Quantifying indirect and direct vaccination effects arising in the SIR model

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.