When outbreaks happen, we have to consider some important questions:
How fast could the disease spread?
How many people could get sick?
What would work best to stop the outbreak?
We can’t run experiments on real people to answer questions like these.
But we can simulate outbreaks to see how different scenarios would play out during an outbreak.
In other words, we can use a model.
In this episode, we’ll have a look at what models are, how they help predict an outbreak’s future,
and their potential to help us manage outbreaks in the real world.
I’m Pardis Sabeti, and this is Crash Course Outbreak Science!
[Theme Music]
Before we get started, it’s worth mentioning that since
outbreaks of infectious disease can be fatal,
we’re going to be talking about deaths in this episode.
We know that loss of life and the other impacts of outbreaks are the kind of tragedy
that math and numbers don’t fully describe.
While graphs and models can seem abstract–
who wants to be reduced to a number?
Real people, from the scientists who make models to the public health officials who use them,
to the people saved by their predictions,
are at the heart of the models we’ll be talking about this episode.
Models can help us make decisions that can save many lives in the real world.
Generally speaking, a scientific model is a description of some features of the world
and their relationship to one another.
As long as you have a good understanding of the relationship between the things you’re studying,
you can model pretty much anything, from economies to ecosystems.
To see this, let’s look at something near to my heart: music!
Besides studying outbreaks, I also write and sing in the rock band Thousand Days.
If our band was going on tour,
one of the things we’d want to know is how many tickets would we sell overall.
And with some simple modelling, we could work out roughly what that number might be.
First off, we’re gonna make an assumption,
a simplification about the world that
gets us into the right ballpark by smoothing over some of the details.
Let’s assume that the number of tickets sold depends on how many gigs we play,
and that at every gig a hundred people come along.
So if we play five gigs, we sell five hundred tickets for the whole tour.
That sounds imprecise, but we can always revisit that assumption later on.
Or, in this case, I might already have data that tells me that about a hundred people
always turn up to our gigs.
Right now, our model is quantitative. Its features describe the number of some particular thing,
like the number of gigs and the number of people who attend per gig.
But we might also be interested in qualities or categories of some kind.
For instance, we could make a more detailed assumption that there are two kinds of gigs.
There are weekday gigs, where only seventy five people attend, and weekend gigs,
where up to two hundred people might show up.
Our model now has a couple of different features.
There are the number of gigs, the category of gigs, weekday or weekend,
and the number of tickets sold.
All of these features are things that vary depending on circumstances,
so we call them variables!
In general, the goal of a model is to describe how variables are related to each other
and the values they have in certain scenarios,
such as the five hundred tickets in our example.
But to do this, our model has to rely on numbers that don’t vary,
which are the minimum and maximum number of tickets sold at weekday and weekend gigs.
These fixed numbers that we put into the model are called parameters.
Parameters are usually numbers that come from data, or our assumptions about the world.
They help us understand the relationship between variables,
like how more gigs means we sell more tickets.
In fact, the relationship between variables is really an equation.
Equations capture relationships like these in a way that lets us plug numbers in,
and get numbers out.
The numerical relationships described by equations are helpful because they tell us
the extent to which variables might affect one another.
In our case, the model tells us that the number of tickets scales linearly
with the number of small or large gigs.
So, if you put the number of ticket sales and the number of gigs played on a graph,
it would look like a straight line.
But there are other kinds of mathematical relationships too,
with curvy lines when you plot them on a graph, which we call non-linear.
Non-linear models are a little trickier but they let us study a broader range of events
with complex behaviors, including outbreaks!
One of the models most commonly used for studying outbreaks is the S-I-R model.
The model gets its name from three groups of people.
The first is “susceptible”, the people who haven’t yet caught the disease but could.
The second group is “infected”, the group who currently have the disease
and could spread it to others.
And finally there’s the “removed” group,
the people who have already had the disease and have either recovered
and gained immunity or, sadly, died of it.
The SIR model describes the relationship between these three groups
and how their interactions affect the fraction of people in each group at a given moment in time.
The proportions of a given population
in each of these three groups are the three variables of interest in our model.
The value of these variables matters to us since
the number of people who do or don’t catch the disease
is one of the most important outcomes of an outbreak!
The basic SIR model makes some assumptions about
how the fraction of people in each group changes over time,
based on real-world observations.
Assumption One:
when susceptible and infectious people interact,
it can cause susceptible people to become infected.
So, the number of susceptible people declines over time,
proportional to how many people are in both of these groups.
Like, if there’s a lot of susceptible and infectious people,
more susceptible people will become infected,
so the number of susceptible people declines more quickly.
Assumption Two:
the number of people removed from the susceptible
group is the same as the number of people added to the infectious group
since, they’re infected now.
And Assumption Three:
after people are infected for a while, they move to the removed group,
since they’ve recovered, or sadly, died, and can’t transmit the infection anymore.
And, the decrease of the number of infectious people over time
is equivalent to the increase in removed people.
If we consider what we know about outbreaks in real life, that all makes sense.
Over some period of time, infected people do infect susceptible people
who then become infected,
and eventually, those infected people recover or die.
The exact relationships in the model depend on a parameter called the reproductive number,
or “R” for short.
