  Vaccination campaigns have both direct and indirect effects that act to
control an infectious disease as it spreads through a population. Indirect
effects arise when vaccinated individuals block disease transmission in any
infection chains they are part of, and this in turn can benefit both vaccinated
and unvaccinated individuals. Indirect effects are difficult to quantify in
practice, but here, working with the Susceptible-Infected-Recovered (SIR)
model, they are analytically calculated in important cases, through pivoting on
the Final Size formula for epidemics. Their relationship to herd immunity is
also clarified. Furthermore, we identify the important distinction between
quantifying indirect effects of vaccination at the "population level" versus
the "per capita" individual level, which often results in radically different
conclusions. As an important example, the analysis unpacks why population-level
indirect effect can appear significantly larger than its per capita analogue.
In addition, we consider a recently proposed epidemiological
non-pharamaceutical intervention used over COVID-19, referred to as
"shielding", and study its impact in our mathematical analysis. The shielding
scheme is extended by inclusion of limited vaccination.
