title: Solvability of implicit final size equations for SIR epidemic models.
abstract: Final epidemic size relations play a central role in mathematical epidemiology. These can be written in the form of an implicit equation which is not analytically solvable in most of the cases. While final size relations were derived for several complex models, including multiple infective stages and models in which the durations of stages are arbitrarily distributed, the solvability of those implicit equations have been less studied. In this paper the SIR homogeneous mean-field and pairwise models and the heterogeneous mean-field model are studied. It is proved that the implicit equation for the final epidemic size has a unique solution, and that through writing the implicit equation as a fixed point equation in a suitable form, the iteration of the fixed point equation converges to the unique solution. The Markovian SIR epidemic model on finite networks is also studied by using the generation-based approach. Explicit analytic formulas are derived for the final size distribution for line and star graphs of arbitrary size. Iterative formulas for the final size distribution enable us to study the accuracy of mean-field approximations for the complete graph.
authors.affiliation: Budapest Semesters in Mathematics, Budapest, Hungary.
authors.forename: Subekshya
authors.initials: S
authors.lastname: Bidari
authors.affiliation: Budapest Semesters in Mathematics, Budapest, Hungary.
authors.forename: Xinying
authors.initials: X
authors.lastname: Chen
authors.affiliation: Budapest Semesters in Mathematics, Budapest, Hungary.
authors.forename: Daniel
authors.initials: D
authors.lastname: Peters
authors.affiliation: Budapest Semesters in Mathematics, Budapest, Hungary.
authors.forename: Dylanger
authors.initials: D
authors.lastname: Pittman
authors.affiliation: Institute of Mathematics, Eötvös Loránd University Budapest, Hungary; Numerical Analysis and Large Networks Research Group, Hungarian Academy of Sciences, Hungary; Budapest Semesters in Mathematics, Budapest, Hungary. Electronic address: simonp@cs.elte.hu.
authors.forename: Péter L
authors.initials: PL
authors.lastname: Simon
keywords: Exact master equation; Final epidemic size; Mean-field model; SIR epidemic