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SIR Model of an Epidemic – Basic Reproductive Number (R0)

February 3, 2020


>>Instructor: Now that
we have the basics down we’re going to look a little more closely at what’s going on with this model. And to do so it helps
us to look at the start of the epidemic because that’s
a really interesting time to see how things evolved, and what happens right at the beginning is also a great interest
when setting health policy as we’ll see in a little bit. So let’s say that we’re
going to define T nought as the start time of epidemic. And at T nought, we have I equal to one. The number of people in the
effected population is one, we have an index case, the first person with this disease. And so if I is one at time T nought, then that means that S,
the number of people in S is approximately N. It’s not quite N, it’s
technically N minus one but if N is large, eh
S is approximately N. So we’re going to approximate S as N, and we’re going to see what
happens when we take a look at this equation at the initial times. So let’s go ahead and look at this. We’re going to say I nought,
rate of change at T zero which I’m just going to
simplify and simply call it I dot at zero, is going to
equal to beta S I over N minus gamma I. But because of our assumptions
of what happens at T nought, I is just one so that
goes away, that goes away. And similarly S is approximately N and look, that means we’ll
get to rewrite this as N and N over N is one so
this just goes away. And what are we left with? We’re now left with beta minus gamma. Well that’s easy. It’s also very important
and extremely convenient. This is going to look a
little weird, but humor me. What I’m going to do next
is I’m going to take this and multiple and divide by gamma
for a very specific reason. Because what this comes
out to then if I do that and I distribute as I want,
is I’m going to get a term gamma times beta over gamma minus one. That was set up intentionally, why? Because this property
right here, this value is R nought. We said this is equal to
gamma R nought minus one. And look at what this is saying here, this is saying that the rate of change, the number of people that
are added or subtracted to the infected population at
the very first time initially as the epidemic is just about
to grow because we only have one infected person is determined entirely by this equation, gamma
beta over gamma minus one. And that is that beta over
gamma’s termed as R nought. If you look at this you can see then that for an infection to be positive, for the rate of change here
to be greater than zero, for new people to be added to this pool R nought here has to be greater than one. And if it’s greater than
one you have an epidemic. That means in time as the
dynamical system evolves across time, the number of
people in I are going to grow. If R nought is less than one, you will not have an epidemic. No epidemic, why? Because the number of
people that are being added, the rate of change of
the infected population is going to be negative. So even from the very first
case the infected pool is going to fall and you
aren’t going to get a spread of the disease. This term R nought is critically
important in epidemiology and health policy, it’s called the basic reproductive number of basically of an infection. And epidemiologists and
people that set health policy are very concerned about
what this R nought is because if it’s greater than one, suddenly that particular type of disease is a health risk in populations. If it’s less than one,
they tend not to care. How much greater it is than one is also of serious concern. If this number is extremely large like in the case of say, measles, which is something when
they were like 13 or 15 then that particular disease
can spread like wildfire throughout a population, because what is it saying? It’s saying that for every
person that’s infected R nought number of additional people are expected to be infected
by that individual person. So it’s the rate at which,
it’s how many multiples of people are infected based
on every individual index case. So measles spreads really quickly, other diseases not so much. We’re really terrified about Ebola and we hear a lot about
that often in the news. But turns out its R nought is on the order of like two or three. So it doesn’t really post
that much of a health risk in a first world society because we have good health policies, we have good quarantine practices, and with an R nought
of around two or three as long as you have maintained
good quarantine policies, you’ll be fine. It won’t really spread throughout
the population very much. However we could never
pull that off with measles. Measles spreads way too quickly for it to simply be solved and addressed by quarantine policies. So for something with
an R nought that large, you really would like
to see a vaccine present because that’s the level
of protection you need to minimize this spread of infection when you have an R nought that large. Mercifully we do have
a vaccine for measles. But at the same time this helps explain why the need or urgency
for an Ebola vaccine at least in the U.S. is
really not that big of a deal because we can organize
the policy procedures such that quarantines are effective. So there it is, there’s
the basic introduction to looking at how you can start
poking around this equation, this dynamical system,
these systems of equations. Look through some very simple analysis at what happens around time nought and derive a very key
principle, a very key parameter called R nought which is the
basic reproductive number of a disease.

2 Comments

  • Reply Divyanshi Karmani April 4, 2019 at 9:01 am

    Thank you for this effectively informative video. Helps with my coursework 🙂

  • Reply khalid elhail December 25, 2019 at 11:46 am

    thank u

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