Good evening. My name is allegorically, and I would like to welcome you all to the Oxford
Mathematics Public Lecture series. Before we start, I want to express our gratitude to
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Obviously, this is not our usual format. We are broadcasting from home
and I very much hope that you are also watching from the comfort of your home and that you are all in
good health. We live extraordinary times. This
is indicated not only by the fact that half of the world population is in one form
or another of lockdown, but also and maybe even more extraordinary.
We are now hearing world leaders talking about mathematics. We hear them talking
about curves, about exponential us and about models. Mathematics
and mathematician are playing a key role in both understanding the current crisis,
but also in looking for possible ways out of it.
Any form of prediction requires a mother. But what is an epidemic, mother? Where does
one start? What data do we need? And more importantly, how do we use
models this evening? I'm very happy that Robin Thompson has
agreed to talk to us about these issues. Robyn is currently
a junior research fellow in mathematical epidemiology at Christchurch College, Oxford,
and a member of the Mathematical Institute. Robyn has been modelling infectious diseases
for several years and since early January he has been working on important questions
about coronavirus forecasting and control. Since then, he has been
extremely busy contributing to the national modelling effort
and making appearances in media about topics including social distancing and covert nineteen
outbreak risk. If you would like to ask a question, please send it in via social media
and we will collate them and send out answers in the next couple of days. Thank you, Robyn.
Thanks, Allan. And hello, everybody, welcome to this Mathematical Institute public lecture, which is coming like
from my home here in Oxford. My name is Robin Thompson. I'm a junior research fellow
at Christchurch, and I also work in the Mathematical Institute in the Wilson Centre of
Mathematical Biology. My specialist research area is infectious disease outbreak modelling.
And that's a topic, of course, that's been in news a lot recently because mathematical models are being used
in real time to inform public health measures against the Kovik 19 outbreak.
So I'm going to do now is when I talk to you about precisely how mathematicians go about modelling infectious
disease outbreaks. So the natural place to start then is, well, what exactly is
a mathematical model? Well, a mathematical model is a mathematical representation
of a system that can be used to explore that system's behaviour.
And the goal of real time infectious disease outbreak modelling is as follows. What we do
is we look at data from an outbreak so far. So, for example, we might have data on the numbers
of cases per day during the outbreak. Then what we do is we construct
a mathematical model that represents the underlying epidemiology of the system. We then use
simulations of the model forwards to predict what might be likely to happen in future.
So, for example, how many cases we might be expected to see per day going forwards.
What we can then do when we've got a model look can make a sensible forecast is we can then introduce control
interventions into the model to look at how different control interventions might affect the numbers
of cases that we might be likely to see in future.
So during this, let me try to focus on two main questions. The first one is how exactly do
we build a mathematical model of an infectious disease outbreak? And then the second question
I'm going to answer as well. Once we've got a mathematical model, how can we use it to inform public
health measures at different stages of an outbreak? So
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
shape of an infectious disease outbreak? And if so, that's something that's interesting because mathematicians
are very interested in shapes. Then going to talk about how we can build a very, very basic
infectious disease outbreak model. I'll then talk about some epidemiological concepts
that you might have heard about during the COVA 19 outbreak so far. For example,
policymakers have been talking a lot about the basic reproduction number or Nort,
and the concept of herd immunity has been in the news a lot. So I'll talk about those two ideas.
Let's talk about five ways that the very basic infectious disease outbreak model that we're going to construct
can be extended to make them more realistic. Then after that, I going to talk about
how mathematical models can be used to inform public health measures. Well, I'm just go to talk about three different stages
of an outbreak. So come mathematical models be used usefully early in an outbreak
to inform public health measures, come off. My models be used when a major outbreak is ongoing.
So, for example, in the scenario that we're in at the moment here in the U.K. and elsewhere around the
world. And finally, I'll talk about how mathematical models can be used usefully at the end
of an outbreak. So the first thing
I want to point out is that most single wave infectious disease outbreaks
tend to have a characteristic shape. And so here's one example of this, what you
can see here is a data set of an outbreak of influenza in a boys boarding school in the north
of England in the 70s. And what you can see is that the virus enters the school.
The numbers of cases gradually starts to increase and it increases until there's quite a large
number of cases. But it doesn't keep going until everyone in the school is infected.
Instead, it slows down again. And the outbreak peaks. And then the numbers of cases comes back down
again to near zero. And this kind of shape is characteristic of lots and
lots of single wave outbreaks. So here's another example. This is the foot and mouth disease outbreak in
the U.K. in 2001. The data here is slightly different. So it's not individual cases
of disease, but rather it's the number of infected farms. But broadly, you see the same kind
of shape. The virus enters the population. It takes off. It goes up to a kind of high level in
which there are lots of farms that are infected. But then it peaks and comes back down again, back down to
near zero. So, again, broadly, you can see the same kind of shape of an outbreak. The graphs are the same
kind of rough shape. Same thing happens if we look at third examples.
This third example is Plake in Mumbai at the beginning of the 20th century. Again,
this is very slightly different data. This is data on the numbers of individuals that were dying during
that plague outbreak. But again, you see a very similar dynamic in which
the number of deaths increases, but it increases up some high level. And when it reaches
that high level, it turns over and falls back down again to near zero. So, again, broadly, the same
shape. His more recent example of this is Ebola in
West Africa, the largest Ebola outbreak in history. And again, we see a very similar
looking outbreak shape. And here's one example
from the ongoing Kobe 19 outbreak. This is data from China, specifically
data on the numbers of individuals currently infected in China each day throughout
the outbreak. And what you can see, again, is that the number of cases gradually increased.
