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SIS Epidemic Containment under Game-theoretic Rational Learning

Urmee Maitra, Keith Paarporn, Ceyhun Eksin

arXiv:2610.03173v1eess.SYcs.SY

TL;DR

The paper asks how bounded rationality changes outcomes in a coupled SIS epidemic and behavioral model, beyond the imitative-learning focus of much prior work. It analyzes logit equilibria and their stability, finding regimes where lower rationality stabilizes disease-free or endemic equilibria that are otherwise unstable.

  • Problem

    Prior epidemic-learning work largely modeled individuals as imitative learners, while bounded-rationality studies had not explicitly tracked SIS disease-free and endemic transitions as rationality changed or compared learning rules.

    Source: §I · p. 1

  • Method

    The paper models behavioral evolution with logit choice in a coupled SIS infection-behavior system and analyzes the stationary equilibria and their stability.

    Source: §I · p. 1, §II · p. 2, B. Behavioral Characterization of SLE · p. 3

  • Results

    For 1 < R0 < 2, low-rationality logit learning makes the disease-free equilibrium locally asymptotically stable although replicator and high-rationality logit dynamics leave it unstable.

    Source: §I · p. 1, §VII · p. 9

  • Takeaways & Limitations

    In some parameter regimes, bounded rationality stabilizes equilibria and is associated with convergence to a lower infection level than higher rationality or replicator dynamics.

    Source: §I · p. 1, §VII · p. 9

    The paper does not completely characterize equilibria across parameter spaces, and a theoretical analysis of the replicator-versus-bounded-rationality convergence comparison remains open.

    Source: §VII · p. 9

Abstract

from arXiv · show

We consider a susceptible-infected-susceptible (SIS) epidemiological model in which individuals dynamically decide whether to follow social distancing or not. Driven by game-theoretic payoffs, their actions influence the spread of infection. Much of the existing work in this area considers individuals that are imitative learners, leading to the well-studied replicator equation. In this paper, we investigate how the dynamical outcomes may fundamentally shift when individuals are instead equipped with logit learning rules, i.e., when individuals possess bounded rationality. We analyze the local asymptotic stability of equilibria that exist in different parameter regimes of the coupled dynamical system. We show that in regimes where the disease-free equilibrium is unstable under imitative learning and sufficiently high-rationality logit learning, sufficiently low rationality can avoid such an undesirable outcome by stabilizing the disease-free equilibrium. We also identify parameter regimes in which an unstable endemic equilibrium stabilizes when rationality falls below a threshold. We numerically illustrate the convergence of infection state and response trajectories to a stable disease-free equilibrium at a lower degree of rationality, whereas the same trajectories otherwise converge to a stable endemic equilibrium as the rationality level rises. We also compare a specific case in which trajectories under replicator dynamics fail to converge, and illustrate convergence when rationality is below an analytically derived threshold.

I. INTRODUCTION

The paper examines how logit learning and bounded rationality shape a coupled SIS infection-behavior system, addressing an underexplored comparison with imitative learning. It characterizes stationary equilibria and shows that rationality can change their stability.

I. INTRODUCTION

  • Low-rationality logit learning can stabilize disease-free equilibria that are unstable under replicator and high-rationality logit dynamics.The paper also identifies a regime where replicator dynamics sustain oscillations, while sufficiently low-rationality logit dynamics converge to a stable endemic equilibrium.Source: §I · p. 1

II. COUPLED EPIDEMIC AND BEHAVIORAL DYNAMICS

  • The SIS model couples infected fraction I(t) and cooperation x(t), with cooperation reducing the effective transmission rate to β(1 − x).The population is modeled as homogeneous, and individuals choose to cooperate with or defect from health recommendations.Source: §II · p. 2
  • Logit learning models behavioral updates, with σ indexing rationality: σ = 0 is completely irrational, and larger σ favors payoff-maximizing actions.The payoff assumptions include defection strictly dominating cooperation in the disease-free environment.Source: §II · p. 2

A. SLE Definition

  • A stationary logit equilibrium is a social state satisfying the epidemiological and behavioral equilibrium conditions; finite σ uses logit response, while σ → ∞ is the complete-rationality limit.The behavioral component is x* and the infection state is I*.Source: A. SLE Definition · p. 2

B. Behavioral Characterization of SLE

  • The complete-rationality SLE characterization differs from the finite-σ characterization; under complete rationality, (1, 0) is not an SLE.The passage excludes this state because its payoff difference contradicts the premise used in the argument.Source: B. Behavioral Characterization of SLE · p. 3

DFE, I∗

The section states conditions for disease-free and endemic epidemic equilibria and contrasts the stationary strategies obtained under replicator and logit learning.

