# Smallpox update

A letter in today’s New York Times offers a serious argument on the anti-vaccination side. Current estimates of vaccination risk come from an era where there were few immuno-compromised people walking around. Now, HIV and cancer chemotherapy mean that the population for which infection with vaccinia could create real health hazards is much larger. Even if they’re not vaccinated themselves, vaccinia is communicable. That observation drives up the threshold at which the threat of a smallpox attack outweighs the cost — in the fullest sense of cost — of the vaccination program

Kieran Healy’s note on the smallpox-vaccination question points up the difference between a measurable risk and true uncertainty. If the risk is known, the calculation of whether a given precaution is worthwhile is conceptually straightforward, thought it may be computationally difficult. But an unknown risk poses a harder problem.

Analytically, the right way to deal with that issue is to work the problem backwards. To simplify, imagine a decision with only two options (V for vaccinate and W for wait) and only two possible states of the world: the event the RAND study calls a large-scale attack happens, or no attack happens. The goal is to minimize the expected value — the probability-weighted sum — of the total damage done by the disease cost (if there is an attack) plus the vaccination cost.

Call the cost of, and damage done by, a vaccination program S, for side-effects.

Call the damage done by an attack on an unvaccinated population X, and the damage done by the same attack if vaccination takes place first Y.

Call the probability of an attack p.

The expected-value cost of option V, e[V], is then:

e[V] = S + (Y x p)

because we bear the fixed cost of vaccination in any case, plus an additional probability p of an attack that will do Y damage, where Y is less than X because of the prior vaccination.

The expected-value cost of W is simply:

e[W] = X x p

We face costs only if an attack happens, but then we face the larger cost X.

Given estimates of S, X and Y, such as those provided in the RAND report, we can then compute the “critical value” of the probability of an attack. The critical value is the value that would make the expected costs equal. (I’m simplifying by ignoring the tricky question of whether we should be risk-neutral, risk-averse, or risk-seeking.) If the actual probability is higher than the critical value, vaccination will have the lower expected cost and is therefore the preferred option; else, not.

Since the critical value is the value that makes the two risks equal, we can set up the equality

e[V] = e[W]

which in this case means

S + (Y x p) = X x p

and then solve for p. The value of p that makes the equality hold is the critical value.

Rearranging, we have

p = S / (X – Y)

Call this critical value p*.

That is, the critical value of the probability of an attack is the damage done by the vaccination divided by the lives saved by the vaccination if an attack occurs. It’s worth spending a dime to protect a dollar if the risk to the dollar is at least one in ten.

Using the numbers in the RAND report, something over 400 deaths from vaccination and about 40,000 lives saved (50,000 + deaths without vaccination, 10,000+ deaths without: I’m deliberately rounding here because the precision of the estimates in the RAND report don’t reflect the imprecision of the assumptions that generated them), we get a critical value p* of something like 1%: a vaccination program is worth undertaking as long as the risk is that high or higher. (If the Times’s letter-writer is correct about the importance of immuno-compromised populations, the critical value would be correspondingly higher; if I’m right about the effects on other countries and the possibility of an attack ten or a hundred times as large as the one hypothesized, the value would be correspondingly lower; decision theory is a way to process facts, not a substitute for them.)

So now, instead of moping around, saying “What do you think the probability of a smallpox attack is?” “I dunno, Marny, what do you think the probability of a smallpox attack is?” we have something to concentrate on: is the probability as high as 1%, or not? If there’s more than one chance in ten that Iraq has weaponized smallpox and the capacity to deliver it, and more than one chance in ten that, assuming Iraq has that capacity, it will use it against us if invaded, then we ought to be getting ready for mass vaccination. (Of course if there are significant non-Iraqi risks, e.g. from al Qaeda or North Korea, those enter the mix as well. And something ought to be figured in for the effect of vaccination in depressing the risk of attack.)

Thinking that the risk of a major smallpox attack from somewhere within, say, the next five years, is as low as 1% requires much more precise knowledge of hostile capacities and intentions than I think we actually have. I don’t insist that my conclusion is right, because I’m fully aware of the paucity of data here, especially on the actual damage done by vaccination. But I’d like to see someone lay out the calculation supporting the current policy.