It increased up some high level again. Then it turned over and then the outbreak began
to fade out. One particularly interesting feature of that specific graph
is that you can see just here. If you look in mid-February, there's a sudden
increase within a single day in the number of infected people.
And the reason for that sudden increase was that precisely what constitutes an infected case
changed the definition of a case changed in China during the coffee nineteen outbreak.
So before that time, in order to be counted in the data, you actually had to be tested
and found to be carrying the novel Corona virus. But obviously, when case
numbers are very high, it's very difficult to go and test everybody. And so what happened then was there was an
assumption that because case numbers were high, if you develop symptoms that were consistent
with infection by the novel coronavirus, then you were included in the data. And
so suddenly, because of this change in definition, you no longer have to be actively tested and found to
be carrying the virus. The numbers of cases shot up very suddenly.
But despite that, broadly, you can see the same kind of shape in this case, the virus invades,
it goes up some high level, then the number of cases turns over and falls back down again to near
zero. And you can see that no matter what kind of outbreak data you're looking at in these graphs, whether
you're looking at deaths or number of infected farms, as in the case of foot and mouth or number
of individual cases like the flu example or like in the coffee 19 example.
Outbreaks of the characteristic shape. And what we want to do is we want to recreate that shape using a mathematical
model. So what we could do if we wanted to is we could make the simplest possible
assumption about how an infectious disease spreads so simple as
possible assumption would be to assume that every single case of disease leads on
to some fixed number of new cases every day that say. So this is one
example. So we're going to assume that each case gives rise to three new cases every day. Well, then at the beginning
of the outbreak. So in day zero, you might just have one infected individual.
On day one. That one infected individual will have gone and infected three other individuals.
So on day one, he'd have three new cases. Ban, according to this very basic
model, we assume that on day two, each of the three eight cases from day one
will have generated three new infections each. So in other words, on day two, you've got nine new cases.
And then on day three, each of those nine would cause three new infections, setoff 27 new infections
on day three and so on and so on. You can imagine this would just carry on going.
So the assumption of this model's this model's called a geometric progression. What you'd find if you looked for
a formula describing the numbers of cases on DADT is you'd see that the formula was just equal
to three to the power of T. That's the number of new cases on dates. Twenty equals zero.
The number of cases is three to the power of zero, which is one. On day one, the number of new cases is three.
The power of one, which is three. On day two, the number of new cases is three. The power of two,
which is nine. And so on. So this is the simplest possible
infectious disease outbreak model. Well, does this actually look anything like real world
data? So what I've done here on this graph is I've plotted the data from the boys boarding
school flu epidemic. They showed you a moment ago and the data plotted in
black. So the black line is what we actually saw in that outbreak dataset.
What I've done then is I've overlaid and read the results of this mathematical model. So a model just
assumes that every individual infects three new individuals every day.
And if we overlay that, what we can see is that actually the model does a pretty good job of representing the very
early parts of the infectious disease outbreak. The red is matching the black very well
early on in the outbreak. But later on in the outbreak, then the
mathematical model isn't doing a good job at all. The numbers of cases in the model is shooting off up to infinity.
Whereas if you look at the infectious disease outbreak data, the data like we just saw
turns over and comes back down again to near zero. It's the model prediction isn't
doing a particularly good job of capturing the data.
So what we have to do next, then, is a very important aspect of mathematical modelling. What we do
next is we have to refine the model. We can't just carry on with this model. We have to make it better,
make sure that the model matches the data more closely. We have to add in more infectious disease
epidemiology. So clearly, one of the issues with this
very basic model is that we've missed out a huge amount of disease biology and in particular
we've missed out the fact that diseases gradually run out of uninfected, susceptible
individuals to infect. That's what we can do instead is we can
consider what's called compartmental modelling. And the idea of compartmental modelling is that you don't just keep
track of how many individuals are infected. You keep track of individuals with all different
infection and symptoms statuses. So the simplest possible
compartmental model you can develop is this model here. This is called the S eye model.
And so in the ESSI model, what you do is you divide individuals according to whether they're susceptible to the disease,
in which case they're in the S compartment. They're in the green circle on the left.
Or an individual can be infected and generating new infections. And if an individual
is infected, then they're in the red circle them in the eye compartments, they're in the red circle on the right.
And as you simulate an infectious disease outbreak using this model, what you would see is you'd see individuals
that are susceptible becoming infected. So in other words, you'd see susceptible individuals from the green circle
transitioning over into the red circle.
So this is the ESSI model I just described. We can write down equations for
this particular model so it doesn't really matter if you know whether that the key
point of this is the idea of compartmental modelling. It's not the precise equations. But for those of you that
are interested in the equations, I'm going to show them. So what we have is we have two separate equations.
One equation describes the rate of change of the number of susceptible individuals,
and the other equation describes the rate of change of the number of infected individuals.
So we write down two equations that look a little bit like this.
So, like I say, it doesn't matter if you haven't met equations like this before. On the left hand side
this term here, this DST by DTT. That just means the rate of change of the
number of susceptible individuals. And this time here is some of negative.
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
of the number of sexual individuals is negative is because the number of susceptible individuals decreases
during the outbreak as susceptible hosts become infected.
Similarly, what we have at the bottom is we have an equation here for the rate of change of the number of infected
individuals. And the number of infants, individuals is something that increases during
the outbreak as infections happen. And so the rate of change of the number
infected individuals is something that is positive. That's why there's no negative sign on the right hand side of this equation.
