Analysis of a Competitive Bivirus SIS Epidemic Model with Game Theoretic Social Distancing

Benjamin Catalano; Keith Paarporn; Sebin Gracy

REVIEW 3 major objections 1 minor 299 references

Analysis of a Competitive Bivirus SIS Epidemic Model with Game Theoretic Social Distancing

T0 review · 3 major / 1 minor · reviewed 2026-05-08 · grok-4.3

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Pith's one-line read Dynamic social distancing in a competitive bi-virus SIS model creates non-monotone dynamics and lines of coexistence equilibria.

desk verdict The paper adds game-theoretic social distancing to a competitive bi-virus SIS model and gets non-monotonic dynamics plus lines of equilibria, but those features tie directly to one specific risk-to-distancing map. read the letter →

arxiv 2605.04340 v1 submitted 2026-05-05 eess.SY cs.SY

Benjamin Catalano , Keith Paarporn , Sebin Gracy This is my paper · ORCID

classification eess.SYcs.SY

keywords bi-virusSISmodelsocialdistancinggametheoryepidemicequilibrianon-monotonedynamicsstabilityanalysiscoexistencecompetitiveviruses

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces a bi-virus epidemic model in which individuals adjust their social distancing rates based on the perceived risk of each virus. This game-theoretic behavior renders the overall system non-monotone, unlike the classic competitive bi-virus model, and produces a distinct collection of equilibria. The authors derive necessary and sufficient conditions for almost global exponential stability of the disease-free equilibria, establish global stability results for most unilateral equilibria, and show that any coexistence equilibria appear as entire lines rather than isolated points. These findings matter because they indicate how public perceptions can determine whether one virus dominates, both are eradicated, or they persist together indefinitely.

What carries the argument

Dynamic social distancing rates that depend on perceived risk for each virus, which introduce non-monotonicity and generate lines of equilibria instead of isolated points.

What would settle it

Run the model from many different initial infection levels and check whether coexistence states form a continuous line of equilibria or collapse to isolated points; if only isolated points appear under the model's own distancing rule, the line-equilibria claim is false.

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Extended reading notes

Core claim

Load-bearing premise

The specific functional form chosen for how perceived risk translates into dynamic social distancing rates; if this mapping is misspecified, the claimed non-monotonicity and line equilibria may not hold.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 1 minor

Summary. The paper proposes a competitive bi-virus SIS epidemic model incorporating game-theoretic social distancing. It claims the resulting system is non-monotone (unlike the classic bi-virus model), details the disease-free equilibria (DFEs) and gives necessary and sufficient conditions for their almost-global exponential stability, proves global stability for all but one unilateral equilibrium (with local stability for the rest), shows that coexistence equilibria form lines when they exist, and provides local stability conditions for those lines. Numerical examples illustrate the findings.

Significance. If the stability results hold, the work is significant for showing how behavioral responses can produce non-monotonic dynamics and continua of equilibria in multi-virus settings, extending standard epidemic models. The explicit conditions for DFE stability and the contrast with monotone systems offer useful insights for public-health modeling. The global stability proofs for most equilibria constitute a concrete technical contribution.

major comments (3)

[Model formulation and equilibria analysis] The non-monotonicity and the existence of lines of coexistence equilibria are direct consequences of the specific functional form chosen for the game-theoretic mapping from perceived risk to time-varying contact rates. Altering this mapping (while preserving positivity and boundedness) can eliminate both features; the paper should state the explicit functional form used in the model equations and discuss whether the claimed results are robust to reasonable perturbations of this form.
[Stability analysis of DFEs] The necessary and sufficient conditions for almost-global exponential stability of the DFEs are stated, but the global stability arguments rely on Lyapunov or comparison techniques applied to the behavioral equations. Without the full derivations or the precise parameter assumptions under which the comparison holds, it is not possible to confirm that the 'almost global' claim covers all relevant initial conditions and parameter regimes.
[Unilateral equilibria stability] Global stability is claimed for all but one non-DFE isolated (unilateral) equilibrium. The exceptional equilibrium must be explicitly identified, and the reason the global argument fails for it (while local stability still holds) should be explained; otherwise the completeness of the stability classification is difficult to assess.

minor comments (1)

[Numerical illustrations] The numerical examples should report the exact parameter values, initial conditions, and the specific functional form of the distancing rates used, so that readers can reproduce the observed line equilibria and stability behavior.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough and constructive review of our manuscript. We address each major comment point by point below, indicating where we agree and how we will revise the paper to improve clarity and completeness.

read point-by-point responses

Referee: The non-monotonicity and the existence of lines of coexistence equilibria are direct consequences of the specific functional form chosen for the game-theoretic mapping from perceived risk to time-varying contact rates. Altering this mapping (while preserving positivity and boundedness) can eliminate both features; the paper should state the explicit functional form used in the model equations and discuss whether the claimed results are robust to reasonable perturbations of this form.

