# Final-Size Solutions for SIRI Models with Vaccination

## TL;DR

This paper extends the SIR model to account for partial immunity and vaccination, deriving formulas for epidemic final size and reinfection dynamics.

## Contribution

The paper introduces analytical expressions for epidemic final size with partial immunity and vaccination, including reinfection thresholds.

## Key findings

- Analytical expressions for cumulative primary and reinfections under all-or-none immunity are derived.
- An implicit equation for final epidemic size is obtained under leaky immunity.
- A reinfection threshold determines whether diseases become endemic or remain transient.

## Abstract

In the classic SIR model, infection gives full immunity against any possible reinfection. However, for many important epidemiological situations, immunity is only partial and reinfection is possible. Though these models are mathematically more complex, we are able to find expressions for the epidemic final size. We also generalise these expressions to include vaccination, with a fraction of the population vaccinated before the epidemic, where vaccinees are less susceptible to primary infections than unvaccinated hosts.

Partial immunity can be interpreted at the population level as providing either full or no protection to each host, in some proportion (all-or-none immunity). In this scenario, we give analytical expressions (mathematically similar to the SIR final-size) for the cumulative primary infections and the cumulative reinfections in unvaccinated and vaccinated hosts. Alternatively, partial immunity can be interpreted as providing homogeneous imperfect protection to each host (leaky immunity). For this other scenario, we again obtain an implicit equation for the final epidemic size. We break down, in terms of the final size, the number of infections in hosts with or without prior immunity (vaccine- or infection- induced), as well as the number of primary infections and reinfections. Under the leaky immunity assumption, we find a form of reinfection threshold. If the relative host susceptibility to reinfection is above this threshold (which is the inverse of the pathogen’s basic reproduction number), transmission rates are high enough to support an endemic disease. Below the reinfection threshold, epidemics are transient. In the all-or-none model, epidemics are always transient.