A Primer on the Deadly Math of Ebola

Liberian Red Cross health workers wearing protective suits carry the body of a victim of the Ebola virus in Monrovia on Sept. 12 Photograph by Zoom Dosso/AFP via Getty Images

If the Ebola virus continued to tear through the world’s population at the rate it has done recently in West Africa, all hell would break loose. The U.S. Centers for Disease Control and Prevention estimated on Sept. 26 that cases in Liberia and Sierra Leone were doubling every 20 days. Hypothetically speaking, if the Ebola outbreak continued to spread at that same pace, which it won’t, the number of cases would surpass 1 million by January, 100 million by June, and 1 billion by August, before infecting the entire world population sometime next fall.

To repeat, this is not going to happen. The point of running out the figures on unconstrained growth is to shed light on what things will probably happen in the coming weeks and months that will constrain Ebola, bending its growth curve downward and eventually causing this outbreak to end. These range from better care to more effective isolation to the simple fact that people who are exposed but don’t die will build up immunity. “It’s going to stop growing. What we can’t say yet is when,” says Jeffrey Shaman, an associate professor of environmental health sciences at Columbia University’s Mailman School of Public Health.

Here’s the math behind Ebola: The critical variable is what epidemiologists call the basic reproductive number, which is the number of secondary infections that a single typical infection can be expected to produce, assuming a population in which everyone is susceptible to infection. It’s known as R-zero. An R-zero above 1 means the number of cases will grow, while an R-zero below 1 means the number of cases will shrink. Each current case produces less than one other case.

The R-zero for seasonal influenza is about 1.3, while the R-zero for measles before the invention of a vaccine for it was a sky-high 17, which meant that almost all children had measles at some point, according to Gerardo Chowell, a professor at Arizona State University’s School of Human Evolution and Social Change. Ebola’s reproductive number assuming no effective intervention is probably 1.4 to 3, with most estimates clustered around 1.8 or 2—high enough to be dangerous.

The reproductive number is a product of the duration of illness—assumed to be six days from symptoms to death or recovery in the case of Ebola—and the number of infectious contacts per day that victims have with susceptible people. Cutting the average number of infectious contacts per day in half would cut Ebola’s reproductive rate from 2 to a manageable 1, says Chowell.

This assumes, of course, that people can’t transmit the disease while it’s latent in them, which seems fair because there have been no confirmed reports of transmission by asymptomatic individuals. A complication with Ebola is that corpses are highly contagious, so even death doesn’t end the risk that the disease poses.

One last thing to know is that any disease’s reproductive number falls over time. (The time-sensitive number is called Rt.) People who are exposed but don’t die gain immunity, so they can’t be reinfected. Even in the absence of quarantine or effective treatment, eventually so many people have been exposed to a disease that the outbreak simply stops, like a forest fire that runs out of fuel.

This background helps explain a statement by CDC officials that probably struck some people as odd. In its Sept. 26 report, the agency said that in theoretical terms, “the epidemic begins to decrease and eventually end if approximately 70% of persons with Ebola are in medical care facilities or Ebola treatment units (ETUs) or, when these settings are at capacity, in a non-ETU setting such that there is a reduced risk for disease transmission (including safe burial when needed).”

Why is 70 percent enough? Because perfect isolation isn’t necessary. There just has to be good enough care and isolation that the reproductive number edges below 1—causing the number of cases to shrink over time. Thomas Gift, an economist at the Centers for Disease Control, says the CDC estimates a reproductive number of 0.12 for victims in hospitals and 0.19 for people in a controlled community setting, vs. 1.8 for no protective measures whatsoever.

The CDC’s Sept. 26 report said that there could be 1.4 million cases of Ebola in Liberia and Sierra Leone by Jan. 20, but Gift points out that the estimate assumed no additional interventions or changes in community behavior. “That was sort of a worst-case scenario,” he says. The CDC hasn’t updated its forecast to take account of the commitment of aid by the U.S. and other donors. The World Health Organization hasn’t made a forecast that far ahead. “In our judgment it’s not valid to go any further than four weeks ahead,” Chris Dye, the WHO’s director of strategy, told Bloomberg News. Arizona State’s Chowell says that since there’s a lag in data-gathering, it’s possible that interventions have already begun to reduce Ebola’s reproductive number.

There has been some speculation that the Ebola virus could mutate into one that can be transmitted through the air, rather than through direct contact. Michael Osterholm, director of the Center for Infectious Disease Research and Policy at the University of Minnesota, raised the possibility in a New York Times op-ed column on Sept. 12. But Chowell thinks that’s unlikely, precisely because of the current virus’s fairly high reproductive rate. A random mutation that made Ebola transmissible by air might result in other changes that would make it less transmissible overall, reducing its chance of displacing the existing strain, Chowell says.

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