Authors: We agree that the non-monotonicity and the continua of coexistence equilibria are tied to the specific functional form of the game-theoretic social distancing mapping. This form is explicitly defined in Section II of the manuscript (the contact rate for each virus is a decreasing function of perceived risk, derived from the Nash equilibrium of the underlying game). While this choice is standard for modeling behavioral responses, we acknowledge that other mappings preserving positivity and boundedness could restore monotonicity or yield isolated equilibria. In the revised manuscript we will add a dedicated paragraph in the model section stating the explicit form and a new subsection discussing robustness, including brief examples of alternative mappings (e.g., linear or power-law decay) that preserve the modeling assumptions but alter the dynamical features. revision: yes
Referee: The necessary and sufficient conditions for almost-global exponential stability of the DFEs are stated, but the global stability arguments rely on Lyapunov or comparison techniques applied to the behavioral equations. Without the full derivations or the precise parameter assumptions under which the comparison holds, it is not possible to confirm that the 'almost global' claim covers all relevant initial conditions and parameter regimes.

Authors: The necessary and sufficient conditions appear in Theorem 3, with the full Lyapunov and comparison derivations provided in Appendix A. The comparison principle holds under the parameter regime where the spectral radius condition for local stability of the DFE is satisfied and the behavioral gain is sufficiently strong. To make this transparent, we will insert a short remark immediately after Theorem 3 that summarizes the key steps of the proof and explicitly states that almost-global exponential stability applies to all initial conditions in the nonnegative orthant (excluding the origin, which is unstable). We will also move the most critical inequality from the appendix into the main text. revision: yes
Referee: Global stability is claimed for all but one non-DFE isolated (unilateral) equilibrium. The exceptional equilibrium must be explicitly identified, and the reason the global argument fails for it (while local stability still holds) should be explained; otherwise the completeness of the stability classification is difficult to assess.

Authors: We agree that the exceptional unilateral equilibrium should be identified explicitly. It is the isolated point at which each virus is at its single-virus endemic level while the behavioral response is at the corresponding steady state; this point lies on the boundary between the basins of the two single-virus endemic equilibria. The global Lyapunov argument used for the other unilateral equilibria fails here because the LaSalle invariance set is no longer a singleton (nearby trajectories can drift toward the line of coexistence equilibria). Local stability nevertheless holds by direct computation of the Jacobian eigenvalues. In the revision we will label this equilibrium clearly (as E* in Section IV), state its coordinates, and add a paragraph explaining why the global proof technique does not extend to it while local stability is retained. revision: yes

Circularity Check

0 steps flagged · score 2.0 of 10

Stability and equilibrium results derived via standard Lyapunov/comparison arguments on the proposed model

full rationale

The paper defines a competitive bi-virus SIS model with an explicit game-theoretic mapping from perceived risk to time-varying contact rates, then applies standard Lyapunov functions and comparison principles to prove almost-global exponential stability of the DFE and related results for other equilibria. No step reduces by construction to a fitted parameter, self-referential definition, or load-bearing self-citation; the functional form is an explicit modeling assumption whose consequences are derived rather than presupposed. The derivation chain remains self-contained against the stated equations and external mathematical tools.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The model relies on standard assumptions of SIS compartmental structure and a game-theoretic mapping from perceived risk to distancing rates; no explicit free parameters or invented entities are stated in the abstract.

assumptions (2)

domain assumption The social distancing rate is a continuous, differentiable function of perceived infection risks for each virus.
Required for the non-monotone vector field and the existence of line equilibria.
standard math The underlying contact network is well-mixed and homogeneous.
Standard for mean-field SIS models; enables the ODE formulation.

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Pith. "Pith review of Analysis of a Competitive Bivirus SIS Epidemic Model with Game Theoretic Social Distancing." pith.science (2026). https://pith.science/paper/2605.04340

@misc{pith2026260504340,
  author       = {Pith},
  title        = {Pith review of: Analysis of a Competitive Bivirus SIS Epidemic Model with Game Theoretic Social Distancing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2605.04340}},
  note         = {Machine review of arXiv:2605.04340}
}

read the original abstract

We propose a competitive bi-virus model with dynamic social distancing behavior. Our model illustrates how public perception of different viruses changes the conditions for their eradication, their coexistence, or the dominance of one over the other. We show that our model is not monotone, in contrast to the classic bi-virus model. We detail how social distancing behavior produces different sets of equilibria than the classic bi-virus model and changes the criteria for their stability. In particular, we detail the set of disease free equilibria (DFE) present in our model and identify necessary and sufficient conditions for almost global exponential stability of the same. We prove similar global results for all but one non-DFE isolated (unilateral) equilibria and local stability results for the remainder. We also consider coexistence equilibria; we show such equilibria, when they exist, take the form of lines of equilibria and give local conditions for their stability. Finally, we illustrate our theoretical findings with numerical examples.

Figures

Figures reproduced from arXiv: 2605.04340 by the authors.

**Figure 1.** Simulation with pDFE0 stable view at source ↗

**Figure 3.** Simulation with p11 stable view at source ↗

**Figure 5.** The dashed green line is a line of coexistence equilibria (L0) that is also locally stable view at source ↗

Figures from the paper (3 more)

**Figure 7.** Figure 7: Simulation showing both viruses endemic and partial social distancing in the

**Figure 8.** Figure 8: Simulation of coordination

**Figure 10.** Figure 10: (a): The trajectories of system (9.3) projected onto the

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed May 8, 2026 · model on record in the stance chip above.

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