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

Xiandong Lin, Jiazhuo Cheng, Qiru Wang

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.

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

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.
Paper Structure (6 sections, 19 theorems, 98 equations)

This paper contains 6 sections, 19 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. The spectrum bound
  3. The weighted eigenvalue
  4. Partially degenerate case
  5. The basic reproduction rate
  6. Application

Key Result

Proposition 2.1

The following statements hold:

Theorems & Definitions (31)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • proof
  • ...and 21 more