DFE, I∗

  • Unlike logit dynamics, replicator dynamics need not have x* = 0.5 at an interior stationary point.The section introduces this comparison with replicator dynamics as the popular imitation-learning rule.Source: DFE, I∗ · p. 3

IV. EXISTENCE OF SLE

  • The disease-free equilibrium always exists and is locally asymptotically stable only when γ ≥ β(1 − x*).The endemic equilibrium exists and is locally asymptotically stable only when γ < β(1 − x*).Source: §IV · p. 3
  • Under complete rationality, defection is uniquely dominant in the disease-free regime, giving x* = 0 and the corresponding social state (0, 0).This follows from the stated positive payoff differences for defection over cooperation.Source: §IV · p. 3

A. Existence of SLE z∗

The section gives conditions for an interior endemic stationary logit equilibrium and establishes its uniqueness when it exists.

B. Existence of SLE z∗

  • The stated existence condition is 2γ(δT R1 + δP S1) + (β − 2γ)(δT R0 + δP S0) = 0.The passage further states that δT R0 + δP S0 must be negative for the interior endemic SLE to exist.Source: B. Existence of SLE z∗ · p. 4
  • When R0 > 2 and the stated existence conditions hold, the interior endemic SLE is unique.The passage identifies R0 > 2 as necessary for the endemic equilibrium.Source: B. Existence of SLE z∗ · p. 3

C. Existence of SLE z∗

The section establishes existence and uniqueness results for endemic stationary logit equilibria. For finite rationality, uniqueness holds in the stated reproduction-number regimes.

C. Existence of SLE z∗

D. Existence of SLE z∗

The results show that rationality can change equilibrium stability: low rationality can stabilize a disease-free equilibrium, while the existence results also characterize a completely rational corner endemic equilibrium. The stated sufficient conditions cover only part of parameter space.

D. Existence of SLE z∗

  • When R0 > 1 and individuals are completely rational, the corner endemic SLE is the only endemic SLE that exists.Source: D. Existence of SLE z∗ · p. 5
  • The sufficient conditions guarantee uniqueness only in part of parameter space; characterizing the remaining spaces is left for future work.Source: D. Existence of SLE z∗ · p. 5

V. STABILITY OF SLE AND IMPACT OF RATIONALITY

  • In some regimes, the disease-free equilibrium is unstable under imitation and high-rationality logit learning but stable at sufficiently low rationality.Source: §V · p. 5

A. LAS of z∗

The disease-free equilibrium’s stability depends on rationality and game parameters. In the specified regime, reducing rationality below a threshold stabilizes it, unlike replicator dynamics.

A. LAS of z∗

  • For 0 ≤ σ < σTH, the disease-free SLE is locally asymptotically stable, and it is unstable for σ > σTH.Source: A. LAS of z∗ · pp. 5–6
  • The condition δT R1 ≥ δP S1 represents a disease-free payoff setting where defection’s advantage is not lower against cooperators than against defectors.Source: A. LAS of z∗ · p. 6
  • At σ = σTH, the equilibrium is non-hyperbolic, so linearization does not establish local asymptotic stability there.Source: A. LAS of z∗ · p. 6
  • When 1 < R0 < 2, the disease-free equilibrium is unstable under replicator dynamics, while low-rationality logit dynamics stabilize it.Source: A. LAS of z∗ · p. 6

B. LAS of z∗

For the unique interior endemic SLE, stability depends on the parameter A and, when A < 0, on rationality relative to a threshold. The paper links instability at high rationality to heightened sensitivity to payoff fluctuations.

