[Paper Review] On a Network SIS Epidemic Model with Cooperative and Antagonistic Opinion Dynamics

Baike She, Ji Liu+2 more|arXiv (Cornell University)|Feb 25, 2021

Opinion Dynamics and Social Influence39 references4 citations

TL;DR

This paper proposes a networked SIS epidemic model coupled with opinion dynamics that captures both cooperative and antagonistic interactions among communities, introducing an Opinion-Dependent Reproduction Number ($R_t^o$) to quantify the mutual influence between epidemic spread and opinion evolution. The key contribution is showing that reshaping community opinions—particularly by targeting stubborn communities—can suppress epidemics by driving $R_t^o < 1$, enabling effective control strategies.

ABSTRACT

We propose a mathematical model to study coupled epidemic and opinion dynamics in a network of communities. Our model captures SIS epidemic dynamics whose evolution is dependent on the opinions of the communities toward the epidemic, and vice versa. In particular, we allow both cooperative and antagonistic interactions, representing similar and opposing perspectives on the severity of the epidemic, respectively. We propose an Opinion-Dependent Reproduction Number to characterize the mutual influence between epidemic spreading and opinion dissemination over the networks. Through stability analysis of the equilibria, we explore the impact of opinions on both epidemic outbreak and eradication, characterized by bounds on the Opinion-Dependent Reproduction Number. We also show how to eradicate epidemics by reshaping the opinions, offering researchers an approach for designing control strategies to reach target audiences to ensure effective epidemic suppression.

Motivation & Objective

• To model the coevolution of epidemic spread and opinion dynamics in a networked community setting, where opinions influence disease transmission and vice versa.
• To capture both cooperative (similar views) and antagonistic (opposing views) interactions in opinion exchange, reflecting real-world polarization during pandemics.
• To develop a control strategy that leverages opinion manipulation to suppress epidemics by driving the system below the epidemic threshold.
• To characterize epidemic outcomes—outbreak or eradication—using a novel Opinion-Dependent Reproduction Number ($R_t^o$) that depends on community opinion states.
• To provide a theoretical foundation for designing targeted interventions that shift opinions in key communities to achieve epidemic control.

Proposed method

• Models epidemic spread using a networked SIS model where infection rates depend on community opinions via an opinion-dependent transmission matrix.
• Introduces a dynamic opinion model on signed networks with positive (cooperative) and negative (antagonistic) edges to represent consensus and dissensus in opinion formation.
• Defines the Opinion-Dependent Reproduction Number $R_t^o$ as a spectral radius of a matrix combining transmission and recovery rates weighted by opinion states.
• Uses stability analysis of equilibria (healthy and endemic) to determine conditions under which epidemics can be eradicated or persist, based on $R_t^o$.
• Applies control theory to identify 'stubborn' communities whose fixed opinions at $0.5$ (strong belief in severity) can drive $R_t^o < 1$, ensuring epidemic extinction.
• Employs Lyapunov-like arguments and matrix analysis (e.g., Gershgorin disk theorem) to prove invariance of compact sets and existence of equilibria under switching opinion topologies.

Experimental results

Research questions

• RQ1 How do cooperative and antagonistic opinion dynamics jointly influence the spread of an epidemic in a networked community?
• RQ2 What is the threshold condition for epidemic outbreak or eradication when opinions about disease severity are dynamically evolving?
• RQ3 Can opinion manipulation—specifically, fixing opinions in key communities—be used to suppress epidemic spread?
• RQ4 How does the Opinion-Dependent Reproduction Number $R_t^o$ characterize the interplay between opinion polarization and epidemic persistence?
• RQ5 Under what conditions is the disease-free equilibrium stable, and when does opinion-driven instability lead to endemic outbreaks?

Key findings

• The system admits at least one endemic equilibrium when $R_{ ext{min}} > 1$, indicating that even under the most pessimistic opinion conditions, an epidemic can persist.
• When $R_t^{o^*} < 1$, the healthy equilibrium is locally asymptotically stable, meaning the epidemic can be eradicated if opinions are sufficiently aligned toward perceiving the disease as severe.
• If $R_t^{o^*} > 1$, the healthy equilibrium becomes unstable, leading to endemic spread, especially when opinion consensus is weak or polarized.
• The consensus-healthy equilibrium at $o^* = -0.5\mathbf{e}$ (pessimistic views) is unstable when $R_{ ext{max}} > 1$, indicating that even strong negative opinions may not prevent outbreaks.
• Targeting stubborn communities with fixed opinions at $0.5$ (strong belief in severity) ensures $R_t^{ar{o}} < 1$, guaranteeing convergence to a disease-free state.
• The system can support multiple endemic equilibria, indicating complex, non-monotonic behavior in epidemic outcomes under opinion dynamics.