And in particular, the overall rate of infection is given by this Bita S.I. This is the rate
at which the number of susceptible individuals decreases and the rate at which the number of infected
individuals increases. And that rate is proportional to both S
and I, so the infection rate is proportional both to how many susceptible individuals we've got, but also
how many infected individuals we've got. And the reason for that is that if there's a large number of susceptible
individuals, then there are lots of people that are targets for the disease. There are lots of people that the disease
could, in fact. And so we'd expect the infection rate to be high. Similarly,
if there are lots of infectious individuals, then there are lots of individuals that can do the infecting. And so, again,
if there are lots of infectious individuals, we'd expect the overall infection rates to be high.
OK, so this is the model. So, like I said, yes. By the time the green term
is just the rate of change of the number of susceptible individuals die by details, the rate of change
of the number of infected individuals, and then we have an overall rate of new infections,
which is given by BITA Times. S Times I. And that depends on exactly how
many sceptical individuals we've got and how many infected individuals we've got.
And the promise of BITA governs the rate of infection that governs the rates at which individuals become
infected. OK, so what happens then if we apply this very basic
outbreak model again to the real data? Well, in black, what we have here is we
have the data from the boys boarding school flu epidemic in the 70s.
And what we can do is we can plot the results of this. S I model on top of the data.
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
line is the model prediction that matches the data pretty closely, but it only
does so for about half the outbreak. What happens, like we saw earlier on, is that the data comes back
down again. The numbers of cases declines back down to near zero, whereas the model output,
the red line just stays flat. So it doesn't capture the decline in the number of cases.
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
infinity anymore. So that's a good thing. But like I said, the model doesn't capture
the decline in the number of infected hosts. So, again, we need to do what we
did before, which is we need to refine the model further to try and get the model to match the outbreak data
even better. So what we gonna do is going to include even
more disease biology here. So before we were looking at the ESSI model, which just accounts professor
sets of individuals and infected individuals, what we're gonna do now is we're
going to include another type of individual. And they're individuals that have recovered and become immune
to the disease. And they're in this, ah, class that you can see in the bottom model where
you have susceptible individuals when they get infected. They become infected individuals. And when
they recover, when the infected individual recovers, they become a recovered individual. And in this
all class and this new model is called the asylum model. OK.
So here's the model. We can again write down a set of equations describing the rate
at which individuals pass from one compartment into another. So this is exactly the same
as the Akseli model. But we have one new feature, and that new feature is that we have this
term here. This new item, not new item, is the rates
of which infectious individuals recover and become immune to the disease.
And that term is only proportional to the number of infected individuals there are. And the reason for that is that if there
is a large number of infected individuals, then we'd expect to see a large number of recoveries
in the near future. OK, so this is the ESSI all model. So, again, we've got three equations,
this time this size for the rate of change of the number of healthy, susceptible individuals, rate of change
of the number of infected individuals, and then rate of change of the number of recovered individuals.
We can write down the rates of new infections, which is the pink term that depends on and on.
And we can write right down the rates which infected individuals recover and become immune. That's
this new I. So now we've got two parameters, we got BITA,
which governs the rate of infection. And we have a new parameter, MEU, which governs
the rate of recovery. The rate at which infectious individuals recover.
So what happens now, then? If we fit this model to the data? So, again, here
we have the boys boarding school flu epidemic data set in black. And we can plot
the results of this new model. And what we see in red is the results of this new model.
What we see is that the red line, the model output, matches the infectious disease outbreak data
pretty closely. I think you'll agree that this actually the model's doing
a pretty incredible job. Given how simple it was, it was pretty simple to construct that model. And
the model only included two parameters. That was a parameter governing the infection rates, which was BITA.
And there was a parameter governing the recovery rates, which was MUE. It was a very simple
outbreak model. And the model output matches the data very well.
So the import conclusion is that by refining best model. So we start with one model, we refined
it twice and eventually we included recoveries. And that allowed us to capture the overall
shape of an infectious disease outbreak.
This basic asylum model is kind of the prototypical infectious disease epidemic model.
It can capture the overall shape of an outbreak. It's a very simple model and involves many
simplifying assumptions. So, for example, there's an assumption that an individual, as soon as they
get infected, is infectious and starts infecting other individuals. That's clearly not something
that's true. In reality, there's an assumption that everyone mixes with everyone else. There are lots of different
assumptions of this very basic outbreak model that may not be realistic. But what
I'm going to come back to later is how we can extend this basic infectious disease outbreak model
to include additional realism. Even this simple model lives, we
even this essi all model can be used to explore different epidemiological concepts. So that's
why I'm going to do now. It's the first concept that I'm going to talk
about is that if the basic reproduction number or zero. This is a quantity
that policymakers have been talking about a lot during the ongoing nineteen
outbreak. So what is the reproduction number? Zero zero zero is the number
of cases of disease arising from each primary case. So
in other words, it's a measure of if I contracts an infection. It's a measure
of the number of people that I'm likely to go on. And in fact.
You can calculate zero by simply taking the infection rate and multiplying it by
how long an infected individual is infectious for. That's how you go about calculating the basic
reproduction number. And diseases that have very different
values, if the reproduction no generate outbreaks with very different shapes.
So, for example, if you look in the bottom, writes what you can see, as you can see, the results
of an outbreak with very high O0 in blue. So what you can see is that the disease
sweeps through the population very quickly. You have large numbers of infections,
whereas if all zero is lower, like the red, then you have a much lower
peak and the disease sweeps through the population a lot more slowly.
If all zero is down below one, well, that means remember, the Ausra is the number of infections caused
by each infectious individual, if I was is below one. Then on average, each
infected individual will go on and infect fewer than one. Other individuals.
So in other words, I'm like, it's going to infect fewer than one person. And so then what you say
is, you see the disease doesn't spread widely in the population. Then you see somebody looks
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,
all zero is less than one. And so the outbreak simply fades out. There's a purple line running
along the x axis just here. OK.
