\[r\]

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.