This is the average number of susceptible people one infected person can infect
and it varies based on the pathogen.
So if R were three, then a single infected person infects three susceptible people on average.
R itself can change over time.
We might start social distancing, for instance,
or the disease might enter a new susceptible population.
So we often define R at the start of the outbreak,
when nearly everyone is in the susceptible group,
and we call it the basic reproductive number or R naught.
Outbreaks happen when R naught is greater than one,
that is, when each infected person can infect more than one person,
making the number of infected people increase over time.
But since R can change, our model variables S, I, and R can, too.
To see how, let’s go to the Thought Bubble.
If we plot the values of our three variables over time, we can see the course an outbreak takes.
On this graph, the horizontal axis follows time,
while the vertical axis shows the proportion of people in each of the three groups.
At the start of our model outbreak,
we have a small proportion of people in the “infected”
group and everyone else in the susceptible group.
R naught is greater than one.
Then, we step ahead in time, and the proportions in each
group change according to the model’s equations:
some susceptible people become infected and some infected people become “removed”,
by recovering or dying when enough time passes.
At first, since the number of infected people is small,
the change in this group is slow and only a few new people become infected.
But as the number of infected people increases,
its rate of change increases too and suddenly there’s a steep rise
in the number of people becoming infected
and a steep drop in the number of susceptible people.
But some people will also be recovering or dying,
which means people will move over time from the infected to the removed group
During this entire time, R is greater than one.
Eventually, at the peak of the outbreak,
since there are fewer susceptible people to become infected,
infections don’t happen as often, which creates a turning point.
At that moment, susceptible people become
infected at exactly the same rate as infected people become removed.
Then, more people will be recovering than becoming infected,
and the number of infected people will begin to drop, since R becomes less than one.
Slowly, the fraction of the population that’s infected tails off towards zero,
and the fractions of susceptible and recovered people reach a steady state,
marking the end of the outbreak.
Thanks Thought Bubble!
There are a few useful observations we can make about these results.
First off, we can subtract the final fraction of susceptible people from the whole,
to find the total number of people who became infected during the outbreak.
We can also predict how long the outbreak could last for,
and at what point the number of infections might peak and potentially strain the healthcare system.
While this version of an SIR model captures the basics of an outbreak,
there are things we can do to make it more accurate.
For example, many infections have a period where a person is infected,
but still can’t transmit them to anyone else.
So we could include another group in our model between susceptible and infected called “exposed”,
which would make the duration of the outbreak more realistic.
There are other details we could include too,
such as the possibility that some recovered people could become infected again,
if immunity isn’t guaranteed.
We can reflect these details in the equations in the model by adding new variables and parameters.
A broader understanding of other factors surrounding an outbreak like these
help improve our model’s ability to predict the course of an outbreak.
Finally, there are other kinds of variables and parameters we
need to include to help make decisions:
the kind that capture our response.
After all, the goal of the model is to tell what could happen so we can act.
We may also include variables and parameters that represent changes during the outbreak
like implementing contact tracing, or social distancing.
These would decrease R since infected people would, on average,
go on to infect fewer susceptible people.
And, as we mentioned, if we can make R become less than one,
an outbreak will begin to decline.
We can run our models with different combinations of these interventions
to forecast the effect they have on the length and severity of the outbreak.
But, at its heart, this model is only a simplified description of the dynamics of an outbreak,
so there are also some challenges to using the SIR model.
For starters, we need to be able to determine parameters like R naught,
which are calculated from real world data.
These might not be available during the early stages of an outbreak.
And even with data, we won’t have a totally exact value for parameters like these.
Instead, they’ll have some uncertainty,
meaning there’s a range of possible numbers that will be compatible with the data.
Since the inputs into our models are uncertain, naturally, their outputs will be uncertain too.
That doesn’t mean the models aren’t helpful at all,
but it means instead of getting a precise prediction like “the outbreak will last 53 days”,
we’ll have estimates, like “the outbreak could last anywhere from 30 to 70 days.”
So models can help our decision making by forecasting which of our actions
could have the greatest impact.
The catch is our ability to predict these outcomes is limited
by the extent to which we can accurately model the effects of the intervention.
But the better we get at developing models, the more useful a model will be.
Before we can add interventions into a model,
we should understand what options are available
to us when an outbreak happens, and how they might help us tackle it.
So, in our next episode, we’ll look at how to intervene in an
outbreak in the context of public health.
We at Crash Course and our partners Operation Outbreak
and the Sabeti Lab at the Broad Institute at MIT and Harvard
want to acknowledge the Indigenous people native to the land we live and work on,
and their traditional and ongoing relationship with this land.
We encourage you to learn about the history of the place you call home
through resources like native-land.ca
and by engaging with your local Indigenous and Aboriginal nations
through the websites and resources they provide.
Thanks for watching this episode of Crash Course Outbreak Science,
which was produced by Complexly in partnership with Operation Outbreak
and the Sabeti Lab at the Broad Institute of MIT and Harvard—
with generous support from the Gordon and Betty Moore Foundation.
If you want to help keep Crash Course free for everyone, forever,
you can join our community on Patreon.