In Which Math is Still Hard


I’ve written here several times, under the category “math is hard,” about Americans’ innumeracy with regard to risk and danger. I have a theory:

  • Americans overestimate the danger (risk times harm) of things that they are willing to do something about (terrorism!), and underestimate the danger of things that they are not willing to do something about (obesity!).
  • The government, meanwhile, has an interest in overstating the danger of things that it is profitable for corporations to do something about (terrorism!) and understating the danger of things that it is unprofitable for corporations to do something about (obesity!).

There are 8,000+—about 2^13—Ebola Zaire cases; that number is doubling monthly. There are 2^33 people on earth, give or take. That’s 33 months from one case to everybody being infected, at the current rate. We’ve used up thirteen of those months; we have twenty months left. Of course the current rate of infection can’t be sustained—the virus might not spread as fast in more affluent countries with their indoor plumbing and their medical care, ((Or it might:

  • 9/24: Duncan symptomatic.
  • 9/25: Duncan goes to hospital. Is sent home.
  • 9/28: Duncan returns to hospital via ambulance.
  • 9/30: Officials confirm that Duncan tested positive for Ebola.
  • 10/8: Duncan dies.
  • 10/11: Pham tests positive for Ebola.
  • 10/15: Vinson tests positive for Ebola.

So the first known Ebola case was known in Dallas on September 28th. Seventeen days later, there were three known cases. That’s equivalent to doubling every ten or eleven days—in a state-of-the-art hospital in a modern city among people who knew that Duncan had Ebola, and how to avoid catching it. We should know in the next week—t+25—whether Duncan infected other people before being admitted to the hospital; we should know in three weeks whether Pham or Vinson infected anyone before testing positive.)) and at some point the population becomes sparse enough that the survivors aren’t hanging out with each other much anymore.

It can spread like the common cold flu, it doubles every month, and it kills 70% of the people it infects. ((It’s not very good at its job, which is to replicate. But it is good at killing its hosts.)) How do you stop this world-changing bug?

If you don’t already have an off-the-shelf solution (and we don’t—see fn1), it seems obvious that you buy some time by slowing it down. You quarantine everyone who wants to enter the U.S. after having been in the most-infected countries in the last thirty days. ((This is different than the cargo-cult solution of barring flights from the most-infected countries. Airplanes don’t carry viruses, people carry viruses.))

But there’s not much money for the corporations in a quarantine, so the government plays down the danger posed by Ebola. And the Americans who are willing to do something about it (quarantine!) overestimate the danger  while the Americans who aren’t willing to do it (it wouldn’t work!) underestimate it.

The raw numbers—doubling every 20-30 days; 70% mortality—seem pretty compelling to me. Do I overestimate the danger? It’s possible, but if Ebola cases double in eleven days in a hospital in Dallas, I think it’s reasonable to be extremely concerned about what’s going to happen when it hits the Harris County Jail, and to look to the government to at least try to delay that catastrophe.


8 responses to “In Which Math is Still Hard”

  1. I would say three things about this:
    1) As a math major, I agree: math is hard.
    2) Again, as a math major, this is one of the most horrifying pictures I’ve ever seen.
    and
    3) I’m not particularly worried that I personally will contract ebola, or that it will become endemic in the United States. Why? Because it appears that, already, Nigeria has completely stomped out its ebola cases due to hard work and having pretty good infrastructure in place at the outset. So while I by no means say “don’t worry at all, we don’t have to do anything and we’ll be fine,” I also would put a considerable amount of money on drastic measures not being needed in America itself.

    • Nigeria stomped out an outbreak starting at a single point. It did so with
      Fast and thorough tracing of all potential contacts
      Ongoing monitoring of all of these contacts
      Rapid isolation of potentially infectious contacts
      This reaction is not scalable: what works when you have one index patient coming into the country (18,500 in-person follow-up visits) doesn’t necessarily work when you have ten infected people coming in (185,000 in-person follow-up visits?).
      Tracing, monitoring, and isolation won’t work in Guinea, Liberia, and Sierra Leone—nobody has the resources to treat 9,000 Ebola patients, much less to trace and monitor the potential contacts of 9,000 infected people. The disease is going to have to burn itself out in those three countries.
      While that’s going on, what will rational asymptomatic people in Guinea, Liberia, and Sierra Leone do? Get the hell out, however they can, to wherever they can. Some of those asymptomatic people will be infected. They can’t go to Cote d’Ivoire, which has closed its western border. How will Senegal or Mali deal with hundreds or thousands of Ebola refugees?

      • I certainly agree that for most countries bordering on a nation where Ebola is still epidemic, there isn’t a great good decent way forward. My point was merely in relation to the US.

      • Ah, but let’s take my vision forward. Senegal gets thousands of Ebola refugees; scores of these refugees unwittingly carry the disease, as emerges three weeks later. The government doesn’t have the resources to trace-monitor-and-isolate all of their contacts. Its disaster-response systems are overwhelmed. Ebola explodes in Dakar. Rational asymptomatic Senegalese run for the border, to Mali or Mauretania, or get on boats or planes for more-distant parts. Three weeks later those countries find themselves dealing with scores of infected people, and trying to trace everyone those people had close contact with before making it to the hospital.

        Any country can deal as Nigeria did with one or two patients. Even Guinea, Liberia, or Sierra Leone might have been able to, but by the time the problem was obvious there the problem was too widespread for tracing-monitoring-and-isolation. I doubt that many countries could deal with 20 simultaneous index patients. If the countries that can’t deal with such an influx fall one at a time, at some point enough asymptomatic infected people will get on planes to Paris or Brussels or New York that we find out whether we can.

        If the virus doesn’t get us, the panic will.

  2. Actually, Ebola isn’t that infectious – it has a baseline reproduction (R) factor of about 2 – very low for most diseases – measles is above 10. So a solid quarantine and treatment plan could drop it below 1, meaning it will burn itself out. Ebola is also only infectious while symptoms are showing – so another mitigating factor. My strong concern is more about the competence of the response – inept responses mean that you aren’t reducing the spread. Even worse, this ineptness is jeopardizing the first responders – key people needed in establishing and maintaining a quarantine.

    • I’d like to know if you have some special expertise. It appears to me that R0 > 1 means the disease is spreading and, per Wikipedia:

      When calculated from mathematical models, particularly ordinary differential equations, what is often claimed to be R0 is, in fact, simply a threshold, not the average number of secondary infections. There are many methods used to derive such a threshold from a mathematical model, but few of them always give the true value of R0. This is particularly problematic if there are intermediate vectors between hosts, such as malaria.
      What these thresholds will do is determine whether a disease will die out (if R0 < 1) or whether it may become epidemic (if R0 > 1), but they generally can not compare different diseases.

      In any case, I don’t see a quarantine, and there is no treatment. Nigeria’s successful response to a very small outbreak comprised:

      • Fast and thorough tracing of all potential contacts
      • Ongoing monitoring of all of these contacts
      • Rapid isolation of potentially infectious contacts

      I’ll discuss this more in response to David’s third point, which brought it to my attention.

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