Variational characterizations of weighted eigenvalue and basic reproduction rate for nonlocal dispersal systems and application

TL;DR

This paper develops variational characterizations for the basic reproduction rate in nonlocal dispersal systems, including cases without a principal eigenvalue, and applies these results to an epidemic model to analyze disease spread.

Contribution

It introduces a novel variational framework for the basic reproduction rate in nonlocal dispersal models, even when principal eigenvalues are absent, and demonstrates its application to epidemic modeling.

Findings

Established a Collatz-Wielandt and Rayleigh-Ritz characterization for spectral bounds.

Derived an explicit expression for the basic reproduction rate in nonlocal models.

Showed different limiting behavior of the reproduction rate in degenerate cases compared to local diffusion.

Abstract

The basic reproduction rate is a crucial threshold parameter in infectious disease models. In nonlocal dispersal systems, its variational characterization is challenging due to the possible absence of a principal eigenvalue caused by non-compactness. In this paper, we aim to establish such a characterization even when the principal eigenvalue does not exist. To this end, we first study the spectral bound of a class of nonlocal dispersal operators, establishing a Collatz-Wielandt characterization as well as a Rayleigh-Ritz characterization when the operator is self-adjoint. Using this, we characterize the unique parameter value at which the spectral bound equals zero, covering both non-degenerate and partially degenerate cases, and subsequently obtain an explicit expression for the basic reproduction rate. To demonstrate the utility of our theoretical framework, we apply it to a nonlocal dispersal SIS epidemic model with saturated incidence rate. The analysis shows that, in the degenerate case of the saturation coefficient, the limiting behavior of the basic reproduction rate as the total population tends to zero is strikingly different from that in local diffusion case.