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Flu Math




GARY ZAMCHICK

A recent short article in Scientific American Mind looked at the following question: suppose you have a 5 percent chance of dying from a flu vaccine but a 10 percent chance of contracting and dying of the flu when an epidemic strikes. Do you take the flu shot? Surprisingly often, people do not. (See "Which Flu Risk Would You Take?," by Nicole Garbarini, in Head Lines, Scientific American Mind, Aug./Sept. 2006).

This apparent irrationality is commonly attributed to "omission bias"-people often prefer inaction to action, even if inaction carries some greater risk. But I think there are other reasons, too. For one thing, authorities commonly have a bias toward action, if only to justify their own existence. If so, the consequences of the flu may be overstated or the risks of the vaccine are understated.

But even if the listed probabilities are to be believed and the bias toward inaction is corrected, there is the observation that the risk of being unvaccinated decreases if most other people take the flu vaccine.

Suppose in fact that your likelihood of dying from the flu when unvaccinated goes down according to the following formula: if a fraction f (excluding you) of the population takes the flu shot, then your probability of dying from the flu is just (1-f)*10%. For example, if 65 percent of the people take it and you are among the 35 percent who do not, then the probability of your dying from the flu is only 0.35 * 10% = 3.5%.

Warm-Up:

Suppose a government official could require 60 percent of the people to take the vaccine. Then what would be the average risk of death for the entire population due to the flu?

Solution to Warm-Up:

Recall that if the official doesn't allow anyone to take the vaccine, then the death toll is 10 percent. If the official forces everyone to take it, the death toll is 5 percent. If 60 percent take the shot, then those people have a 5 percent chance of dying, but the others have only a 40 percent chance of dying from the flu, so the death toll among them is 4 percent. The overall death toll therefore is (0.6 * 5%) + (0.4 * 4%) = 4.6%.

1. As long as the government is in a compelling mode, what fraction should be required to take the vaccine to minimize the average risk overall?

On the other hand, suppose that the government feels it cannot compel people to take the shot. Instead, the government offers each person the flu shot in turn. If a person refuses, then there is no second chance. Each person knows how many people took the flu shot among those already offered. Each person believes and knows that everyone else believes the government's risk figures (5 percent if a person takes it; 10 percent modified by the f formula above if a person doesn't). Each person will take the flu shot if and only if it helps him or her. There is no regard for the greater good.

2. Under those conditions, what percentage of people will take the flu shot?

The government looks at the results and decides that a little benevolent disinformation is in order. That is, the government will inflate the publicized risk of death from the flu to a number R that is greater than 10 percent, but the decrease in risk will also be adjusted to use R instead of 10 percent-that is, to (1-f)*R. The exaggerated risk strategy is a carefully guarded secret, so everyone believes the government and knows that everyone else does too.

3. Again, the government will compel nobody. To which percentage should the risk of disease be inflated to achieve the optimum vaccination level you determined in your answer to question 1?

Disclaimer: The actual death rates associated with the flu and flu vaccines are typically far smaller than the numbers used in these examples. The scenario of benevolent disinformation is purely invented. Purely.

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