The Basic Reproduction Number for Bounded Linear Operators on Ordered Banach Spaces

TL;DR

This paper extends the concept of the basic reproduction number, R0, from non-negative matrices to more general cone-preserving bounded linear operators on Banach spaces, aiding stability analysis in ecological and epidemiological models.

Contribution

It generalizes the R0 concept to cone-preserving operators on Banach spaces, broadening its applicability beyond matrices.

Findings

Spectral radius bounds for cone-preserving operators

Extension of R0 properties to infinite-dimensional spaces

Implications for stability analysis in complex models

Abstract

A basic reproduction number, $R_0$, is a concept encountered frequently in the study of ecological and epidemiological models. It is routinely used to determine the stability of an extinction or a disease-free fixed point or steady state. It is well-known that for linear models described by non-negative matrices, the spectral radius of the matrix is always contained in an interval with endpoints $1$ and $R_0$. Here we extend these results to more general cone-preserving bounded linear operators acting on Banach spaces.