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On a Discrete-Time Networked SIV Epidemic Model with Polar Opinion Dynamics

Qiulin Xu, Hideaki Ishii

TL;DR

This work studies a discrete-time, bilayer network model where a susceptible–infected–vigilant (SIV) epidemic spreads over a physical network while polar opinions evolve over a social network, capturing how public health concerns influence behavior. By introducing an opinion-dependent reproduction number $R_o^V$, the authors derive sufficient conditions for global stability of disease-free and endemic equilibria and reveal the mutual influence between epidemic control and opinion consensus or dissensus. The analysis combines well-posedness results, equilibrium characterizations, and Lyapunov-based stability arguments, supplemented by simulations on a real-world Japan prefecture network that illustrate control strategies via opinion dynamics. The findings suggest that strategic social interventions can effectively suppress epidemics, offering a pathway to public health policies that leverage opinion dynamics alongside traditional biomedical measures.

Abstract

This paper studies novel epidemic spreading problems influenced by opinion evolution in social networks, where the opinions reflect the public health concerns. A coupled bilayer network is proposed, where the epidemics spread over several communities through a physical network layer while the opinions evolve over the same communities through a social network layer. The epidemic spreading process is described by a susceptible-infected-vigilant (SIV) model, which introduces opinion-dependent epidemic vigilance state compared with the classical epidemic models. The opinion process is modeled by a polar opinion dynamics model, which includes infection prevalence and human stubbornness into the opinion evolution. By introducing an opinion-dependent reproduction number, we analyze the stability of disease-free and endemic equilibria and derive sufficient conditions for their global asymptotic stability. We also discuss the mutual effects between epidemic eradication and opinion consensus, and the possibility of suppressing epidemic by intervening in the opinions or implementing public health strategies. Simulations are conducted to verify the theoretical results and demonstrate the feasibility of epidemic suppression.

On a Discrete-Time Networked SIV Epidemic Model with Polar Opinion Dynamics

TL;DR

This work studies a discrete-time, bilayer network model where a susceptible–infected–vigilant (SIV) epidemic spreads over a physical network while polar opinions evolve over a social network, capturing how public health concerns influence behavior. By introducing an opinion-dependent reproduction number , the authors derive sufficient conditions for global stability of disease-free and endemic equilibria and reveal the mutual influence between epidemic control and opinion consensus or dissensus. The analysis combines well-posedness results, equilibrium characterizations, and Lyapunov-based stability arguments, supplemented by simulations on a real-world Japan prefecture network that illustrate control strategies via opinion dynamics. The findings suggest that strategic social interventions can effectively suppress epidemics, offering a pathway to public health policies that leverage opinion dynamics alongside traditional biomedical measures.

Abstract

This paper studies novel epidemic spreading problems influenced by opinion evolution in social networks, where the opinions reflect the public health concerns. A coupled bilayer network is proposed, where the epidemics spread over several communities through a physical network layer while the opinions evolve over the same communities through a social network layer. The epidemic spreading process is described by a susceptible-infected-vigilant (SIV) model, which introduces opinion-dependent epidemic vigilance state compared with the classical epidemic models. The opinion process is modeled by a polar opinion dynamics model, which includes infection prevalence and human stubbornness into the opinion evolution. By introducing an opinion-dependent reproduction number, we analyze the stability of disease-free and endemic equilibria and derive sufficient conditions for their global asymptotic stability. We also discuss the mutual effects between epidemic eradication and opinion consensus, and the possibility of suppressing epidemic by intervening in the opinions or implementing public health strategies. Simulations are conducted to verify the theoretical results and demonstrate the feasibility of epidemic suppression.
Paper Structure (17 sections, 6 theorems, 54 equations, 8 figures)

This paper contains 17 sections, 6 theorems, 54 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Modelling and Problem Formulation
  3. Epidemic Dynamics
  4. Opinion Dynamics
  5. Coupled Epidemic-Opinion Dynamics
  6. Problem Statement
  7. Analysis of SIV-Opinion Dynamical Model
  8. Well-Posedness
  9. Equilibria of the Coupled Model
  10. Stability Analysis of Disease-Free Equilibrium
  11. Stability Analysis of Endemic Equilibrium
  12. Simulations
  13. Real-world Network
  14. Mild Epidemics
  15. Severe Epidemics
  16. ...and 2 more sections

Key Result

Proposition 1

For the model in (SIV-discrete) and (opinion-coupled), the states satisfy $x_i^S(k), x_i^I(k),$$x_i^V(k), o_i(k) \in[0,1]$ for all $i \in[n]$ and $k \geq 0$.

Figures (8)

  • Figure 1: SIV epidemic model with three states and various transition parameters.
  • Figure 2: Network structures. (a) Physical interactions. (b) Opinion interactions.
  • Figure 3: Under a mild epidemic with $R_o^V = 0.9956$, the evolution of the coupled SIV-opinion system for the $46$ communities network in Fig. \ref{['fig1']} and \ref{['fig2']}. (a) The infected states converge to zero. (b) The vigilant states converge to $0.3333$. (c) The opinion states reach consensus and converge to zero.
  • Figure 4: Under the same mild epidemic as Fig. \ref{['fig3']}, the evolution of the infected populations under the SIS model without the effect of opinions.
  • Figure 5: Under a severe epidemic with $R_o^V = 1.1827$, the evolution of the coupled SIV-opinion system for the $46$ communities network in Fig. \ref{['fig1']} and \ref{['fig2']}. (a) The infected states reach an endemic equilibrium. (b) The vigilant states converge to an equilibrium. (c) The opinion states reach dissensus.
  • ...and 3 more figures

Theorems & Definitions (14)

  • Proposition 1
  • proof
  • Definition 1
  • Definition 2: SIV-opinion reproduction number
  • Lemma 1
  • proof
  • Remark 1
  • Theorem 1
  • proof
  • Proposition 2
  • ...and 4 more