Explainer: The basic maths mistake anti-vaxxers make when criticising the COVID-19 vaccine's effectiveness

• 25/01/2022

Despite unequivocal evidence the COVID-19 vaccine significantly reduces the chances of dying or being hospitalised from the virus, anti-vaxxers are still clinging to an age-old argument:

"If the vaccine is so effective, then why are more vaccinated people currently in hospital in New Zealand with COVID?"

Dr Leighton Watson, an Auckland-based geophysicist and mathematician with the University of Oregon, said while the question may seem confronting at face value, it's actually just a mathematical error.

Dr Watson based his calculations on the most up to date figures as of Monday morning, which came via the Sunday Ministry of Health media release.The data showed there were eight people in hospital with COVID-19. Since then two more people have been hospitalised, one of which is unvaccinated. The other's vaccine status hasn't been reported. For the eight cases Watson used, vaccination data is only reported for seven. Of those, one is unvaccinated (14 percent), 5 are fully vaccinated (71 percent), and one has an unknown vaccination status (14 percent).

Dr Watson said while the data appears to show vaccinated people are more likely to be hospitalised than unvaccinated people at face value, that's not the case. He said there are several factors that make this assertion incorrect.

Firstly the vast majority of New Zealand's population is vaccinated. Ministry of Health figures show of those aged 12 and up, 94 percent are fully vaccinated and 96 percent have had one jab. Additionally, more than 1 million Kiwis have also received a booster jab.

"We will see more cases and hospitalisations in vaccinated people simply because there are so many more vaccinated people," Dr Watson said. "This does not mean the vaccine is not working or is not effective."

And accounting for high vaccination rates shows unvaccinated people are still much more likely to end up in hospital, even using Monday's statistics which showed more vaccinated people in hospital in total.

"Accounting for this [New Zealand's high vaccination rates] gives a hospitalisation rate of 0.00128 per 1000 people for vaccinated and 0.0034 for unvaccinated. This calculation shows that unvaxxed are 2.7 times more likely to be hospitalised."

He noted the number is significantly lower than previous modelling which showed unvaccinated people were 25 times more likely to be hospitalised. But he said the limited data skews the results and it's better to look at the entire outbreak.

"Eight individuals is a very, very small number of data points to do any statistics on and hence estimates from this data cannot be reliably extrapolated to the population as a whole.

"Because the numbers are so small, having one more or one less unvaccinated or vaccinated individual drastically changes the relative likelihood."

Dr Watson said the data could also be skewed because the Ministry of Health doesn't report why people are in hospital, just that they have COVID.

"MoH does not report on if the case is in hospital due to COVID-19 or in hospital for an unrelated medical case and has COVID-19. Therefore, some of the vaccinated individuals could be in hospital for other issues."

He also pointed out the modelling assumed unvaccinated and vaccinated people were equally likely to be exposed to COVID, but that's not the case.

"Government mandates mean that most of the people in high-risk settings (MIQ, healthcare) have to be vaccinated. So vaccinated people are more likely to be exposed to COVID-19.

"In the COVID-19 Protection Framework, unvaccinated individuals are barred from high-risk places and so less likely to be exposed to COVID-19."

Dr Watson said looking at data from the Delta outbreak shows unvaccinated people are 18 times more likely to be hospitalised with the virus than vaccinated people.

He said overall in the Delta outbreak there were 3907 unvaccinated cases and 2159 vaccinated cases. But there were only 49 vaccinated hospitalisations compared to 399 unvaccinated. He said it's tricky to calculate the relative number of cases and hospitalisations as normalised by population because the vaccination rate changed during the outbreak.

But he said if you assume a vaccination rate of 70 percent and normalise the cases and hospitalizations by the size of the vaxxed/unvaxxed populations, it's clear unvaccinated people were significantly more at risk.

Dr Watson said looking at cases per 1000 people, 3.1 were unvaccinated but just 0.73 were vaccinated showing unvaccinated people were 4.2 times more likely to be infected.

And it's even more startling with hospitalisations. Per 1000 people, 0.32 were unvaccinated and 0.017 were vaccinated meaning unvaccinated people are 18 times more likely to be hospitalised.

"The result depends on the exact vaccination rate you choose but you can see unvaccinated individuals are more likely to be infected and hospitalised at all vaccination levels."