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A New Perspective on Determining Disease Invasion and Population Persistence in Heterogeneous Environments

Poroshat Yazdanbakhsh, Mark Anderson, Zhisheng Shuai

Abstract

We introduce a new quantity known as the network heterogeneity index, denoted by $\mathcal{H}$, which facilitates the investigation of disease propagation and population persistence in heterogeneous environments. Our mathematical analysis reveals that this index embodies the structure of such networks, the disease or population dynamics of patches, and the dispersal between patches. We present multiple representations of the network heterogeneity index and demonstrate that $\mathcal{H}\geq 0$. Moreover, we explore the applications of $\mathcal{H}$ in epidemiology and ecology across various heterogeneous environments, highlighting its effectiveness in determining disease invasibility and population persistence.

A New Perspective on Determining Disease Invasion and Population Persistence in Heterogeneous Environments

Abstract

We introduce a new quantity known as the network heterogeneity index, denoted by , which facilitates the investigation of disease propagation and population persistence in heterogeneous environments. Our mathematical analysis reveals that this index embodies the structure of such networks, the disease or population dynamics of patches, and the dispersal between patches. We present multiple representations of the network heterogeneity index and demonstrate that . Moreover, we explore the applications of in epidemiology and ecology across various heterogeneous environments, highlighting its effectiveness in determining disease invasibility and population persistence.

Paper Structure

This paper contains 12 sections, 13 theorems, 87 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Notation and Terminology
  3. Perron Root of Perturbed $-L$
  4. Network Heterogeneity Index: Definition and Properties
  5. Application to a Multi-Patch SIS Model
  6. Star network with one hot spot
  7. Path network with one hot spot
  8. Network with two hot spots
  9. Application to a Multi-Patch Single Species Model
  10. A bridge connecting two heterogeneous landscapes
  11. Different allocations of resources on a path network
  12. Discussion

Key Result

Lemma 2.1

Let $L$ be an irreducible Laplacian matrix and $L^\#$ the corresponding group inverse. Then,

Figures (11)

  • Figure 1: (\ref{['star']}) Flow diagram of an n-patch star network. (\ref{['lap-star']}) An associated Laplacian matrix $1$.
  • Figure 2: Flow diagram of the star network with the hotspot patch located (\ref{['star1']}) at the hub, and (\ref{['star2']}) on leaf $2$.
  • Figure 3: Demonstration of of a flow diagram of a 4-patch path network with symmetric movement among patches.
  • Figure 4: Flow Diagrams of one hot spot in the 4-patch path network with symmetric movement on patch 1 (\ref{['path1']}) and path 2 (\ref{['path2']}). Comparison of the values of the network disease growth rates (\ref{['h1r']}) and the basic reproduction numbers (\ref{['h1R0']}) of both scenarios with respect to the dispersion rate $\mu_ I$.
  • Figure 5: Four scenarios of two hot spot arrangements in the 4-patch path network
  • ...and 6 more figures

Theorems & Definitions (26)

  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.2.1
  • Lemma 3.3
  • Theorem 3.4
  • proof
  • ...and 16 more