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Game-theoretic Social Distancing in Competitive Bi-Virus SIS Epidemics

Benjamin Catalano, Keith Paarporn, Sebin Gracy

TL;DR

This work develops a bi-virus SIS model in which individual social distancing evolves through replicator dynamics and jointly shapes infection spread for two competing strains. The authors derive a five-dimensional dynamical system and perform a comprehensive fixed-point analysis, revealing disease-free, unilateral, and line-coexistence equilibria. Crucially, endemic coexistence can occur only when the reproduction numbers satisfy $R_0=eta_1/oldsymbol{ abla}_1=eta_2/oldsymbol{ abla}_2$, in which case lines of coexistence exist and are locally stable under specific threshold conditions on $q$, costs, and risk parameters. Theoretical results are complemented by numerical simulations illustrating various asymptotic regimes, highlighting how social behavior can fundamentally alter multi-strain epidemic outcomes and potentially enable or prevent coexistence.

Abstract

Numerous elements drive the spread of infectious diseases in complex real-world networks. Of particular interest is social behaviors that evolve in tandem with the spread of disease. Moreover, recent studies highlight the importance of understanding how multiple strains spread simultaneously through a population (e.g. Delta and Omicron variants of SARS-CoV-2). In this paper, we propose a bi-virus SIS epidemic model coupled with a game-theoretic social distancing behavior model. The behaviors are governed by replicator equations from evolutionary game theory. The prevalence of each strain impacts the choice of an individual to social distance, and, in turn, their behavior affects the spread of each virus in the SIS model. Our analysis identifies equilibria of the system and their local stability properties, which reveal several isolated fixed points with varying levels of social distancing. We find that endemic co-existence is possible only when the reproduction numbers of both strains are equal. Assuming the reproduction number for each virus is the same, we identify suitable parameter regimes that give rise to lines of coexistence equilibria. Moreover, we also identify conditions for local exponential stability of said lines of equilibria. We illustrate our findings with several numerical simulations.

Game-theoretic Social Distancing in Competitive Bi-Virus SIS Epidemics

TL;DR

This work develops a bi-virus SIS model in which individual social distancing evolves through replicator dynamics and jointly shapes infection spread for two competing strains. The authors derive a five-dimensional dynamical system and perform a comprehensive fixed-point analysis, revealing disease-free, unilateral, and line-coexistence equilibria. Crucially, endemic coexistence can occur only when the reproduction numbers satisfy , in which case lines of coexistence exist and are locally stable under specific threshold conditions on , costs, and risk parameters. Theoretical results are complemented by numerical simulations illustrating various asymptotic regimes, highlighting how social behavior can fundamentally alter multi-strain epidemic outcomes and potentially enable or prevent coexistence.

Abstract

Numerous elements drive the spread of infectious diseases in complex real-world networks. Of particular interest is social behaviors that evolve in tandem with the spread of disease. Moreover, recent studies highlight the importance of understanding how multiple strains spread simultaneously through a population (e.g. Delta and Omicron variants of SARS-CoV-2). In this paper, we propose a bi-virus SIS epidemic model coupled with a game-theoretic social distancing behavior model. The behaviors are governed by replicator equations from evolutionary game theory. The prevalence of each strain impacts the choice of an individual to social distance, and, in turn, their behavior affects the spread of each virus in the SIS model. Our analysis identifies equilibria of the system and their local stability properties, which reveal several isolated fixed points with varying levels of social distancing. We find that endemic co-existence is possible only when the reproduction numbers of both strains are equal. Assuming the reproduction number for each virus is the same, we identify suitable parameter regimes that give rise to lines of coexistence equilibria. Moreover, we also identify conditions for local exponential stability of said lines of equilibria. We illustrate our findings with several numerical simulations.

Paper Structure

This paper contains 13 sections, 11 theorems, 27 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Payoff functions
  4. Coupled evolutionary dynamics
  5. Analysis: Identification of fixed points
  6. Stability analysis of DFE and unilateral FPs
  7. Stability analysis of DFE
  8. Stability analysis of unilateral FPs
  9. Stability of line of coexistence equilibria
  10. Stability of line $\mathcal{L}_0$
  11. Stability of line $\mathcal{L}_1$
  12. Simulations
  13. Conclusion

Key Result

Lemma 1

Consider system eq:main under Assumption assume:betadelta. The set $\Gamma$, where $\Gamma$ is as defined in eq:gamma, is positively invariant.

Figures (4)

  • Figure 1: Simulation showing both viruses dying out and the system converging to $\boldsymbol{p}_{\text{DFE} 0}$.
  • Figure 2: Simulation showing virus $1$ endemic and virus $2$ dying out with partial social distancing in the healthy population, i.e., the system converging to $\boldsymbol{p}_{1\text{S}}$.
  • Figure 3: Simulation of $\mathcal{L}_{0}$.
  • Figure 4: Simulation of $\mathcal{L}_{1}$.

Theorems & Definitions (14)

  • Lemma 1
  • Lemma 2: Disease-Free Equilibria (DFE)
  • Lemma 3: Unilateral Equilibria
  • Lemma 4: Coexistence equilibria
  • Proposition 1
  • Remark 1
  • Proposition 2
  • Remark 2
  • Proposition 3
  • Proposition 4
  • ...and 4 more