B. LAS of z∗

  • When A ≥ 0, the unique interior endemic SLE is locally asymptotically stable for every σ ≥ 0.Source: B. LAS of z∗ · pp. 6–7
  • When A < 0, the interior endemic SLE is locally asymptotically stable for σ < σc and unstable for σ > σc.Source: B. LAS of z∗ · pp. 6–7
  • At σ = σc, the linearization is non-hyperbolic and the theorem makes no stability claim.Source: B. LAS of z∗ · p. 7
  • For the specified payoff condition, high rationality makes logit responses more sensitive to small payoff differences, so infection perturbations can drive the strategy away from its equilibrium value.Source: B. LAS of z∗ · p. 7

C. LAS of z∗

The section establishes stability conditions for endemic equilibria under finite and complete rationality, and distinguishes the pointwise complete-rationality limit from sufficiently large finite rationality. It also derives a rationality level for targeting a steady-state cooperation fraction.

C. LAS of z∗

  • Under uniqueness condition (16), the finite-σ endemic SLE is locally asymptotically stable.Source: C. LAS of z∗ · pp. 7–8
  • For R0 > 1 and inequality (19), the fully rational corner endemic SLE is locally asymptotically stable; with condition (20), it is the unique stable SLE.Source: C. LAS of z∗ · p. 8
  • The authors derive σ to attain a target steady-state cooperation fraction ε; increasing σ reduces the finite-σ equilibrium cooperation fraction.Source: C. LAS of z∗ · p. 8
  • When equality (9) fails under the stated conditions, no endemic SLE exists in the pointwise complete-rationality limit, although one exists for every sufficiently large finite σ.Source: C. LAS of z∗ · p. 8

VI. SIMULATION RESULTS

The simulations show that rationality thresholds govern whether disease-free or endemic equilibria are stable. In the selected interior-endemic regime, bounded rationality stabilizes the equilibrium and yields a lower time-averaged infection level than replicator dynamics.

Figure 1

  • For σTH=1.6894, the disease-free SLE is stable below the threshold, whereas a stable finite-σ endemic equilibrium exists and the disease-free SLE is unstable above it.Figure 1 compares σ=0.6894 and σ=4.6894, respectively.Source: §VI · p. 8, Figure 1 · p. 8

Figure 2

  • As σ rises, cooperation falls and infection rises; infection is zero below σTH=1.6894 and positive beyond it.At σ=0, exactly half the population cooperates and the equilibrium is disease-free.Source: §VI · pp. 8–9, Figure 2 · p. 9

Figure 3

  • In the selected parameter regime, trajectories under high rationality converge numerically to a stable finite-σ SLE close to z*, while in the limit σ→∞ the stable SLE is (0, 0.4).Source: §VI · p. 9

Figure 4

  • Below σ_c=14, the unique interior SLE is stable; above it, it is unstable, while replicator trajectories oscillate and bounded rationality yields lower time-averaged infection.Under replicator dynamics, the averages after 600 time units are x(t)=0.39 and I(t)=0.27.Source: §VI · p. 9

VII. CONCLUSIONS AND FUTURE WORK

The paper finds that bounded rationality can stabilize equilibria that are unstable under replicator or high-rationality logit learning. Its equilibrium analysis is incomplete, and a full theoretical comparison with replicator dynamics remains open.

VII. CONCLUSIONS AND FUTURE WORK

  • In the regime 1 < R0 < 2, the disease-free equilibrium is unstable under replicator and high-rationality logit dynamics but locally asymptotically stable under low-rationality logit learning.The authors also illustrate parameter regimes where bounded rationality stabilizes an existing endemic equilibrium and trajectories converge to a lower infection level.Source: §VII · p. 9
  • The paper does not completely characterize equilibria across parameter regimes, leaving the remaining parameter spaces for future work.A complete theoretical analysis of the illustrated comparison with replicator dynamics also remains open.Source: §VII · p. 9

APPENDIX

The appendix establishes positive invariance of the state region and gives conditions for endemic stationary logit equilibria.

APPENDIX

  • For every finite σ, the state region Γ is positively invariant because the vector field does not point outward at its boundary.At I=0 the infection derivative is zero, at I=1 it is negative, and the response dynamics point inward or tangent at x=0 and x=1.Source: APPENDIX · p. 11
  • R0 > 1 implies β > γ, which is required for the endemic infection level I*=1−γ/β to exist.Source: APPENDIX · p. 11
  • Under conditions (19) and (20), the endemic stationary logit equilibrium z*EE(cor) is unique.Condition (20) ensures a strictly positive payoff difference for x* in the stated interior range.Source: APPENDIX · p. 11
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