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The Misunderstood Number That Predicts Epidemics

'Contagion' got it wrong.
October 6, 2014, 10:55am
Image: Pressmaster/Shutterstock

R-__nought, also known as R__0 or the Basic Reproduction Number, is already famous. A scene in the movie Contagion finds a blogger character working out the math for some virus, calculating what an R__0 value might mean for its spread.

In the film, the value is explained as the number of people that a given infected person can expect to infect before becoming non-infectious (by dying or recovering, for example).

In more proper terms, r-nought can be explained as the number of secondary cases that can be expected to result from a given primary case. That's the literal textbook definition, as laid out in the volume Mathematical and Statistical Estimation Approaches in Epidemiology.

The goal in containing a given outbreak can be simply defined in terms of r-nought as the intensity and type of interventions necessary to limit the r value for a given outbreak to less than 1. If r < 1, on average, it means that the epidemic is declining and can be considered to be officially under control.

R-nought is a popular number right now for obvious reasons. It's become a popular way to tell people to either panic or chill out about Ebola in the United States, where the disease's magic number hovers somewhere between 1.5 and 2. Next to the measles r-nought of 12-18, Ebola's transmissibility in the US makes it look like trying to conduct electricity through bark. For those in the panic camp, however, it can be taken to mean exponential growth, where each person is necessarily infecting two other people.

That's not how it works, however, as r-nought isn't a fixed number and doesn't exist within a fixed relationship. It changes over time as interventions are put into place. As NPR pointed out, if all of the people that came into high-risk contact with the Dallas Ebola patient-zero are isolated, we can expect r < 1 without much of a fight.

If all of the people that came into high-risk contact with the Dallas Ebola patient-zero are isolated, we can expect r < 1 without much of a fight

It's a more difficult task in West Africa, for reasons that include poor healthcare infrastructure, relatively intimate methods of handling the dead, and basic fear of the sort that keeps potentially infected patients away from medical establishments.

The West Africa r value is more like a steady 2, and wrestling it downward is a much, much greater challenge than in an American city.

It turns out that r-nought isn't nearly as simple as it might seem. It also ties in with the SIR model, which models the progress of an epidemic by relating the relative proportions of sick (infectious), recovered, and susceptible people in a given population to each other. This relationship is defined in a few clean differential equations that tell us the rate of change in the infected/infectious/sick fraction of the total population.

In the SIR model, we're given a variable B. B is generalized as the number of contacts a person is likely to have in a given day. That's obviously a kind of goofy way to look at things, because what difference would the number of contacts even make if we don't know how easily the disease is transmitted? So we have to come up with a B that takes into account the ease and method of transmission.

If we accept "contacts" in everyday terms, the whole thing is meaningless. Think about the different meanings of everyday contact for, say, HIV or chicken pox. One needs the direct transmission of non-saliva bodily fluids and the other is spread through virus particles in the air.

The bigger picture in epidemiology is a dynamical system.

This is where our r-nought comes back in. You can't really understand either description of an epidemic, its transmissibility (B or r-nought) or force (the SIR model), without the other. That's why the Contagion explanation—and indeed the popular explanation of r-nought—is so misleading. We can't look at an r-nought value of 2 and say that this means there will be two people infected today, four tomorrow, eight the next day, and so on. It's just a smaller part of a bigger picture.

The bigger picture in epidemiology is a dynamical system. The population in which a disease is being transmitted determines its transmissibility and the transmissibility determines the population. Nothing is static, with all of these different pieces changing and swirling about.

A key part of this interrelationship is how many susceptible people are in a population vs. those who are infected and recovered (or immune). A very large portion of susceptible people means that an illness has a lot more "fuel" than if the population had some preexisting immunity (such as that conferred by a vaccination program).

If you have near-zero infected or recovered individuals, you can expect an r-nought value to mean something like exponential, but as the number of susceptible individuals declines, the overall change in the rate of infections declines as well.

So, if you hear someone throwing around r-nought numbers like a Hollywood epidemiologist-blogger, just tell them to chill out and look at some differential equations already.

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