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Game-Theoretic Protection Adoption Against Networked SIS Epidemics

Abhisek Satapathi, Ashish R. Hota

TL;DR

The paper tackles how agents should strategically adopt partially effective protection against networked SIS epidemics in large, heterogeneous networks. It combines a degree-based mean-field (DBMF) epidemic model with replicator dynamics for protection decisions, using a timescale separation to derive equilibrium protection strategies and a reduced slow epidemic dynamics. The main theoretical contribution is a rigorous existence and uniqueness result for the endemic equilibrium of the coupled dynamics, supported by a switched-system analysis and an endemic-equilibrium algorithm. Numerical experiments show how degree heterogeneity, infection rates, and protection costs shape the equilibrium prevalence, offering insights for designing decentralized interventions to curb epidemics in complex networks.

Abstract

In this paper, we investigate game-theoretic strategies for containing spreading processes on large-scale networks. Specifically, we consider the class of networked susceptible-infected-susceptible (SIS) epidemics where a large population of agents strategically choose whether to adopt partially effective protection. We define the utilities of the agents which depends on the degree of the agent, its individual infection status and action, as well as the the overall prevalence of the epidemic and strategy profile of the entire population. We further present the coupled dynamics of epidemic evolution as well as strategy update which is assumed to follow the replicator dynamics. By relying on timescale separation arguments, we first derive the optimal strategy of protection adoption by the agents for a given epidemic state, and then present the reduced epidemic dynamics. The existence and uniqueness of endemic equilibrium is rigorously characterized and forms the main result of this paper. Finally, we present extensive numerical results to highlight the impacts of heterogeneous node degrees, infection rates, cost of protection adoption, and effectiveness of protection on the epidemic prevalence at the equilibrium.

Game-Theoretic Protection Adoption Against Networked SIS Epidemics

TL;DR

The paper tackles how agents should strategically adopt partially effective protection against networked SIS epidemics in large, heterogeneous networks. It combines a degree-based mean-field (DBMF) epidemic model with replicator dynamics for protection decisions, using a timescale separation to derive equilibrium protection strategies and a reduced slow epidemic dynamics. The main theoretical contribution is a rigorous existence and uniqueness result for the endemic equilibrium of the coupled dynamics, supported by a switched-system analysis and an endemic-equilibrium algorithm. Numerical experiments show how degree heterogeneity, infection rates, and protection costs shape the equilibrium prevalence, offering insights for designing decentralized interventions to curb epidemics in complex networks.

Abstract

In this paper, we investigate game-theoretic strategies for containing spreading processes on large-scale networks. Specifically, we consider the class of networked susceptible-infected-susceptible (SIS) epidemics where a large population of agents strategically choose whether to adopt partially effective protection. We define the utilities of the agents which depends on the degree of the agent, its individual infection status and action, as well as the the overall prevalence of the epidemic and strategy profile of the entire population. We further present the coupled dynamics of epidemic evolution as well as strategy update which is assumed to follow the replicator dynamics. By relying on timescale separation arguments, we first derive the optimal strategy of protection adoption by the agents for a given epidemic state, and then present the reduced epidemic dynamics. The existence and uniqueness of endemic equilibrium is rigorously characterized and forms the main result of this paper. Finally, we present extensive numerical results to highlight the impacts of heterogeneous node degrees, infection rates, cost of protection adoption, and effectiveness of protection on the epidemic prevalence at the equilibrium.
Paper Structure (14 sections, 5 theorems, 34 equations, 3 figures, 4 tables)

This paper contains 14 sections, 5 theorems, 34 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Networked SIS Epidemic under Partially Effective Protection
  3. Degree-Based Mean-Field Approximation
  4. Game-Theoretic Protection Adoption
  5. Analysis of Equilibrium
  6. Numerical Results
  7. Convergence to Endemic Equilibrium
  8. Heterogeneous Degree Distribution and Infection Rates
  9. Comparison among Degree Distributions
  10. Conclusion
  11. Omitted Proofs
  12. NIMFA of the SIS Epidemic Model
  13. Proof of Proposition \ref{['proposition:tnse_reduced_dynamics']}
  14. Proof of Theorem \ref{['theorem:tnse_main']}

Key Result

Lemma 1

Equation eq:theta_equilibrium_identity admits a nonzero solution $\Theta^\star(\mathbf{z})$ if and only if Furthermore, $\Theta^\star(\mathbf{z})=1$ is not a solution of eq:theta_equilibrium_identity.

Figures (3)

  • Figure 1: Evolution of infected proportion (left), $\Theta(\mathbf{x}(t)) = \Theta(\mathbf{y}(t))$ and degree-specific thresholds (middle), and proportion of unprotected susceptible agents (right) for $c_\mathtt{P} = 10$ (top row) and $c_\mathtt{P} = 8$ (bottom row).
  • Figure 2: Variation of expected fraction of infected nodes ($y^{\mathtt{avg}}$) at the endemic equilibrium for different values of $m_d$ with $d=4$ and for two possible choice of infection rates stated in Section \ref{['section:tnse_casestudy1']}.
  • Figure 3: Expected fraction of infected agents (top row) and probability of becoming infected from a randomly chosen neighbor (bottom row) at the equilibrium as a function of effectiveness of protection (left panel), infection probability (middle panel) and cost of protection (right panel). We have assumed $\beta_\mathtt{P}^d=\beta_\mathtt{P}$ for all degrees in this example.

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Remark 1
  • Proposition 1
  • Lemma 2
  • proof
  • Theorem 1
  • Theorem 2: mei2017dynamicskhanafer2016stability
  • proof
  • proof