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# The apparent decline in vaccine effectiveness might be just a statistical thing.

The anti-vaxxers are bound to jump on stories like this (headline about declining vaccine effectiveness chosen at random – but no blog post is complete without a hyperlink).

Simplified assumptions

1. In the unvaccinated population, ten per cent of the population will catch the disease each month.
2. You either recover and are immune for several years, or you die.
3. The vaccine is “60% effective”, i.e. in the vaccinated population, only four per cent will catch it each month.

So the sensible decision for any individual ex ante is to get vaccinated.

(It appears that in real life, the Covid-19 vaccines are only about “60% effective” against infection, but about “75% effective” against having a severe case and about “90% effective” against dying, but that then makes comparisons more complicated, so let’s just use the simplified assumptions and look at total cases.)

Worked example, first 24 months of the epidemic

Country A and Country B both have a population of one million.
Country A doesn’t do vaccinations; Country B (miraculously) vaccinates its whole population on Day One.
So in month 1, there are 100,000 cases in Country A and 40,000 cases in Country B.
That’s a clear win for Country B.

As time goes on, it becomes less clear who’s winning…

By month 15, there will be about 23,000 cases in each country, there being ever fewer people in Country A who can catch it.
After that, there will be more cases in Country B, there being still plenty of people who haven’t caught it yet.
By month 24, there will have been 920,000 cases in Country A (nearly everybody has had it) and in Country B there will have been 625,000 cases.
So if we calculate vaccine effectiveness after 24 months, looking at total figures, the vaccine was only 32% effective.

But it was still a good decision to get the vaccine ex ante, even if ex post it doesn’t seem to have made that much of a difference. Even if the worst happens, you have still gained another few months or years of healthy life.

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