A lot of discussion around “living with Covid” starts from the premise that, as long as vaccination rates are high (say 80 per cent of the population), we don’t need to worry about high case numbers. That’s because vaccinated people are less likely to suffer bad outcomes (hospitalization and death). The problem with this claim is that, because the primary function of vaccines is to protect against infection, unvaccinated people will be over-represented among cases. Let’s try some simple arithmetic.

Suppose the vaccination rate is *x*%, and that vaccination gives *y*% reduction in infection risk and *z*% reduction in bad outcomes (hospitalization and death) conditional on infection. Denote the percentage of bad outcomes for unvaxed by *b* %. Assume all unvaccinated will be infected if exposed.

Then, for each 100 people exposed, (100-*x*) unvaccinated and *x**(100-*y*)/ 100 vaccinated will be infected. Example, if *x* = *y* = 80%, there will be 20 unvaccinated and 16 vaccinated. That is, even though unvaccinated people are only 20 % of the population, they will account for more than half the cases.

The number of outcomes will be

*b** = (100-*x*)**b*⁄ 100 + *x* *(100-*y*)*(100-*z*)**b*⁄ (100*100 )

Say *b* = 5%, *z* = 80%, Then we get 1.16 bad outcomes, compared to 5 in the absence of vaccination. So the good news is vaccination works,

But as a proportion of cases, things don’t look nearly so good. In the absence of vaccination *b=*5 % of cases have bad outcomes, but with vaccination *b**=1.16/36 = 3.2 per cent, which is only marginally lowe.

So, the idea that we don’t need to worry about high case numbers if vaccination rates are high doesn’t really stand up. We have to keep case numbers down. Taking the vaccination rate as given, that can only be done with measures like mask mandates, social distancing and vaccine passports.