British regulators have shut down Chiron Corp.’s flu vaccine plant in England, taking half this year’s vaccine supply for the U.S. with it. We’re going to be short. Lots of people are going to get sick, some of them are going to get very sick, and a small number of them will die, as a result.

(Wth a shortfall of 40 million doses, I think we’re looking at a couple of million flu cases, tens of thousands of bad complications, and tens of deaths. However, these are guesses; it’s not my field and I haven’t seen any published numbers.)

Questions: Should we have planned for a bigger supply of flu vaccine? Should we do so in the future?

The quick answers, I think, are yes and yes. Here’s why I think so, subject of course to correction by those with more facts.

Flu vaccine is still grown in hens’ eggs, and therefore takes months to make. If can’t be stored from year to year. So we face the classic grocer’s inventory problem from a first course in operations analysis: Given uncertain demand, how much stuff should you buy, given that you can’t sell what you don’t have but that whatever you don’t sell has to be thrown away?

The answer turns out to depend on two things beyond the estimated demand: the cost of running out (“stockout cost”) and the cost of having too much (“overstock cost”).

For any given level L of inventory, imagine buying one additional unit. The probability distribution of demand gives you some probability x that you will need that additional unit (be able to sell it, in the grocer’s case) and the complement ary probabilty 1-x that you won’t need it.

The overstock cost is the what you paid for the extra unit, plus any costs associated with getting it, moving it, storing it, and discarding it.

The stockout cost is the net benefit of having it (what the grocer could sell it for, plus the value to him of not disappointing a customer who might therefore not return, minus the costs of acquiring, moving, and storage).

The solution to optimal inventory problem is surprisingly simple: the ratio of stockout risk (not having enough) to overstock risk (having too much) should the the inverse of the ratio of the costs. If it’s nine times as bad, per unit, to run out as it is to have too much, then at the optimal level of inventory there will be only one chance in ten of running out.

A quick calculation suggests that the ratio for flu vaccine is at least in that range.

The stuff costs a few dollars a dose to make, and one chance in twenty of avoiding three days’ lost work and the associated suffering, plus the small chance of getting really sick, must be worth tens of dollars.

(Say the risk of getting the flu if unvaccinated were one in ten, and the vaccine reduced that risk by about two-thirds, which is the published number. That means that a flu shot has about one chance in fifteen of preventing a case. If the typical flu case costs a few days’ lost work, the cost of that loss alone is a few hundred dollars, giving us an expected value in the tens of dollars. Add in something what the victim would pay not to be sick, and something more for the small risk of bad complications, and the expected value of the vaccine must be at least ten times its cost. The numbers are presumably even stronger for high-risk groups.)

But it’s not the case that we only run out once in ten years. And the Chiron experience points up the fact that having only two suppliers greatly increases the risk of a shortage. So an optimally run single-payer health-care system would have multiple suppliers — even at a somewhat higher cost per dose — and routinely buy considerably more vaccine that there’s expected to be demand for.

Query: Is there any way to achieve that result, or something close to it, given the actual industry and regulatory structure?