits in terms

603
00:54:06,970 --> 00:54:12,370
of health care services is that if you can reduce your Nords, then the peak also becomes later.

604
00:54:12,370 --> 00:54:17,540
And that's really important because it allows health care services more time to prepare

605
00:54:17,540 --> 00:54:23,020
for the peak number of infections. And that's a very good thing because it allows health care services

606
00:54:23,020 --> 00:54:28,420
to increase their hospital bed capacity, for example. It allows them to distribute personal

607
00:54:28,420 --> 00:54:33,490
protective equipment to health care workers. It allows them to make sure that they have ventilators

608
00:54:33,490 --> 00:54:38,620
to increase the number of ventilators they have. In other words, it allows health care services

609
00:54:38,620 --> 00:54:45,850
to better be able to deal with the number of individuals that they might have to treat.

610
00:54:45,850 --> 00:54:50,890
So clearly, this idea of flattening the curve is reducing Arnaut, making sure that the peak number of

611
00:54:50,890 --> 00:54:56,170
individuals that are simultaneously infected because later on at that peak is lower. And that's clearly

612
00:54:56,170 --> 00:55:01,350
good for health care services. Something else we talked

613
00:55:01,350 --> 00:55:07,020
about a little bit earlier on is the idea of herd immunity. So the resistance to the spread of disease, the results,

614
00:55:07,020 --> 00:55:12,200
if a sufficiently high proportion of individuals are immune. Well,

615
00:55:12,200 --> 00:55:17,450
vaccinations who go the vaccination. This is actually allows you to increase the number

616
00:55:17,450 --> 00:55:22,640
of immune individuals. So remember that we talked about that earlier in the context of Arnaut. So

617
00:55:22,640 --> 00:55:27,890
Arnaut was given by this formula here where N is the number of individuals that are susceptible

618
00:55:27,890 --> 00:55:33,320
to the disease. And if you go and vaccinate lots of individuals, well, that reduces

619
00:55:33,320 --> 00:55:38,660
the value of n it reduces the number of individuals that are susceptible to the disease. And in turn,

620
00:55:38,660 --> 00:55:43,850
that reduces our Nords. And if we reduce our N again, that changes the shape

621
00:55:43,850 --> 00:55:49,040
of the outbreak curve as a mistake or on the left, you can go from something that looks like the blue to something

622
00:55:49,040 --> 00:55:54,440
that looks a little bit more like the red. And if you can continue to decrease

623
00:55:54,440 --> 00:56:00,050
Arnalds, well then the immunity in the population becomes even higher and eventually

624
00:56:00,050 --> 00:56:05,780
you get to a point where Arnaut falls below one. And outbreaks can't happen at all.

625
00:56:05,780 --> 00:56:11,030
So hopefully. So I think a vaccine is of the order of a year away. But if a vaccine

626
00:56:11,030 --> 00:56:16,520
can be deployed widely in the population covered 19, that can prevent this disease recurring

627
00:56:16,520 --> 00:56:22,810
in future. And that's clearly something that's very important.

628
00:56:22,810 --> 00:56:29,230
These two concepts I've just been talking about were in the context of very basic epidemiological models.

629
00:56:29,230 --> 00:56:34,450
Of course, as I showed you earlier, you can extend these models to become much more complex.

630
00:56:34,450 --> 00:56:39,820
And with these more complex models, you can also test very complex control strategies.

631
00:56:39,820 --> 00:56:45,140
So here are a couple of examples for Kovar 19. And these are examples of output's for models

632
00:56:45,140 --> 00:56:50,260
that have been developed by Imperial College London for the graph on the right and the London School of Hygiene

633
00:56:50,260 --> 00:56:55,420
and Tropical Medicine for the graph in the bottom left. And the idea

634
00:56:55,420 --> 00:57:00,640
of these graphs is to test a more complex control strategy. I should say that the London School of Hygiene

635
00:57:00,640 --> 00:57:05,770
and Tropical Medicine model in the bottom left. It includes all five extensions

636
00:57:05,770 --> 00:57:11,260
to the basic ESSI all model that I told you about earlier. So it's much more complex model.

637
00:57:11,260 --> 00:57:16,910
And what you can see here is you can see quite complex dynamics.

