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arxiv: 2604.10614 · v1 · submitted 2026-04-12 · 🧮 math.AP · physics.soc-ph· q-bio.PE

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Kinetic models of opinion-driven epidemic dynamics modulated by graphons

Andrea Bondesan , Jacopo Borsotti , Mattia Fontana
Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:05 UTC · model grok-4.3

classification 🧮 math.AP physics.soc-phq-bio.PE
keywords kinetic modelsepidemic dynamicsopinion dynamicsgraphonsreproduction numberrelative entropyconvergence to equilibriumstructure-preserving scheme
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The pith

Kinetic models with graphon opinion dynamics prove convergence to equilibrium and show a reproduction analog whose oscillations generate epidemic waves without external forcing.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops kinetic models coupling epidemic spread to individuals' opinions on protective behaviors, with opinions exchanging across a graphon that represents the social network and may include opinion leaders. Convergence of the solutions to equilibrium is shown in the strong L1 norm by relative entropy arguments and in homogeneous Sobolev spaces dot H^{-s} for s between 1/2 and 1 by Fourier methods. A structure-preserving numerical scheme is constructed to explore the coupled system, revealing that leaders favoring protection curb spread while susceptible individuals can drift toward riskier views and amplify outbreaks. The authors define a time-dependent quantity analogous to the reproduction number and establish that its oscillations alone suffice to produce recurrent epidemic waves.

Core claim

We introduce kinetic models to simulate epidemic spread while accounting for individuals' opinions on protective behaviors. Opinion exchanges occur on a social network represented by a graphon, leading to scenarios with or without opinion leaders. We prove convergence to equilibrium in the strong L1 norm via relative entropy methods and in homogeneous Sobolev spaces dot H^{-s}, s in (1/2,1), using Fourier-based techniques. We then design a structure-preserving scheme for the coupled opinion-epidemiological system, highlighting graphon effects: opinion leaders supporting protective behaviors limit disease spread, whereas influenceable individuals may shift toward opposing views, worsening the

What carries the argument

The graphon-modulated kinetic system that couples opinion exchange with epidemiological compartments, together with the time-dependent reproduction-number analog whose oscillations drive waves.

If this is right

  • Opinion leaders favoring protective behaviors reduce overall disease spread through the graphon-mediated network.
  • Individuals who are easily influenced can shift toward opposing protective views and thereby increase epidemic severity.
  • The numerical scheme preserves key structural properties and accurately reproduces graphon-driven effects in simulations.
  • Solutions converge strongly in L1 and in dot H^{-s} for s in (1/2,1), independent of the specific graphon shape under the model's assumptions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Targeted messaging aimed at a small set of opinion leaders on real networks approximated by graphons could measurably slow epidemic growth.
  • Recurrent waves arising purely from internal opinion feedback imply that seasonal external drivers are not always required to explain multi-wave outbreaks.
  • Replacing the abstract graphon with empirical adjacency data from contact-tracing studies would test whether specific network motifs amplify or suppress the observed oscillations.

Load-bearing premise

The coupled kinetic system admits the stated equilibria and is well-posed under the chosen graphon modulation and opinion-exchange rules.

What would settle it

Running the structure-preserving scheme on a fixed graphon with constant parameters and no external forcing, then checking whether epidemic waves appear precisely when the time-dependent reproduction analog oscillates.

Figures

Figures reproduced from arXiv: 2604.10614 by Andrea Bondesan, Jacopo Borsotti, Mattia Fontana.

