Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Epidemic outbreaks in structured host populations

Horst R Thieme

TL;DR

The paper develops a rigorous framework for epidemic outbreaks in structurally heterogeneous host populations by recasting the basic reproduction number ${\mathcal{R}}_0$ as the spectral radius of a next generation operator $K$ defined via measure kernels. It uses Hammerstein-type equations with measure kernels to express the final size $w$ and establishes threshold results through positive eigenfunctionals, avoiding reliance on compactness. The work analyzes a spectrum of kernel classes—dominated, semi-separable, and Feller (tight, strong, comparability)—and proves existence and properties of minimal solutions that are epidemiologically relevant, including under minimal latency constraints. The approach unifies the derivation of the final-size equations with the underlying operator theory, providing conditions for pandemics, uniqueness, and positivity of outbreak trajectories. The results have potential for broad applicability to non-compact trait spaces and provide analytic tools for understanding how host-structure shapes outbreak dynamics and final outcomes.

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.

Epidemic outbreaks in structured host populations

TL;DR

The paper develops a rigorous framework for epidemic outbreaks in structurally heterogeneous host populations by recasting the basic reproduction number as the spectral radius of a next generation operator defined via measure kernels. It uses Hammerstein-type equations with measure kernels to express the final size and establishes threshold results through positive eigenfunctionals, avoiding reliance on compactness. The work analyzes a spectrum of kernel classes—dominated, semi-separable, and Feller (tight, strong, comparability)—and proves existence and properties of minimal solutions that are epidemiologically relevant, including under minimal latency constraints. The approach unifies the derivation of the final-size equations with the underlying operator theory, providing conditions for pandemics, uniqueness, and positivity of outbreak trajectories. The results have potential for broad applicability to non-compact trait spaces and provide analytic tools for understanding how host-structure shapes outbreak dynamics and final outcomes.

Abstract

For a heterogeneous host population, the basic reproduction number of an infectious disease, , 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 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 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.

Paper Structure

This paper contains 36 sections, 53 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Scent of the model
  3. Overview
  4. Some basic concepts
  5. The final size of the epidemic and measure kernels
  6. Next generation operator and basic reproduction number
  7. Preview of threshold results
  8. A fundamental result from positive operator theory
  9. A general but weak threshold result
  10. Dominated measure kernels
  11. Semi-separable measure kernels
  12. Metric spaces of traits and Feller kernels
  13. Tight Feller kernels
  14. Strong Feller kernels
  15. Comparability kernels
  16. ...and 21 more sections

Key Result

Theorem 3.1

If ${\mathcal{R}}_0 >0$, there exists a bounded linear positive eigenfunctional $\theta: M^b(\Omega) \to { \mathbb{R}}$ of $K$ (and of $\kappa$) such that $\theta (K g)= {\mathcal{R}}_0 \, \theta(g)$ for all $g \in M^b(\Omega)$, $\theta(g) > 0$ if $g \in M^b_+(\Omega)$ and $g \ge \delta u$ for some

Theorems & Definitions (107)

  • Remark 2.1
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Remark 3.4
  • proof
  • Theorem 3.5
  • Corollary 3.6
  • Theorem 3.7
  • ...and 97 more