My Hull is So Big and the Leak is So Small

A Missouri doctor made news the other day by checking in at the airport wearing a Hazmat suit to dramatize his belief that the CDC is “lying” about the dangers of Ebola by downplaying it.  He subsequently made a number of headline-grabbing statements which some may regard as scaremongering or alarmist.  CBS talked to Dr. Gil Mobley to get his point of view. Mobley said:


“For months, doctors in my community — since we had a meeting six weeks ago — have been convinced that the United States will be importing clusters regularly. Right now, on the continent of West Africa, there are a million people in isolation, in quarantine, because of Ebola, and ten thousand passengers leave West Africa every single day. It’s just a matter of time before this disease is carried to every corner of the world.”

“They said the chance of importing a cluster — just two weeks ago — was extremely small, yet we knew that it was a sure thing. And the very same day that the President echoed [Director Tom] Frieden’s sentiment at the CDC that it’s very small, that very same day, they made the misdiagnoses in Dallas and sent this infectious guy home to infect these other people.

“They said the chance of importing a cluster — just two weeks ago — was extremely small, yet we knew that it was a sure thing. And the very same day that the President echoed [Director Tom] Frieden’s sentiment at the CDC that it’s very small, that very same day, they made the misdiagnoses in Dallas and sent this infectious guy home to infect these other people.”

Is Dr. Mobley crazy? After all the CDC asserts that the “chances of importing a cluster [are] very small”. But that may be true without contradicting Dr. Mobley’s argument because they are talking about different things.  The CDC is talking about probabilities (or “chances”) as expressed in percentages while Mobley is talking about expected values which is the number of Ebola patients you can predict will arrive over time.

What is expected value? “In probability theory, the expected value of a random variable is intuitively the long-run average value of repetitions of the experiment it represents.” Think of it as the long run value of a bet. If you have a ten percent chance of winning $500 the expected value of the bet is 0.1 x $500 or $50.  If you’re playing 10 percent for a million the bet is more valuable.  It is .1 x $1M or $100,000.


The chances of an Ebola infectee coming to the US  is the sum of the probabilities it can arrive by all possible paths.  If node 5 in the diagram below represents the US, then it can be reached from node 1 via 1-2-6-5, 1-2-3-5 or 1-3-5.  If the chance of the first two are .0001 and the odds of  the latter are .001 then the total chance of getting from 1 to 5 by all possible routes is .001 + .0001 + .0001 or .0012.

Expected Values and Isolation

Expected Values and Isolation

When the CDC says the probability of a diseased person reaching the US is “very small” they are talking about the betting odds, which in our hypothetical example was 12/10,000 or .0012. But remember that expected value has a second term: the stake at the table.  If there is one Ebola patient with a .0012 chance of reaching America that is one thing, but if there 10,000 people each with a .0012 chance of getting through that is another.  The CDC is saying that they have a revolver with thousands of chambers and only of them has a cartridge in it. Mobley is saying that’s true but if you keep playing Russian Roulette long enough you’ll eventually hit upon a loaded chamber.

Spin the wheel enough times and eventually Ebola gets lucky. The more Ebola patients there are worldwide the more times you spin the wheel.  Dr. Mobley is saying that if the disease keeps exploding in West Africa the wheel will get spun too often.  Someone will get through, because even though the chance of arriving is individually small, there are just too many individuals who might potentially try to get through.

Wikipedia has been following the rate of Ebola infection.  It is evident that the experts are not even agreement except in that their old estimates have been too low. The newer estimates are  higher estimates.


On 28 August, the WHO released its first estimate of the possible total cases (20,000) from the outbreak  … When the WHO released its first estimated projected number of cases, a number of epidemiologists presented data to show that the WHO’s projection of a total of 20,000 cases was likely an underestimate.

