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A Networked SIS Epidemic--Opinion Model with Higher-Order Interactions

Saba Samadi, José I. Caiza, Sebin Gracy, Philip E. Paré

arXiv:2608.31032v1eess.SY

TL;DR

Pairwise SIS models do not capture group-level transmission or feedback from epidemic prevalence to community opinions. The paper formulates a coupled higher-order SIS–opinion model and establishes stability, eradication, and bistability results. Its analysis shows that quadratic higher-order transmission can sustain endemic equilibria and produce healthy–endemic bistability under conditions where the pairwise-only mechanism cannot.

  • Problem

    Standard networked SIS models describe pairwise contagion, while real contagion can involve group-level exposures and opinion feedback.

  • Method

    The paper formulates a single-virus SIS–opinion model with pairwise and quadratic higher-order transmission, opinion-dependent rates, and state-dependent signed opinion dynamics.

  • Results

    The paper establishes local stability and instability conditions, a sufficient condition for global exponential eradication, and bistability conditions on a synchronous invariant set.

  • Takeaways & Limitations

    Quadratic higher-order transmission does not affect disease-free linearization but can sustain endemic equilibria and enable healthy–endemic bistability under the same healthy-stability condition.

Abstract

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This paper studies a susceptible--infected--susceptible (SIS) epidemic model coupled with opinion dynamics over a network of communities with higher-order interactions. Unlike standard networked SIS models, which account only for pairwise transmission, the proposed model incorporates group-level infection mechanisms and feedback between epidemic prevalence and community opinions. We establish local stability and instability conditions for a particular healthy equilibrium, derive a sufficient condition for global exponential eradication of the infection state, and identify conditions under which higher-order interactions induce bistability in the reduced dynamics on a positively invariant synchronous set. These results characterize how higher-order interactions alter the dynamics of opinion-dependent epidemic systems. Numerical simulations illustrate the predicted eradication, bistable, and endemic regimes.

I. INTRODUCTION

The paper develops an SIS–opinion model that combines pairwise and quadratic higher-order transmission with opinion-dependent epidemic rates and signed opinion dynamics. It addresses limitations of pairwise contagion models by analyzing how higher-order interactions affect epidemic regimes.

  • Motivation: Pairwise network SIS models cannot faithfully represent group-level exposures, motivating hypergraphs and simplicial complexes.Prior higher-order SIS studies associate simplicial interactions with discontinuous phase transitions and bistability.
  • Motivation: Opinion feedback links perceived epidemic severity to protective behavior and effective transmission and recovery rates.The effective recovery rate aggregates treatment, adherence, testing, isolation, and community intervention rather than purely biological recovery.
  • Model and novelty: The HOI-SIOS model combines pairwise and quadratic higher-order transmission with opinion-dependent rates and state-dependent signed opinion dynamics.It focuses on a single-virus networked SIS setting with feedback between infection prevalence and opinions.
  • Contributions: The paper establishes positive invariance, local stability and instability of a maximally skeptical healthy equilibrium, and a sufficient condition for global exponential eradication.It also studies healthy–endemic bistability after reducing homogeneous synchronous dynamics to two dimensions.
  • Model and novelty: Quadratic higher-order interactions are selected as the lowest-order nonlinear terms capable of producing the studied bistability.Higher-order products can be added analogously, but their equilibrium analysis leads to higher-degree equations.

A. Opinion-Dependent Rates

Opinion states determine effective recovery and transmission rates through affine interpolation between skeptical and concerned regimes. The model assumes nonnegative epidemic interactions and a symmetric unsigned opinion matrix without requiring connectivity or irreducibility.

  • Opinion-dependent rates: D(o) represents opinion-dependent effective recovery rates, while Bq(o) represents opinion-dependent transmission rates.As concern increases, effective recovery rises from γmin to γi and transmission falls from βiq to βmin_q.
  • Opinion-dependent rates: The affine rate dependence interpolates between maximally skeptical and maximally concerned opinion regimes.The effective recovery rate includes treatment seeking, adherence, testing, isolation, and community intervention.
  • Assumptions: The assumptions require A ≥ 0, Bi ≥ 0, and a symmetric nonnegative unsigned opinion matrix, with no connectivity assumption.The analysis therefore permits disconnected pairwise and opinion graphs.

B. Coupled HOI-SIOS Dynamics

The coupled dynamics combine epidemic transmission, signed opinion interactions, and local prevalence–opinion feedback within a positively invariant state domain. The analysis distinguishes healthy and endemic regimes using opinion-dependent reproduction numbers.