So another concept you'll have heard about during this outbreak is the concept of herd immunity.
So what is herd immunity? Well, it's the resistance to the spread of a disease. The results,
if a sufficiently high proportion of individuals in the population are in mean.
So in other words, maybe you've got a disease which has been around before and it's infected lots
of people and those individuals have become immune. Well, then that creates resistance to the spread of disease
when it reappears in the population. We can actually see how her community works
by looking at the effects of zero swine zero, remember, was the product of an infection rate her?
And the duration of infection term, which is this term here? Well, the infection
rate term is equal to betore times by N. And in this equation, N represents the
number of susceptible individuals. The number of individuals that are available to be infected.
So if there's a high amount of immunity in the population, what that does is it reduces
the number of individuals that are available to be infected. So in other words, it reduces and.
And so if N is reduced, well, then that means that the basic reproduction number on north is also reduced because
of this formula here. That's what we can do then, is we can look at those graphs
again in the context of immunity. Well, if there's no immunity at all,
then O0 is high. And so you get an outbreak that looks a little
bit like the blue curve. If there's some immunity in the system, though, if N reduces
on wards effectively, then there's some immunity in the system and you instead see an outright that looks a
little bit like the red. And if lots of individuals are immune, then you instead see an
outbreak that looks a little bit like the purple. So, again, immunity being
in the system significantly changes the dynamics of an infectious disease outbreak.
OK, so we talk now about a very basic epidemiological model, the ESSI Amahl, and we
talked about a few concepts relating to infectious disease outbreaks. So concepts like the basic
reproduction number and the concept of herd immunity. I'm going to go on and do now is talk about how
we can extend the basic asylum model. And these extensions are ones
that had uncommonly by infectious disease outbreak models. So the first extension
is to note that infectious disease outbreaks are inherently random.
So in other words, all of the curves that we've seen so far for the number of infected individuals to
a time have been these very smooth curves. Right. The numbers of cases went up. It peaks
and then it comes back down again. And it does so in a very smooth fashion. But real infectious
disease outbreak dates just doesn't look like that. So here are a couple of examples. So
what we have at the top is we have the number of new coronavirus cases every
day in South Korea since February 15th. And what you can see is
that we get the general overall shape of the outbreak. The number of cases goes up, it peaks, and then it comes
back down again in this particular wave that the outbreak.
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.
Similarly, if we look at the bottoms, this is data from Italy. We see not a
complete wave of the outbreak, but we see, again, a similar looking pattern where the numbers
of cases goes up. It then looks to peak and it looks like it's starting to come back down again.
But again, it does it by sort of jagging around a little bit. Not like the smooth curves that we've seen so far
for the essi all model. And we can include this randomness in simulations of an infectious
disease outbreak by using what's called stochastic models.
So the idea of a sarcastic model is that an infectious disease outbreak
isn't simply a deterministic process. Instead, what happens? So if you want
to simulate a stochastic S.A. model, you can simply flip a coin lots and lots of times.
And then according to the results of each coin flip, you can either generate a new infection. So
a susceptible individual becomes infected or you can make one of your infected individuals
recover and become immune. And just to be clear, when we're flipping this coin lots and lots
of times, this isn't a fair coin. It's not 50/50 whether or not there's an infection events
or or recovery events. Instead, the coin is waited and it's weighted according to
the number of sceptical individuals and the number of infectious individuals that are in the population
at that time. So as an example, if you imagine there are lots of healthy individuals
in the population at moment. Well, that means that there are lots of targets for infection. So you'd
expect the chance that the next event is an infection event to be high because there are lots of potential
individuals that could be infected. And so you'd wipe the coin when the number of susceptible
individuals is high so that the chance that the next event is an infection event is high compared
to the chance that the next event is a recovery event.
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
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
the idea. So we flip a coin lots and lots of times and we see this graph, the number of infected
individuals, three time. We can then repeat that if we
want to. We can generate a new simulation of an infectious disease outbreak by
simply flipping a coin lots and lots of times again. And if we do that in a second
simulation on drop suit, identical conditions, we might see an outbreak that instead
looks like the blue curve there. So qualitatively, the blue curve and
the black curve. Pretty much identical. The only difference. So the reason they don't look exactly the same
is that we've got a slightly different sequence of coin flips. We've got a slightly different sequence
of infection and recovery events. So what we can do if we want to
is we can actually buy another simulation again under absolutely identical conditions. So again, we generate
a sequence of coin flips. And if we do that, we might instead see something that looks a little
bit like the red line there. So what's happened this time is we started with one infected individual.
We've generated a sequence of coin flips. But the first coin flip, as indicated,
is the one infected individual has recovered without infecting anyone else. And because they've
been because they've recovered without insects getting anyone else, the disease isn't in the population anymore.
And so the outbreak simply fights out. So by including this randomness in the model,
then you can get very different qualitative to behaviour each time you want a simulation. You can either get
a big outbreak like the blue or the black, or you can get a very small outbreak like the
red. So another
thing you can do to extend the basic asylum model is you can include all sorts of different infectious
disease epidemiology. So I'm going to talk about one particular example in the context
of some of the current government guidance for Kovar 19. It's the current government guidance,
Povey 19 is that if you live with others and you're the first in your household to have symptoms
of a coronavirus infection, then you should stay at home for seven days. Well,
the other members of your household, on the other hand, must stay at home for 14 days.
So what you can see there is that the other members of your household, if you will, the infected one initially,
the other members of your household may not even be carrying the virus yet. They are expected to stay at home
for longer than you are. But this rule, in fact, makes complete sense. And the
reason for that is that there's a delay between an individual becoming infected and an individual
showing symptoms and starting to infect other individuals. So what that means
is that we're supposed to. I get the infection. Well, let's suppose that I'm symptomatic,
let's say for seven days, within seven days, then I might go in, infect someone else in my household
and they may only show symptoms and start infecting other individuals in the second set of seven days.