638
00:57:16,910 --> 00:57:22,220
So the idea here. So the thing that's being tested is the idea of reducing our zero

639
00:57:22,220 --> 00:57:27,770
periodically by having lockdowns. So having a lockdown for a period of time to reduce the reproduction

640
00:57:27,770 --> 00:57:32,810
number, that then reduces the numbers of cases and therefore reduces the pressure on

641
00:57:32,810 --> 00:57:37,940
health care services. And then you can release the lockdown so that people can

642
00:57:37,940 --> 00:57:42,970
go about their daily lives in a more normal fashion. And when you reduce the lockdown, then the numbers

643
00:57:42,970 --> 00:57:48,200
of cases goes up again. So, for example, if we look at this graph,

644
00:57:48,200 --> 00:57:53,280
is this one from Imperial College London. The blue lines represent whether or not

645
00:57:53,280 --> 00:57:58,740
the lockdown is on or off. So in this period here where the blue line is high, then the lockdown is on.

646
00:57:58,740 --> 00:58:03,780
And in this period here, the blue line is low and the lockdown is off. So what you can

647
00:58:03,780 --> 00:58:09,060
see in this graph here is that the numbers of cases increases three time.

648
00:58:09,060 --> 00:58:14,490
Then the lockdown gets implemented, and so then the numbers of cases eventually starts to decline

649
00:58:14,490 --> 00:58:20,250
again and it falls back down again to low levels. But when the numbers of cases are low,

650
00:58:20,250 --> 00:58:26,110
then the lockdown can be taken off. But then the numbers of cases increases again just here.

651
00:58:26,110 --> 00:58:31,380
When the numbers of cases increases again, you might put the lockdown back on again to try and manage the numbers of cases

652
00:58:31,380 --> 00:58:36,400
you've got and therefore the pressure on health care services. That's when the lockdown is back on again.

653
00:58:36,400 --> 00:58:41,490
Then the number of cases falls back down again and so on. And you see this repeating pattern where you put

654
00:58:41,490 --> 00:58:46,650
the lockdown on cases fall down again. You take the lockdown off, cases increase. You put

655
00:58:46,650 --> 00:58:52,380
the lockdown on again. Cases fall down again and so on. And the benefit of this kind of strategy

656
00:58:52,380 --> 00:58:57,390
is, is it allows you to manage the number of cases of disease at

657
00:58:57,390 --> 00:59:02,790
any particular time. And by doing that, you can keep the need for intensive

658
00:59:02,790 --> 00:59:07,950
care unit beds low enough so you can keep the number of infects individuals low

659
00:59:07,950 --> 00:59:13,110
enough. The hospitals are able to cope with the number of individuals requiring intensive care

660
00:59:13,110 --> 00:59:18,290
unit beds. So that's kind of more complex model and a more complex

661
00:59:18,290 --> 00:59:23,330
control strategy. And you see a similar thing from the London School of Hygiene and Tropical Medicine model in the

662
00:59:23,330 --> 00:59:28,470
bottom left. You put a lockdown on the cases, fall down again. You take a lockdown off the people

663
00:59:28,470 --> 00:59:33,920
who go about their everyday lives in a more normal fashion and then cases go up again

664
00:59:33,920 --> 00:59:39,020
and so on. And you repeat and you get this kind of oscillatory behaviour, which in theory might be able

665
00:59:39,020 --> 00:59:44,060
to keep the bad demand in intensive care units

666
00:59:44,060 --> 00:59:50,780
below the number of intensive care unit beds that the health care set this house.

667
00:59:50,780 --> 00:59:56,790
OK. So that's something a little bit about how metals can be used when outbreaks are ongoing.

668
00:59:56,790 --> 01:00:01,830
When you can't see any more cases. So when we're right towards the end of an infectious disease,

669
01:00:01,830 --> 01:00:07,230
outbreaks models are still useful. So this is an example of some work that we did for Ebola

670
01:00:07,230 --> 01:00:12,270
outbreaks. And the idea here is that we have this model, which I showed

671
01:00:12,270 --> 01:00:17,400
you earlier, this model in which it's an ASIO model. But there are also individuals that may not

672
01:00:17,400 --> 01:00:22,450
be reporting disease. So in other words, you have asymptomatic infections

673
01:00:22,450 --> 01:00:27,700
which don't get recorded in routine surveillance data. So in other words,

674
01:00:27,700 --> 01:00:32,740
you can be in a situation where you haven't seen any cases for the last five days. But there may still be cases

675
01:00:32,740 --> 01:00:39,450
out there because there are these asymptomatic infection individuals that you simply don't see.