  • Coupled dynamics: The signed opinion graph uses fixed unsigned interaction strengths while signs switch with current opinion signs.Same-sign communities interact cooperatively, opposite-sign communities antagonistically, and each fixed sign pattern is structurally balanced.
  • Coupled dynamics: Local infection fractions drive perceived severity, while the signed-network term couples each community’s response to neighboring communities.Concern increases when xi > o′_i and decreases when xi < o′_i.
  • Positive invariance: The state domain D = [0, 1]^n × [−0.5, 0.5]^n is positively invariant under the model assumptions.The epidemic and opinion subsystems point inward at their respective boundaries.
  • Equilibrium regimes: Healthy equilibria have x* = 0_n, whereas endemic equilibria satisfy x* ≫ 0_n.The extremal reproduction numbers Rmin and Rmax bound the reproduction number over admissible opinion profiles.

III. STABILITY OF A HEALTHY EQUILIBRIUM

The maximally skeptical healthy equilibrium pairs zero infection with the lowest-opinion boundary. Its local analysis uses a combined infection term and a piecewise-smooth Jacobian away from opinion switching surfaces.

  • Healthy equilibrium: The equilibrium (xh, oh) = (0_n, −0.5 1_n) represents disease eradication with the maximally skeptical opinion profile.Healthy equilibria may be nonunique because the zero-infection opinion subsystem can have multiple equilibria.
  • Linearization: The infection analysis combines pairwise and quadratic transmission through y(x, o) = B1(o)Ax + B2(o)A2(x).For each target community, the higher-order component is [A2(x)]i = x⊤Bi x.
  • Linearization: Jacobian-based stability arguments apply only where opinion signs are fixed and every equilibrium opinion coordinate is nonzero.The vector field is piecewise smooth because the gauge map Φ(o) is discontinuous at opinion switching surfaces.

A. Local Stability

At the maximally skeptical healthy equilibrium, the infection-opinion Jacobian is block triangular: Rmax < 1 gives local exponential stability, while Rmax > 1 gives instability.

  • A. Local Stability: The maximally skeptical healthy equilibrium is locally smooth because all opinion components remain negative nearby.This permits Lyapunov’s indirect method.
  • A. Local Stability: Rmax < 1 makes the maximally skeptical healthy equilibrium locally exponentially stable.The opinion block is Hurwitz because ¯Lu is a Laplacian matrix with nonnegative eigenvalues.
  • A. Local Stability: Rmax > 1 makes the same healthy equilibrium unstable because the epidemic block has an eigenvalue with positive real part.The full Jacobian inherits this unstable eigenvalue through its block-triangular structure.
  • A. Local Stability: The Jacobian is lower block triangular, with epidemic block −Γmin + β1A and opinion block −(In + ¯Lu).Its eigenvalues are those of the two diagonal blocks.

B. Global Disease Eradication

A linear upper bound on quadratic transmission yields a sufficient spectral-radius condition for global exponential eradication of infection. The theorem guarantees convergence of x(t), not necessarily convergence or uniqueness of the opinion state.

  • B. Global Disease Eradication: The global theorem concerns exponential eradication of infection only, because the zero-infection opinion subsystem may have multiple equilibria.Convergence of o(t) requires additional assumptions and is not pursued.
  • B. Global Disease Eradication: Quadratic higher-order transmission satisfies A2(x) ≤ Cx componentwise on [0, 1]n.Here cik is the sum of the corresponding quadratic-interaction coefficients, and the bound uses xj ≤ 1.
  • B. Global Disease Eradication: Define M := Γmin^-1(β1A + β2C); if ρ(M) < 1, every admissible solution has x(t) → 0n exponentially.The condition is sufficient and applies uniformly over admissible opinion states through the bounds on D(o) and Bq(o).
  • B. Global Disease Eradication: The proof uses a positive linear Lyapunov function constructed from w := (In − M⊤)^−1 1n and a comparison argument.Because w ≫ 0n, the Lyapunov function is positive definite on the nonnegative infection domain and equivalent to a norm.
  • B. Global Disease Eradication: ρ(M) < 1 implies Rmax < 1, but the converse need not hold because higher-order transmission enlarges the comparison matrix.The inequality Rmax ≤ ρ(M) follows from β2C ≥ 0 and monotonicity of spectral radius.

IV. NON-HEALTHY EQUILIBRIA AND BISTABILITY

Beyond the disease-free regime, the homogeneous model shows that higher-order interactions can create healthy–endemic bistability even when Rmax < 1, while Rmin > 1 yields endemic behavior on the synchronous set.

  • IV. NON-HEALTHY EQUILIBRIA AND BISTABILITY: Under homogeneous parameters, higher-order interactions can induce healthy–endemic bistability even when Rmax < 1.The analysis is conducted on a synchronous invariant set where the dynamics reduce to a lower-dimensional system.
  • IV. NON-HEALTHY EQUILIBRIA AND BISTABILITY: When Rmin > 1, an endemic equilibrium exists on the synchronous set.The general heterogeneous model also has unstable healthy equilibria away from opinion switching surfaces.

A. Bistability on a Synchronous Invariant Set

Under homogeneous assumptions, the model admits a positively invariant synchronous set whose dynamics reduce to a planar system. Higher-order transmission can create two endemic equilibria and bistability relative to this set, while pairwise-only transmission cannot.