So that's why it's very important that other household members stay at home for a longer period, for a period
of 14 days. So like I say, the key thing here is that there's a delay between
an individual becoming infected and that individual starting to show symptoms or infects other individuals.
And we can include that in the basic CSI all model by simply including another class
in between being susceptible and being infectious.
So in other words, we can build a model that looks like this one on the screen now where you have two separate individuals
and after infection, they end this E class and each class represents individuals that are infected,
but they're not yet generating new infections. And then only some time later to infect
individuals, become infectious and enter the AI class. And then eventually, after they've been infectious,
they recover and enter the all class. So this novel here includes
this sort of additional bit of epidemiology, the fact that there's a delay between being infected
and then starting to generate new infections. And you could include other types of epidemiology
in compartmental models in a similar way. You can simply introduce new compartments into the
model to represent different aspects of the underlying epidemiology.
So another thing that you might like to include in the infectious disease outbreak model is the idea
that individuals of different ages contacts other individuals at different rates.
So here's an example. This is a graph that shows this quite clearly, I think. This is data
from a paper by Premiss Al and plus computational biology. And this graph was created by one of my Ph.D.
students. And what you can see is so on the x axis, you've got
the age of an individual. And then on the Y axis, you've got ages of
the individuals contacted by that first individual. And then the colour
in the graph represents the number of contacts per day between individuals of those ages.
So what you can see, as you can see, this diagonal here, which is quite dog. So this diagonal
represents individuals on the x axis, contacting individuals on
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
that an individual of a certain age will contact other individuals of the same age.
You can also see some other kind of dark areas as well on this on this particular graph. So here are some dark
areas up here and there were also some dark areas down here. This dark area down
here represents an adult that's roughly of an age of parents
contacting someone that's the age of a child. So this particular streak represents
parents contacting their children. This kind of dark
area up here, which this represents individuals that are at the age of the children of a child.
Contacting individuals that are the age of an adult. So this particular area up here represents
children contacting their parents.
She can really clearly see that there are some very strict age structure in who contacts whom within a
population. You can also look at contacts in different settings. So that's what we have
here. So this middle graph here represents home contacts and you can see very similar
patterns to the ones that I just described in the left graph. So in particular, what you can see
here is that at home it's very likely that individuals contacts individuals of the same age.
And it's also very likely that individuals so parents contact children and children
contact parents. But if you look in another setting, then contacts will look different.
So if you look, for example, at the right graph, well, the right graph represents school age contacts.
And what you see in schools is that the majority of contacts, children contacting individuals
of the same age. So this kind of
age structure wasn't represented in the basic essi all model that we looked at earlier. The assumption
in the I all model was that everyone of every age contacts other individuals that a kind
of constant rates, whereas the data not shown on this particular slide here
shows that that clearly isn't the case. But age structure can very straightforwardly
be inclusion and model like the asylum model. So here's one way to do it. So in this
particular model I'm showing here, we just have individuals of two ages. So we just have children
and we have adults. This approach, though, is the approach I'm showing you here can easily be extended to
any number of age groups you want. So in particular, the data on the previous slide can all be incorporated
in an essay type model. If we want to do the. But here in
this very basic extension, we have individuals that are children and adults. And what we essentially have is we have an essay
all model for children. So that's this one on the top where you have susceptible children, infected
children and recovered and immune children. And then we also have an
essay model on the bottom for adults. So, again, we got susceptible adults, infected adults
and recovered and immune adults. Something to notice there is if we
look at the infection rates term, which of these terms in here, the infection rates depend
on. So this is the rate at which children become infected. That depends on the rate at which children
infects other children. This beta CCE term. And it also depends
on the rate at which adults infects children. So it depends on this beta AC term.
So if children infect others. If children have lots of contact with other children, then you might
expect the B to C, C to the infection rate between children would be high. And
so then the rate at which children become infected will also be high. Similarly, if adults have a lot of contact
with children, well, then the beta AC term would be high because the number of contacts is
high and the infection rate is also high. And then again, children would be infected.
That's a very high rate. So in other words, the data from the previous slide,
the data on the numbers of contacts between individuals of different ages, is included in this model
by the various different beta terms. So via the various different infection rates.
Another thing that we can include in other extensions, the basic S.A.M. is to include asymptomatic
transmission or transmission from individuals with very few symptoms. This has been something
that's been talked about a lot in the context of kov. At Nineteens of Cauvin, 19
infected individuals could have any of a wide spectrum of symptoms.
So an individual could have very clear symptoms. But it is also possible that they have very few symptoms.
And if an infected individual has very few symptoms, then that makes the outbreak difficult to control
because an individual can be spreading the virus without even knowing that they're doing so.
With that in mind, I want to show you this figure here in the. On the left. So this is the figure from
a paper by Christophe Fraser in PNAS in 2004.
Christophe Fraser is now based in Oxford on what you can see as. If so, the value
on the x axis is the proportion of new infections that arise from
individuals that have very few symptoms. So it occurs from individuals either prior to them developing
symptoms or from individuals that never develop symptoms. And what you can see
is that this source cluster here, they source cluster is on the left hand side of this graph.
So what that means is that almost all Saar's infections were from individuals
that were displaying clear symptoms. So that's very important because
what it means is so infections are arising from individuals who are displaying clear symptoms. So
in order to control the outbreak of Saar's in 2003, what could be done is
we could go out and actually find individuals that are displaying clear symptoms. We can make sure
that we isolate those individuals to make sure that those individuals with clear symptoms don't transmit it to anyone
else. And so by isolating individuals with clear symptoms, you can actually bring the outbreak
under control. That's something is unlikely to be possible for Kovik 19, because
it isn't the case that the only infection's out there are driven by individuals with clear symptoms.
