### How to Test a Vaccine

I have been making some calculations on the alternative ways of testing
a vaccine, and unless I misunderstand something, the current procedure
not only takes longer, it probably kills more people. Here are my
calculations:

Method 1: Give the vaccine to N1 people. Wait a month. If none of them get the disease, conclude that the vaccine works.

Method 2: Give the vaccine to N2 people. Deliberately expose all of
them to the disease. If none of them get the disease, conclude that the
vaccine works.

The following calculations assume:

A: We select N1 and N2 to reduce the chance of a false positive to no more than .05 .

B: Someone not already immune who is deliberately exposed has a .5 chance of catching the disease.

C: The probability that the vaccine works is .1, but if it works it works perfectly — probability of catching the disease zero.

D: The probability that the vaccine not only does not work but gives the recipient the disease is .01 .

In the U.S. at present, about one person in a thousand gets the
disease each month, so with method 1, in the U.S., if the vaccine does
not work each test subject has a .001 probability of getting the
disease. So if it does not work, the probability that none of them get
the disease is .999^N1. If we set N1=3000, that comes to about .05.

With method 2, if the vaccine does not work, the probability that
nobody gets the disease is .5^N2. We set N2=5, giving us a probability
of about .03.

With method 1, the expected number of people who get the disease
because of the vaccination is .01xN1=30. The number who get it because
because they are in the test and the vaccination doesn’t work is zero,
since their exposure is the same as if they were not in the test. The
number who avoid getting the disease as a result of being in the test
and the vaccine working is .3 . Net increase in disease due to Method 1
is 29.7 .

With method 2, the expected number of people who get the disease
because of the vaccination is .01xN2=.05. The number who get it because
of the exposure (and the vaccine doesn’t work) is .9x.5xN2= 2.25 . The
number who don’t get the disease as a result of being in the test and
the vaccine working is .0005. So the net increase in disease due to
Method 2 is 2.3.

For simplicity, I am calculating the number of people in the test who
don’t get the disease as a result of the vaccine over a month in both
cases. It’s small with Method 1, trivially small with Method 2.

Adding all of this up, Method 1 results in 29.7 people getting the
disease as a result of the vaccine trial, Method 2 results in 2.3 people
getting the disease as a result of the vaccine trial. Method 2 also
gives a somewhat lower chance of a false positive and produces a result
about a month faster.

This is obviously a simplified analysis — a vaccine doesn’t have to
work perfectly to be worth using, and my particular numbers were
invented. But given how much larger the first figure is than the second,
the argument that we must use the first because the second is too
dangerous looks implausible unless one believes that the chance the
vaccine gives people the disease is lower than the chance that it
prevents the disease by substantially more than an order of magnitude.

Also, even if there is no chance that the vaccine causes the disease,
the downside of Method 2 is tiny. A small number of people, two or
three with my numbers, get the disease as a result of the test. Since
you will be using healthy young adult volunteers, the chance of death
for each is about one in a thousand. Getting a vaccine out a month
sooner, on the other hand, saves about 20,000 lives in the U.S. alone.

Am I missing anything? Is there any plausible set of assumptions
under which Method 1 is better than Method 2? Alternatively, have I
misunderstood what the methods are?

## 4 Comments:

https://www.iflscience.com/health-and-medicine/this-lab-will-pay-you-to-get-infected-with-coronavirus/

The main drawback, of course, is that while Method 1 saves lives overall, at insignificant additional risk to the study participants, Method 2 entails serious risk

for the participants.You mention that one would use "young and healthy volunteers" -- presumably because they're the most likely to recover if they do get sick. However, that means the study doesn't tell you whether the vaccine works in older or sicker populations. This is a problem on both the false-negative and false-positive sides: you might approve a vaccine that doesn't work (or is actively dangerous) for older people, and you might miss out on a vaccine that only works (or only makes a perceptible difference) among older people. Both of which seem to be realistic possibilities, since we know this particular disease affects people of different ages differently.

Still, an intriguing idea....

@SB A vaccine that was only approved for young healthy people could help us get herd immunity.

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