Epidemic outbreaks in structured host populations

TL;DR

This paper explores the spectral properties of the next generation operator in structured populations, providing general conditions for the basic reproduction number and employing measure kernels for analysis.

Contribution

It introduces a measure kernel approach to analyze the spectral radius of the next generation operator, broadening the conditions under which $ _0$ can be characterized.

Findings

Established conditions linking $ _0$ to positive eigenfunctionals.

Applied measure kernel methods for spectral analysis.

Extended threshold properties for heterogeneous populations.

Abstract

For a heterogeneous host population, the basic reproduction number of an infectious disease, $\cR_0$, is defined as the spectral radius of the next generation operator (NGO). The threshold properties of the basic reproduction number are typically established by imposing conditions that make $\cR_0$ an eigenvalue of the NGO associated with a positive eigenvector and a positive eigenfunctional (eigenvector of the dual of the NGO). More general results can be obtained by imposing conditions that associate $\cR_0$ just with a positive eigenfunctional. The next generation operator is conveniently expressed by a measure kernel or a Feller kernel which enables the use of analytic rather than functional analytic methods.