So, in fact, a large number of infections are driven by individuals who don't have clear
symptoms. And so that means that we have to introduce control strategies that don't only target
symptomatic hosts, but also target all of us because we might be spreading
the virus without knowing that we're doing so. That's why we need to implement measures like social distancing
that have a huge impact on everybody. So this idea
that about asymptomatic transmission, well, that can be included in a model like the ESSI all model.
So here's an example. So basically, this is exactly the same as the S.A.M. The only thing I want to point
out is that we have this additional class in the model, this a class, and this represents individuals
that are infected and transmits it, transmitting the underlying disease, but
they're not showing any clear symptoms. These are the asymptomatic carriers of the disease.
Something else that can be included in the basic compartmental modelling
framework is the idea of spatial structure. So here on the left, there's
a graph which represents the population density throughout England and Wales.
And there are two features that I want to point out. The first feature is that the population is very wide
spread. And actually, it's pretty unlikely that if you've got an infected individual on the south coast,
let's say that they go on and directly infect someone living in the very north of England. That
is very unlikely. Similarly, so another thing you can notice from this graph
is that the population density vary substantially throughout the country.
And these things can all be included in infectious disease outbreak models like the Asylum
Armidale, on the right hand side is a graph of air traffic routes over
Eurasia. And what you can see here is that some areas are very well connected
and other areas are a lot less well connected. And again, that's something that we can include in an exile
type model if we want to. And this is how we do it. So what I'm considering here
is I'm considering a model which only has two regions and it's a two spatially distinct areas.
You can extend this idea to any number of regions you want. And so what we have in
the first region is we have an essay, although in the second region we also
have an essay or model. And we can include individuals moving between these two regions.
There's some kind of coupling between these two regions, a particular right, which in this model is represented
by Lambda. And if you have regions, some of which are better connected than others,
then you would simply change the value of LAMDA between two different regions. So a higher value of lambda
would represent a better connexion between two particular regions.
You can include different population densities in these models by simply having different numbers of
individuals in total within each region. And in regions
that are a long way apart. So you could include that if you want to in the model by again, assuming a small value
of lambda. Because transmission is perhaps less likely if regions are alone apart from each other.
So that's how one of the lines, the spatial structure can be included in a model.
Yes, I will. So we talked about so far
as we talked about how a very basic infectious disease outbreak model can be developed. That outbreak
model matches the kind of shape of most infectious disease outbreaks.
We talked about various concepts like the reproduction number and like herd immunity. And we talked about
how models can be extended from the very basic model I showed you near the beginning to much more complex
settings that are much more realistic. One to talk about now is how
models can be used to inform control during outbreaks. So clearly
one example of this is the ongoing KOVA 19 pandemic. I first heard about this
pandemic right back in early January. So I think it was something like the April the 9th January
when I saw the note that's on the right hand side of this figure. So I saw
this note. It was it was posted on Twitter. And this is a public health notice that was posted
in the city of Wuhan in China. And this public health notice says
that there have been a number of cases of atypical pneumonia in the city of Wuhan
and thing that these cases appear to all have in common is travel to the Juan Island seafood
market in that city. And so one of the first things that I did
was I started to go about collecting information about Cauvin 19. And I started to go about
developing mathematical models for this particular outbreak. What we
saw as January went on so as January this year went on, is that the numbers of cases starts to accumulate
within China. So on the 20th of January, there were 291 reported cases,
only a couple of days later. There were already four hundred and forty six cases. And by the time we got to the
twenty sixth of January, there were over two thousand reported cases.
And so the question that I want to answer as the outbreak was spreading within China is, well, what
is the risk of getting outbreaks in other countries? How likely is it we see an outbreak
like the outbreak that we're seeing in China, but in the U.K.?
And again, models can be very useful to explore questions like that. So again, this particular graph is
not specifically for Cauvin 19, but it demonstrates a key epidemiological principle.
And that key epidemiological principle is that of the epidemic risk.
So the epidemic risk. So what is that? That means every time you get an imported case, it's simply
the risk that that imported case in a new location generates a large outbreak
there. So do they start chains of transmission that lead on to a large outbreak?
The graph that I'm showing here is the graph. I showed you a bit earlier in that graph.
What I showed you was that when you run up to Capstick epidemic model. You
can either see a large outbreak like the blue or the black, and you could or you can see a small outbreak
like the red. So if the epidemic risk is zero,
what that means is that every time you get an imported case, you're not going to get a large outbreak.
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.
If, on the other hand, the epidemic risk is one, well, then a major epidemic is definitely going to occur.
So in other words, every time you get an imported case, they're going to start chains of transmission that lead to a large
outbreak like the blue or the black, rather than a small outbreak like the red.
Usually the epidemic risk isn't zero one. Instead, the epidemic risk takes a value between
zero and one, and that value represents the chance that any single imported
case will lead onto a large outbreak.
So the economic risk is the probability that an imported case leads onto a major epidemic.
And if you have a higher value of the basic wheat production number like we saw before, that means that the
disease is more transmissible. You have a higher value of the basically production number. Then you have a higher
epidemic risk. In other words, it's more likely that you're going to go on and see an outbreak. It looks like the
blue or black as opposed to an outbreak that simply fades out like the red.
And remember the Arnaut? So the basic reproduction number could be calculated by taking
the infection rate and multiplying it by the duration of infection.
So if you can reduce Donalds, then you can reduce the epidemic risk. And
so right back at the beginning of the Cove, it 19 outbreak when there were very few cases in the UK.
