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Global and Distributed Reproduction Numbers of a Multilayer SIR Model with an Infrastructure Network

José I. Caiza, Junjie Qin, Philip E. Paré

TL;DR

The paper tackles epidemic spreading in a multilayer setting by coupling a population SIR network with an infrastructure network that supports pathogen transmission. It introduces a global effective reproduction number $R(t)$ to describe network-wide dynamics and develops distributed reproduction numbers (DRNs) and local effective reproduction numbers (LERNs) to capture node-level behavior. The authors prove that, while healthy equilibria abound and there is no endemic steady state, the transient dynamics can be forecast via $R(t)$ and the per-node DRNs/LERNs; they further relate these local measures to the global spectral radius of an infection-transport matrix. Simulations on a ten-population-ten-node-scale network illustrate how $R(t)$ tracks global trends, whereas DRNs/LERNs provide richer insight into local spread, especially in the infrastructure layer, underscoring the practical value for targeted interventions.

Abstract

In this paper, we propose an SIR spread model in a population network coupled with an infrastructure network that has a pathogen spreading in it. We develop a threshold condition to characterize the monotonicity and peak time of a weighted average of the infection states in terms of the global (network-wide) effective reproduction number. We further define the distributed reproduction numbers (DRNs) of each node in the multilayer network which are used to provide local threshold conditions for the dynamical behavior of each entity. Furthermore, we leverage the DRNs to predict the global behavior based on the node-level assumptions. We use both analytical and simulation results to illustrate that the DRNs allow a more accurate analysis of the networked spreading process than the global effective reproduction number.

Global and Distributed Reproduction Numbers of a Multilayer SIR Model with an Infrastructure Network

TL;DR

The paper tackles epidemic spreading in a multilayer setting by coupling a population SIR network with an infrastructure network that supports pathogen transmission. It introduces a global effective reproduction number to describe network-wide dynamics and develops distributed reproduction numbers (DRNs) and local effective reproduction numbers (LERNs) to capture node-level behavior. The authors prove that, while healthy equilibria abound and there is no endemic steady state, the transient dynamics can be forecast via and the per-node DRNs/LERNs; they further relate these local measures to the global spectral radius of an infection-transport matrix. Simulations on a ten-population-ten-node-scale network illustrate how tracks global trends, whereas DRNs/LERNs provide richer insight into local spread, especially in the infrastructure layer, underscoring the practical value for targeted interventions.

Abstract

In this paper, we propose an SIR spread model in a population network coupled with an infrastructure network that has a pathogen spreading in it. We develop a threshold condition to characterize the monotonicity and peak time of a weighted average of the infection states in terms of the global (network-wide) effective reproduction number. We further define the distributed reproduction numbers (DRNs) of each node in the multilayer network which are used to provide local threshold conditions for the dynamical behavior of each entity. Furthermore, we leverage the DRNs to predict the global behavior based on the node-level assumptions. We use both analytical and simulation results to illustrate that the DRNs allow a more accurate analysis of the networked spreading process than the global effective reproduction number.
Paper Structure (9 sections, 27 equations, 4 figures)

This paper contains 9 sections, 27 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Model and Problem Formulation
  3. Network-level Analysis
  4. Equilibrium
  5. Effective Reproduction Number
  6. Distributed Reproduction Numbers
  7. Global Behavior from LERNs
  8. Simulations
  9. Conclusions

Figures (4)

  • Figure 1: Evolution of the global effective reproduction number $R(t)$ (top) and the weighted average $v(0)^\top z(t)$ (bottom).
  • Figure 2: Evolution of $x_i(t)$ (top) and $R_i(t)$ (bottom) for $i\in\{2,3,6,8,9\}$ of the population network. Note that $R_i(t)$ can be non-monotonic and only crosses one once. Moreover, the peak infection time $\tau_{p_i}$ satisfies $R_i(\tau_{p_i})=1$ for all $i\in\mathcal{V^P}$.
  • Figure 3: Evolution of the contamination level $w_j(t)$ (top) and the LERNs $R_j(t)$ (bottom) for $j\in\{2,3,4\}$ in the infrastructure network. All the claims in Theorem \ref{['theo:Rt_local']} hold. However, $R_j(t)$ can cross one more than once.
  • Figure 4: Evolution of the LERNs for all $i\in \mathcal{V}$. Theorem \ref{['theo:node_to_network']}\ref{['theo:all_great_1']} is depicted in the blue region. Theorem \ref{['theo:node_to_network']}\ref{['theo:all_less_1']} is depicted in the yellow region.

Theorems & Definitions (8)

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