Figure 1
Figure 1. Figure 1: (A) Plot of the graphon B given by (16). (B) Plot of the interaction function P defined by (17). Values of the parameters: ξ = 0.05 and χ = 2. and of ξ < (1 + χ) −1 is such that p attains its maximum value in the interior of Ω. This means that the agents who are the most inclined to interact have an intermediate number of connections. Indeed, highly connected individuals (i.e., the opinion leaders, corresp… view at source ↗
Figure 2
Figure 2. Figure 2: Graph of the propensity to interact p defined by (18), for different values of ξ and χ. The choice of parameters ξ = 0.25, χ = 2 (left scale) corresponds to the presence of opinion leaders (p → 0 as x → 0), while the choice ξ = 0.05, χ = 0.5 (right scale) corresponds to their absence. Remark 5. The choice (16) for the graphon B satisfies the integrability condition (9) under a reasonable and realistic assu… view at source ↗
Figure 3
Figure 3. Figure 3: Beta distributions (23) for different values of P˜(x). Agents that are not inclined to interact (P˜(x) = 0.05) are characterized by opinion polarization, while agents that are inclined to do so (P˜(x) = 0.95) experience consensus formation. We have chosen m∞ P˜ = 0 in (A) and m∞ P˜ = −0.2 in (B), while the other parameters are given by ρP˜ = 0.6 and νJ = 0.032. Moreover, the function cJ (x) is such that th… view at source ↗
Figure 4
Figure 4. Figure 4: Inverse gamma distributions (40) for different values of F ∞. We have set µ = 1.5, ζ = 1, and θ = 1, corresponding to an equilibrium h ∞ with finite variance. Finally, we remark that it is possible to replace the function I[w, ˆ 1](w) in (31) with its opposite I[−1,wˆ](w). In this case, one could measure how not seriously the disease is perceived by the population and we could determine analogous results t… view at source ↗
Figure 5
Figure 5. Figure 5: Test 1. Evolution of the simplified model (21) with epidemiological transition rate βT defined by (5) and graphon B given by (16). Opinion distri￾butions of susceptible (A), infected (B), and removed (C) individuals at different time instants and different fixed positions x ∈ Ω on the graphon. Note that we only plot f in S , since f in I and f in R are uniform distributions with a very small mass. The dash… view at source ↗
Figure 6
Figure 6. Figure 6: Test 1. Evolution of the simplified model (21) with epidemiological transition rate βT defined by (5) and graphon B given by (53). Opinion distri￾butions of susceptible (A), infected (B), and removed (C) individuals at different time instants and different fixed positions x ∈ Ω on the graphon. Note that only f in S is displayed, since f in I and f in R are uniform distributions with very small mass. The da… view at source ↗
Figure 7
Figure 7. Figure 7: Test 2. Evolution of the popularity model (32) depending on that of f, which varies as in [PITH_FULL_IMAGE:figures/full_fig_p043_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Test 3. Evolution of the simplified model (21) with epidemiological transition rate βT defined by (5), graphon B given by (16), and opinion interaction function G of the form (52). Figure (A) shows the initial distribution of susceptible agents. At the final simulation time T = 450, the distributions of susceptible (B) and infected (C) populations display a polarization of their highly connected individu￾a… view at source ↗
Figure 9
Figure 9. Figure 9: Test 4. Evolution of the original model (3) with epidemiological tran￾sition rate βT defined by (5), graphon B given by (16), and opinion interaction function G of the form (52). The initial population of susceptible individuals (A) is split between two opposite opinions, depending on the connectivity levels, with the unfavorable part being poorly connected and four times larger than the highly connected, … view at source ↗
Figure 10
Figure 10. Figure 10: Test 4. Evolution of the original model (3) with epidemiological tran￾sition rate βT defined by (5), graphon B given by (16), and opinion interaction function G of the form (52). The initial population of susceptible individuals (A) is split between two opposite opinions, depending on the connectivity levels, with the unfavorable part being poorly connected and four times larger than the highly connected,… view at source ↗
Figure 11
Figure 11. Figure 11: Test 4. Evolution of the effective reproduction number (13), for the original model (3) with epidemiological transition rate βT defined by (5), graphon B given by (16), and opinion interaction function G of the form (52). Temporal variations of Reff for the model without leaders (A) and for the model with leaders (B). The parameters are those used for [PITH_FULL_IMAGE:figures/full_fig_p048_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: (A) Plot of the graphon B given by (53). (B) Plot of the interaction function P defined by (54). Values of the parameters: r = 0.2 and χ = 2. fast timescale). Hence, the subsequent waves of the epidemic will be affected by such equilibrium of the opinion. Obviously, this analysis must be mainly carried out with numerical simulations. Appendix A. A spatial adjacency graphon Define the bounded symmetric gra… view at source ↗
Figure 13
Figure 13. Figure 13: Graph of the propensity to interact p defined by (55), for r = 0.2 and different values of χ [PITH_FULL_IMAGE:figures/full_fig_p050_13.png] view at source ↗
read the original abstract

We introduce kinetic models to simulate epidemic spread while accounting for individuals' opinions on protective behaviors. Opinion exchanges occur on a social network represented by a graphon, leading to scenarios with or without opinion leaders. We prove convergence to equilibrium in the strong $L^1$ norm via relative entropy methods and in homogeneous Sobolev spaces $\dot{H}^{-s}$, $s \in \big(\frac{1}{2},1\big)$, using Fourier-based techniques. We then design a structure-preserving scheme for the coupled opinion-epidemiological system, highlighting graphon effects: opinion leaders supporting protective behaviors limit disease spread, whereas influenceable individuals may shift toward opposing views, worsening epidemics. Finally, we introduce a time-dependent quantity, analogous to the reproduction number, whose oscillations can generate epidemic waves without explicit external forcing. The MATLAB code implementing our algorithms is made publicly available.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript introduces kinetic models coupling opinion dynamics on graphons with epidemic spread, accounting for protective behaviors and opinion leaders. It proves strong L1 convergence to equilibrium via relative entropy methods and convergence in homogeneous Sobolev spaces dot H^{-s} (s in (1/2,1)) via Fourier techniques. A structure-preserving numerical scheme is developed for the coupled system, with numerical illustrations of graphon effects on disease spread. Finally, a time-dependent quantity analogous to the reproduction number is introduced, whose oscillations are claimed to generate epidemic waves without external forcing. Public MATLAB code is provided.