Only cases that were imported from outside. The question was, well, how can we reduce our Nords
and therefore reduce the chance of getting a large outbreak in this country?
And the answer to that was, well, either you could reduce the infection rate in some way or you
can reduce the length of time that individuals are infectious. Four.
And so the initial policy in the U.K. aimed at controlling the Kovar 19 outbreak was to
reduce the length of time that individuals are infectious four. So, in other words, to go out
there and find imported cases and find all of their contacts
and make sure that those individuals were isolated quickly by selecting those individuals quickly.
You're reducing the duration of infection for those individuals who will therefore reducing
are zero. And in turn, your reducing the chance of a large epidemic in the U.K.
So that was the initial policy. This is a paper that I wrote on this topic right back in January.
And the key conclusion is fast isolation of imported cases can reduce the epidemic
risk in countries other than China. So what you need to do is you really need to rapidly
isolate any imported case and their contacts if you want to reduce
the chance of having a large outbreak in other countries. Unfortunately,
what then went on to happen is we did see large outbreaks elsewhere. The epidemic risk wasn't reduced
sufficiently. And so what then happens is we enter a different phase of the outbreak.
So then the question for modellers is, well, what can we do when a major outbreak is ongoing? And so this is the
kind of procedure that we go through. What we do is we observe data from the ongoing outbreak.
We then develop an epidemiological model, so a sort of compartmental model of the type that I
showed you earlier. The model I've shown here in step number two is a model that I developed for Ebola
virus disease. What we then do is we estimate the parameters of the model.
So we estimate the values of parameters like the infection rate that we looked to earlier or like
the recovery rates. So we choose those parameters so that the output of the model
is consistent with the data that we've observed. And then once we've got a mathematical
model, we can run simulations of it forwards to make a forecast
and then we can take those simulations and introduce different control interventions in the model. And look at what happens
in the model under different possible control interventions. And the aim there isn't to predict
precisely how many cases there are going to be or precisely when the outbreak is going to peak.
Instead, the aim is to look at different possible controlled interventions and look at which are most
likely to have beneficial effects and which are most likely to have the largest beneficial
effects in terms of reducing case numbers. So these
are the kinds of things that one might find. So, again, this is not for Cauvin 19. This is just general
infectious disease outbreak model. Something that you might have heard policymakers,
the prime minister's talked about this a lot, is the idea of flattening the curve. One of the things that we want to
do is we want to flatten the curve. What does that actually mean? Well, we can think about that
in terms of social distancing. So social distancing involves reducing
the rates that we contact other individuals. If we reduce the rate at which we contact individuals,
we reduce the overall infection rates because everyone's having a lot fewer contacts. And so we're likely
to pass the disease on to fewer people. And so if we reduce the infection rate,
what we do is we reduce this parameter BCO. And if we reduce this parameter BITA, the knock
on effects of that is to reduce are zero because our zero depends on this parameter.
Bita. And if we reduce our zero, we'll epidemiological
models tell us. That we can go from something like the blue curve in this graph
to something that's a bit more like the red curve. And this is highly desirable
during an infectious disease outbreak. And the reason for that is not. Well,
one of the key things that you might want to do during an infectious disease outbreak is make sure that the number of
infected individuals at any single time remains below the capacity for
treatment. So in other words, try and manage the number of infected individuals in such
a way that health sector health care services can cope.
So what you can see here is that in the blue case. So when honour is high. So without something like a social distancing
intervention, the disease sweeps through the population very fast. The peak
of this outbreak is very high. What that means is that there's a large number of individuals infected at any
one time. And because of that, it's very difficult for healthcare services to cope
because they have to deal with a huge amount of cases all at a single time.
In contrast, if you introduce an intervention that reduces our Nords, that has two effects.
Firstly, it reduces the size of the peak. So if you look at the red curve, then the peak is much lower
than for the blue curve. What that means is that at any one time, health
care services aren't having to treat as many infected individuals. And that's
clearly a beneficial thing. But the other thing, the other benefits in terms
of health care services is that if you can reduce your Nords, then the peak also becomes later.
And that's really important because it allows health care services more time to prepare
for the peak number of infections. And that's a very good thing because it allows health care services
to increase their hospital bed capacity, for example. It allows them to distribute personal
protective equipment to health care workers. It allows them to make sure that they have ventilators
to increase the number of ventilators they have. In other words, it allows health care services
to better be able to deal with the number of individuals that they might have to treat.
So clearly, this idea of flattening the curve is reducing Arnaut, making sure that the peak number of
individuals that are simultaneously infected because later on at that peak is lower. And that's clearly
good for health care services. Something else we talked
about a little bit earlier on is the idea of herd immunity. So the resistance to the spread of disease, the results,
if a sufficiently high proportion of individuals are immune. Well,
vaccinations who go the vaccination. This is actually allows you to increase the number
of immune individuals. So remember that we talked about that earlier in the context of Arnaut. So
Arnaut was given by this formula here where N is the number of individuals that are susceptible
to the disease. And if you go and vaccinate lots of individuals, well, that reduces
the value of n it reduces the number of individuals that are susceptible to the disease. And in turn,
that reduces our Nords. And if we reduce our N again, that changes the shape
of the outbreak curve as a mistake or on the left, you can go from something that looks like the blue to something
that looks a little bit more like the red. And if you can continue to decrease
Arnalds, well then the immunity in the population becomes even higher and eventually
you get to a point where Arnaut falls below one. And outbreaks can't happen at all.
So hopefully. So I think a vaccine is of the order of a year away. But if a vaccine
can be deployed widely in the population covered 19, that can prevent this disease recurring
in future. And that's clearly something that's very important.
