## Thee r Rate andthe Vaccination Rate

A virus can only spread if on average each infected person infects one other person before they recover. The number of people an infected person infects before they recover is called the reinfection of r - rate.

Obviously, is some people are vaccinated against the virus, those people will be less likely to be infected and the r - rate will drop. As more people are vaccinated the r - rate will drop further. What proportion of the population will need to be infected for the r - rate to fall below 1, for a given unhindered r - rate?

Suppose each infected person infects
$r$
people. If
$r-1$
of these people were vaccinated then only the remaining unvaccinated person could catch the disease. Hence if a fraction greater than
$\frac{r-1}{r}$
were vaccinated, then the infection would die out. This would be the case if people could be reinfected after having recovered. If people acquired some natural immunity after infection, the vaccinate rate could in fact be lower than this.

This vaccination levvel gves "herd immunity. In practice some groups - will resist vacillation add the infection will continue to infect those groups after the above vaccination threshold haven reached.