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

TL;DR

The paper addresses extending the basic reproduction number $R_0$ to cone-preserving bounded linear operators on Banach spaces. It develops a functional-analytic framework using generating and normal cones, resolvent-positivity, and convexity of the spectral-radius map to relate $R_0$ to the spectral radius $r(A)$ for $A=T+F$ with $r(T)<1$, proving that $rig(T+rac{1}{R_0}Fig)=1$ when $R_0>0$. It then establishes a nonstrict and a strict trichotomy between $R_0$ and $r(A)$ under progressively stronger assumptions, generalizing classical matrix results to infinite-dimensional settings. The Leslie model example on $oldsymbol{ extell_p}$ demonstrates how the theory applies to age-structured populations, with $R_0$ given by an explicit rank-one expression and acting as a bound for $r(A)$ to predict population extinction or growth. Collectively, the work broadens the utility of the basic reproduction number as a spectral-radius proxy in Banach spaces and provides concrete criteria for stability in infinite-dimensional dynamics.

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.

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Theorem 2.2
• Corollary 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Corollary 2.8