These two concepts I've just been talking about were in the context of very basic epidemiological models.
Of course, as I showed you earlier, you can extend these models to become much more complex.
And with these more complex models, you can also test very complex control strategies.
So here are a couple of examples for Kovar 19. And these are examples of output's for models
that have been developed by Imperial College London for the graph on the right and the London School of Hygiene
and Tropical Medicine for the graph in the bottom left. And the idea
of these graphs is to test a more complex control strategy. I should say that the London School of Hygiene
and Tropical Medicine model in the bottom left. It includes all five extensions
to the basic ESSI all model that I told you about earlier. So it's much more complex model.
And what you can see here is you can see quite complex dynamics.
So the idea here. So the thing that's being tested is the idea of reducing our zero
periodically by having lockdowns. So having a lockdown for a period of time to reduce the reproduction
number, that then reduces the numbers of cases and therefore reduces the pressure on
health care services. And then you can release the lockdown so that people can
go about their daily lives in a more normal fashion. And when you reduce the lockdown, then the numbers
of cases goes up again. So, for example, if we look at this graph,
is this one from Imperial College London. The blue lines represent whether or not
the lockdown is on or off. So in this period here where the blue line is high, then the lockdown is on.
And in this period here, the blue line is low and the lockdown is off. So what you can
see in this graph here is that the numbers of cases increases three time.
Then the lockdown gets implemented, and so then the numbers of cases eventually starts to decline
again and it falls back down again to low levels. But when the numbers of cases are low,
then the lockdown can be taken off. But then the numbers of cases increases again just here.
When the numbers of cases increases again, you might put the lockdown back on again to try and manage the numbers of cases
you've got and therefore the pressure on health care services. That's when the lockdown is back on again.
Then the number of cases falls back down again and so on. And you see this repeating pattern where you put
the lockdown on cases fall down again. You take the lockdown off, cases increase. You put
the lockdown on again. Cases fall down again and so on. And the benefit of this kind of strategy
is, is it allows you to manage the number of cases of disease at
any particular time. And by doing that, you can keep the need for intensive
care unit beds low enough so you can keep the number of infects individuals low
enough. The hospitals are able to cope with the number of individuals requiring intensive care
unit beds. So that's kind of more complex model and a more complex
control strategy. And you see a similar thing from the London School of Hygiene and Tropical Medicine model in the
bottom left. You put a lockdown on the cases, fall down again. You take a lockdown off the people
who go about their everyday lives in a more normal fashion and then cases go up again
and so on. And you repeat and you get this kind of oscillatory behaviour, which in theory might be able
to keep the bad demand in intensive care units
below the number of intensive care unit beds that the health care set this house.
OK. So that's something a little bit about how metals can be used when outbreaks are ongoing.
When you can't see any more cases. So when we're right towards the end of an infectious disease,
outbreaks models are still useful. So this is an example of some work that we did for Ebola
outbreaks. And the idea here is that we have this model, which I showed
you earlier, this model in which it's an ASIO model. But there are also individuals that may not
be reporting disease. So in other words, you have asymptomatic infections
which don't get recorded in routine surveillance data. So in other words,
you can be in a situation where you haven't seen any cases for the last five days. But there may still be cases
out there because there are these asymptomatic infection individuals that you simply don't see.
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
models to say, well, if we haven't seen any cases for the last 10 days, let's say,
how likely is it that the outbreak really is over? And so we developed this figure for
Ebola outbreaks, which shows that if you wait a long time without seeing
any symptomatic cases when you can be very confident that the infectious disease
outbreak is over. But if, in contrast, you only wait for a short time. So let's say it's been five
days since you last saw a symptomatic case. Well, then you can't be very confident that your Ebola
outbreak is over. If you wait for this sort of W8 show guideline period
before declaring an Ebola outbreak over, then this very simple
outbreak model suggests that, well, after 42 days, which is the WHL guideline, before declaring
an Ebola outbreak to a finished, there's approximately eight to two percent chance that that Ebola outbreak
really is over. And about 18 percent chance that the outbreak, in fact, isn't over
and that there are still hidden cases out there that you just can't see.
OK. So it's conclude then. So we talked about infectious disease outbreak modelling and we said precisely
what an infectious disease outbreak model is. We've shown that infectious disease outbreaks have a characteristic
shape. The numbers of cases in a single wave outbreak goes up, it peaks, and then it comes
back down again to near zero. Even very basic infectious disease outbreak models
can capture that kind of characteristic shape. Those models can then be extended to include
additional realism. So things like, for example, transmission from individuals aren't showing
symptoms or things like different transmission rates between individuals in different
locations or individuals of different ages. We then talked
about how models can be used at different stages of an outbreak for doing various things, including making
forecasts and predicting the effects of different potential control interventions. For example,
interventions like social distancing that affect the reproduction number and
the idea behind using mathematical models for making forecasts and predicting the effects of different interventions
is shown at the bottom. So you have some data from an ongoing outbreak. You then develop
a mathematical model that can replicate, that can reproduce those data. You were following
the model to make sure that the model can reproduce the data more accurately. And you also
refine the model as more data come in during the outbreak. Once you've got the model,
you can then use simulations forward to generate a forecast as to how many cases you might expect to
see you going forwards. And then you can introduce different public health measures in your mathematical
model to look at how different interventions are likely to change the numbers of cases you might
be expected to see in future. And in that way, mathematical modelling can help you
to prioritise public health measures during an outbreak. So I'm going to stop
there. I'm gonna say thanks. Thanks again very much for joining me for this Mathematical Institute's
public lecture. Live from my home. Please send in any questions you've got via social
media. And we'll be answering a selection of those over the next couple of days.
Thanks again.