On August 31, the journal Science quoted Christian Althaus, a mathematical epidemiologist at the University of Bern in Switzerland, as saying that if the epidemic were to continue in this way until December, the cumulative number of cases would exceed 100,000 in Liberia alone. According to a research paper released in early September, in the hypothetical worst-case scenario, if a BRN of over 1.0 continues for the remainder of the year we would expect to observe a total of 77,181 to 277,124 additional cases within 2014.

On 9 September, Jonas Schmidt-Chanasit of the Bernhard Nocht Institute for Tropical Medicine controversially announced that the containment fight in Sierra Leone and Liberia has already been “lost” and that the disease would “burn itself out”.

Writing in the NYT on 12 September, Bryan Lewis, an epidemiologist at the Virginia Bioinformatics Institute at Virginia Tech, said that researchers at various universities who have been using computer models to track the growth rate say that at the virus’s present rate of growth, there could easily be close to 20,000 cases in one month, not in nine. …

In a 23 September CDC report, a projection calculates a potential underreporting which is corrected by a factor of 2.5. With this correction factor, approximately 21,000 total cases are the estimate for the end of September 2014 in Liberia and Sierra Leone alone. The same report predicted that total cases, including unreported cases, could reach 1.4 million in Liberia and Sierra Leone by 20 January 2015


Dr. Mobley’s estimate that a “million people” are now under isolation in Africa is not quite true. But it might reach the million mark  by year’s end if some of the estimates are to be believed.

Why does he advocate isolation?  Because the effect of isolation is to slice up the graph and cut off the pathways. In the example above the righthand  graphic shows that cutting up the graph makes it impossible to go from node 1 to node 10 because there is no path.  Of course on the real planet earth, total isolation is impossible.  Even if all airline routes were canceled to the affected countries, there would still be some chance that someone could walk to a place where a flight could be obtained.  But the probability goes down and with it, the expected value of an infected passenger arriving.

One result from this analysis is that no one is really safe from Ebola until the the locus of the disease has been beaten down.  No matter how you isolate, the plague is out there.  You are then like Prince Prospero in his castle, safe for so long as the Red Death cannot scale your fortress wall.  But if he does the problem starts again.

In a submarine at depth, the tiniest opening in the hull can be fatal.  The size of the epidemic represents the pressure of water on the hull. It is no more comforting to say “the chances of someone infected coming to America are small” than it is to reassure a submarine captain that the “hole in your hull is tiny”.  That might be OK at keel depth, but if the submarine dives to 1,000 feet that little hole will give you big trouble.

Parenthetically, what is really interesting is where the “Precautionary Principle” has gone to.  The administration used to be really worried about the long shot odds that Michael Mann’s climate model might be right.  But it doesn’t worry about Ebola odds.  Maybe we should apply the Precautionary Principle all of the time or none of the time.  But it’s altogether too confusing to apply it in this uneven manner.


When I was a child I remembered reading the about the exploits of Barton and Beebe who explored  the depths of the ocean in a bathysphere. Their stories impressed upon me forever the power of a little hole under great pressure.

When conducting an unmanned test of the Bathysphere with the third window installed, after pulling the Bathysphere back up they found it almost entirely full of water. Realizing the immense pressure that the water must be under, Beebe ordered his crew to stand clear and began loosening the hatch’s bolts to remove the hatch himself. Beebe described the experience that followed this in his book Half Mile Down:

“Suddenly, without the slightest warning, the bolt was torn from our hands, and the mass of heavy metal shot across the deck like the shell from a gun. The trajectory was almost straight, and the brass bolt hurtled into the steel winch thirty feet away across the deck and sheared a half-inch notch gouged out by the harder metal. This was followed by a solid cylinder of water, which slackened after a while into a cataract, pouring out the hole in the door, some air mingled with the water, looking like hot steam, instead of compressed air shooting through ice-cold water.”

All that came through a hole the diameter of a bolt.  Well, you can cut metal with a tiny, high pressure jet. It’s true the chances of an Ebola patient comingn to America are small, but that doesn’t mean Dr. Mobley isn’t right to worry.

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