  • Synchronous reduction: Synchronous states require equal infection levels and equal shifted opinions across all communities; the homogeneous assumptions equalize pairwise and quadratic infection inputs.These assumptions define the invariant setting but do not imply synchronization from arbitrary heterogeneous initial conditions.
  • Synchronous reduction: The synchronous set S is positively invariant, so trajectories initialized with common infection and shifted-opinion levels remain synchronous and follow a two-dimensional reduction.The reduced coordinates satisfy ξ̇ = ξψ(ξ, η) and η̇ = ξ − η within [0, 1]^2.
  • Equilibria: Positive equilibria of the reduced system correspond to roots of g(s), with each root yielding an endemic equilibrium (s1_n, (s − 0.5)1_n) in S.At equilibrium, η* = ξ*, so positive equilibria lie on the diagonal (s, s).
  • Bistability: Without the quadratic higher-order term, g(s) remains negative under āβ_1 < γ_min, so the reduced system has no endemic equilibrium or healthy–endemic bistability.Thus, the bistable mechanism requires nonlinear higher-order transmission in this setting.
  • Bistability: If g(ŝ) > 0 while g(0) < 0 and g(1) < 0, the reduced system has at least two endemic equilibria; with exactly two simple roots, the lower is unstable and the upper is locally exponentially stable.The lower equilibrium is a hyperbolic saddle, whereas the upper equilibrium is Hurwitz under the stated trace condition.
  • Interpretation and scope: The quadratic higher-order term vanishes from the disease-free linearization, allowing local healthy stability under subcritical pairwise transmission while supporting a stable endemic state at positive infection levels.These stability and bistability conclusions are relative to trajectories in S, not arbitrary initial conditions.

B. Endemic Regime

When Rmin > 1, the model admits endemic behavior and healthy equilibria are unstable under the stated conditions. The higher-order term vanishes at zero infection, so healthy-equilibrium instability is determined by the pairwise epidemic linearization.

  • Healthy-equilibrium instability: Rmin > 1 makes the pairwise epidemic linearization supercritical even at the most concerned opinion profile.This yields a positive spectral abscissa for the epidemic block.
  • Endemic equilibria: Rmin > 1 guarantees at least one endemic equilibrium on the synchronous set S.The endemic equilibrium has strictly positive infection components.
  • Healthy-equilibrium instability: Every healthy equilibrium away from opinion switching surfaces is unstable when Rmin > 1.The full Jacobian is lower block triangular and inherits an eigenvalue with positive real part from the epidemic block.
  • Healthy-equilibrium instability: The quadratic higher-order infection term vanishes at x = 0n, leaving the healthy-equilibrium linearization governed by pairwise transmission.The epidemic Jacobian block is Metzler, and spectral-radius monotonicity establishes the instability condition.

V. SIMULATIONS

The simulations illustrate three regimes: exponential eradication, bistability between healthy and endemic states, and an endemic-dissensus trajectory when Rmin > 1. The outcomes match the paper’s stability and eradication results within the simulated settings.

  • Eradication regime: Rmax = 0.20 and ρ = 0.93 produce exponential infection eradication and a dissensus opinion profile.Theorem 1 guarantees eradication for this parameter set, illustrated by Fig. 1.
  • Bistable regime: Figure 2 shows trajectories approaching either the healthy equilibrium or the larger endemic equilibrium from selected initial conditions in S.The red and blue trajectories separate according to the attracting equilibrium, with the smaller endemic equilibrium marking the unstable boundary.
  • Bistable regime: Two synchronous equilibria are locally exponentially stable, while the smaller endemic equilibrium is unstable relative to S.The stable equilibria are the healthy state and (0.51131n, 0.01131n); the unstable state is (0.12811n, −0.37191n).
  • Endemic regime: Rmin = 1.10 > 1 yields a trajectory numerically approaching an endemic-dissensus state.This is consistent with instability of healthy equilibria away from opinion switching surfaces, but does not establish global persistence or endemic stability.

VI. CONCLUSION

The paper develops the HOI-SIOS model and analyzes how opinion feedback and quadratic higher-order transmission affect epidemic outcomes. It establishes eradication and stability results, while showing that synchronous dynamics can exhibit healthy–endemic bistability.

  • The HOI-SIOS model couples epidemic prevalence and signed opinion dynamics through opinion-dependent effective transmission and recovery rates.
  • The analysis establishes local stability and instability conditions, a sufficient condition for global exponential eradication, and bistability conditions on a synchronous invariant set.
  • Quadratic higher-order transmission does not affect the disease-free linearization but can sustain an endemic equilibrium at positive infection levels.Under the same healthy-stability condition, the pairwise-only case rules out this mechanism.
  • When Rmin > 1, every healthy equilibrium away from opinion switching surfaces is unstable.
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