676
01:00:39,450 --> 01:00:44,540
So what you can do, in fact, I won't go into too much detail about that, about this, but what you can do is you can use my Basco

677
01:00:44,540 --> 01:00:49,610
models to say, well, if we haven't seen any cases for the last 10 days, let's say,

678
01:00:49,610 --> 01:00:54,680
how likely is it that the outbreak really is over? And so we developed this figure for

679
01:00:54,680 --> 01:00:59,690
Ebola outbreaks, which shows that if you wait a long time without seeing

680
01:00:59,690 --> 01:01:04,760
any symptomatic cases when you can be very confident that the infectious disease

681
01:01:04,760 --> 01:01:09,980
outbreak is over. But if, in contrast, you only wait for a short time. So let's say it's been five

682
01:01:09,980 --> 01:01:15,200
days since you last saw a symptomatic case. Well, then you can't be very confident that your Ebola

683
01:01:15,200 --> 01:01:20,420
outbreak is over. If you wait for this sort of W8 show guideline period

684
01:01:20,420 --> 01:01:25,670
before declaring an Ebola outbreak over, then this very simple

685
01:01:25,670 --> 01:01:31,520
outbreak model suggests that, well, after 42 days, which is the WHL guideline, before declaring

686
01:01:31,520 --> 01:01:36,860
an Ebola outbreak to a finished, there's approximately eight to two percent chance that that Ebola outbreak

687
01:01:36,860 --> 01:01:41,960
really is over. And about 18 percent chance that the outbreak, in fact, isn't over

688
01:01:41,960 --> 01:01:48,330
and that there are still hidden cases out there that you just can't see.

689
01:01:48,330 --> 01:01:53,840
OK. So it's conclude then. So we talked about infectious disease outbreak modelling and we said precisely

690
01:01:53,840 --> 01:01:59,570
what an infectious disease outbreak model is. We've shown that infectious disease outbreaks have a characteristic

691
01:01:59,570 --> 01:02:04,670
shape. The numbers of cases in a single wave outbreak goes up, it peaks, and then it comes

692
01:02:04,670 --> 01:02:10,340
back down again to near zero. Even very basic infectious disease outbreak models

693
01:02:10,340 --> 01:02:15,800
can capture that kind of characteristic shape. Those models can then be extended to include

694
01:02:15,800 --> 01:02:21,050
additional realism. So things like, for example, transmission from individuals aren't showing

695
01:02:21,050 --> 01:02:26,210
symptoms or things like different transmission rates between individuals in different

696
01:02:26,210 --> 01:02:31,260
locations or individuals of different ages. We then talked

697
01:02:31,260 --> 01:02:36,270
about how models can be used at different stages of an outbreak for doing various things, including making

698
01:02:36,270 --> 01:02:41,490
forecasts and predicting the effects of different potential control interventions. For example,

699
01:02:41,490 --> 01:02:46,650
interventions like social distancing that affect the reproduction number and

700
01:02:46,650 --> 01:02:52,410
the idea behind using mathematical models for making forecasts and predicting the effects of different interventions

701
01:02:52,410 --> 01:02:57,630
is shown at the bottom. So you have some data from an ongoing outbreak. You then develop

702
01:02:57,630 --> 01:03:02,730
a mathematical model that can replicate, that can reproduce those data. You were following

703
01:03:02,730 --> 01:03:07,740
the model to make sure that the model can reproduce the data more accurately. And you also

704
01:03:07,740 --> 01:03:13,080
refine the model as more data come in during the outbreak. Once you've got the model,

705
01:03:13,080 --> 01:03:18,150
you can then use simulations forward to generate a forecast as to how many cases you might expect to

706
01:03:18,150 --> 01:03:23,610
see you going forwards. And then you can introduce different public health measures in your mathematical

707
01:03:23,610 --> 01:03:28,620
model to look at how different interventions are likely to change the numbers of cases you might

708
01:03:28,620 --> 01:03:33,720
be expected to see in future. And in that way, mathematical modelling can help you

709
01:03:33,720 --> 01:03:38,880
to prioritise public health measures during an outbreak. So I'm going to stop

710
01:03:38,880 --> 01:03:44,160
there. I'm gonna say thanks. Thanks again very much for joining me for this Mathematical Institute's

711
01:03:44,160 --> 01:03:49,260
public lecture. Live from my home. Please send in any questions you've got via social

712
01:03:49,260 --> 01:03:54,840
media. And we'll be answering a selection of those over the next couple of days.

713
01:03:54,840 --> 01:04:18,100
Thanks again.