Significance. If the convergence results hold with appropriate regularity, the framework connects kinetic theory, graphon networks, and epidemiology in a novel way, offering mechanistic insight into how opinion leaders and network structure modulate epidemics. The structure-preserving scheme and open code support reproducibility and potential extensions. The oscillating reproduction-number analog, if rigorously tied to the dynamics, could explain endogenous waves, though its impact depends on precise justification.

major comments (2)
  1. [Well-posedness and convergence sections (around the statements of Theorems on L1 and dot H^{-s} convergence)] The relative entropy dissipation for L1 convergence and the Fourier multiplier estimates for dot H^{-s} convergence both require the system to be globally well-posed with equilibria whose stability can be quantified. The manuscript does not explicitly list the functional assumptions on the graphon (e.g., measurability and L^infty boundedness on [0,1]^2) or the opinion-exchange kernel (e.g., Lipschitz continuity or boundedness) needed to close these estimates; without them the central convergence claims cannot be verified.
  2. [Section introducing the time-dependent reproduction-number analog and the associated numerical experiments] The time-dependent quantity introduced as analogous to the reproduction number is used to argue that oscillations can produce epidemic waves without external forcing. It is not shown to be derived from the linearization around the disease-free equilibrium or to control the growth rate via a precise threshold relation; if it remains phenomenological, the claim that it generates waves rests on numerical observation rather than analysis.
minor comments (3)
  1. [Model formulation] Notation for the graphon-modulated interaction terms and the opinion variable should be introduced with explicit functional forms early in the model section to aid readability.
  2. [Numerical scheme] The structure-preserving properties of the numerical scheme (positivity, conservation) are asserted but would benefit from a short theorem or explicit verification in the scheme section.
  3. [Numerical results] Figure captions should specify the graphon choices (e.g., constant vs. leader-dominated) and parameter values used in each panel for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major comment point by point below.

read point-by-point responses

  1. Referee: [Well-posedness and convergence sections (around the statements of Theorems on L1 and dot H^{-s} convergence)] The relative entropy dissipation for L1 convergence and the Fourier multiplier estimates for dot H^{-s} convergence both require the system to be globally well-posed with equilibria whose stability can be quantified. The manuscript does not explicitly list the functional assumptions on the graphon (e.g., measurability and L^infty boundedness on [0,1]^2) or the opinion-exchange kernel (e.g., Lipschitz continuity or boundedness) needed to close these estimates; without them the central convergence claims cannot be verified.

    Authors: We agree that the functional assumptions must be stated explicitly to allow verification of the convergence results. In the revised manuscript we have added a new subsection 2.1 'Assumptions' that lists: the graphon W is measurable and essentially bounded in L^infty([0,1]^2), and the opinion-exchange kernel is Lipschitz continuous and bounded. These are the minimal conditions used throughout the proofs of global well-posedness and of the relative-entropy and Fourier-multiplier estimates in Theorems 3.1 and 3.2. A short well-posedness argument has also been included in the appendix. revision: yes

  2. Referee: [Section introducing the time-dependent reproduction-number analog and the associated numerical experiments] The time-dependent quantity introduced as analogous to the reproduction number is used to argue that oscillations can produce epidemic waves without external forcing. It is not shown to be derived from the linearization around the disease-free equilibrium or to control the growth rate via a precise threshold relation; if it remains phenomenological, the claim that it generates waves rests on numerical observation rather than analysis.

    Authors: We acknowledge that the time-dependent quantity is introduced as a direct analog to the classical reproduction number rather than being obtained from linearization around the disease-free equilibrium. Its oscillatory behavior and correlation with wave-like incidence are demonstrated numerically. In the revision we have added a clarifying remark in Section 5 stating that a rigorous threshold analysis via linearization is outside the present scope and left for future work. The numerical evidence, obtained with the structure-preserving scheme, still provides mechanistic insight into endogenous wave generation. revision: partial

Circularity Check

0 steps flagged

No circularity: standard analytical proofs and model introduction are independent of conclusions.

full rationale

The paper introduces a coupled kinetic system for opinion-epidemic dynamics on graphons, proves strong L1 convergence via relative entropy dissipation and dot H^{-s} convergence via Fourier multipliers, designs a structure-preserving numerical scheme, and defines a time-dependent reproduction-number analog whose oscillations produce waves. These steps rely on external mathematical techniques (entropy methods, Fourier analysis) applied to the stated system; no equation reduces to a prior fitted parameter or self-referential definition, no load-bearing self-citation chain appears, and the new quantity is explicitly introduced rather than derived from data fits. The derivation chain is self-contained against the paper's own assumptions and standard PDE tools.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; specific free parameters, axioms, and invented entities are not enumerated in the provided text. The graphon representation of the social network and the well-posedness of the coupled kinetic system are implicit domain assumptions.

axioms (2)
  • domain assumption Opinion exchanges occur on a social network represented by a graphon
    Central modeling choice enabling continuous modulation of interactions.
  • domain assumption The coupled opinion-epidemiological system admits equilibria reachable via relative entropy and Fourier methods
    Required for the stated convergence proofs.

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