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Epidemic Population Games And Perturbed Best Response Dynamics

Shinkyu Park, Jair Certorio, Nuno C. Martins, Richard J. La

TL;DR

The paper addresses how to design a dynamic payoff mechanism to mitigate epidemic spread by steering a population of strategic agents toward safer behaviors under a budget. It couples a modified $SIRS$ epidemic model with a perturbed best-response (PBR) evolutionary dynamic and uses system-theoretic, passivity-based Lyapunov analysis to guarantee global convergence to an endemic equilibrium that minimizes infection prevalence under budget constraints. A key contribution is the incorporation of noise in agent decision-making via a one-parameter logit-type choice function $C^\\mu$, together with learning methods to estimate $\\mu$ from data and to compute optimal stationary rewards $r^*$ and transmission rate $\\beta^*$ under budget limits; the framework also yields anytime upper bounds on the infectious fraction and on the instantaneous cost. The results demonstrate that the time-scale parameter of the payoff dynamics and the learning-parameter estimation combine to enable practical, data-driven control of epidemics with provable stability, offering a principled approach for adaptive incentives in public health. The work has implications for designing affordable, robust incentive schemes that account for bounded rationality and noisy reward perception in large populations.

Abstract

This paper proposes an approach to mitigate epidemic spread in a population of strategic agents by encouraging safer behaviors through carefully designed rewards. These rewards, which adapt to the evolving state of the epidemic, are ascribed by a dynamic payoff mechanism we seek to design. We use a modified SIRS model to track how the epidemic progresses in response to the agents' strategic choices. By employing perturbed best response evolutionary dynamics to model the population's strategic behavior, we extend previous related work so as to allow for noise in the agents' perceptions of the rewards and intrinsic costs of the available strategies. Central to our approach is the use of system-theoretic methods and passivity concepts to obtain a Lyapunov function, ensuring the global asymptotic stability of an endemic equilibrium with minimized infection prevalence under budget constraints. We leverage the Lyapunov function to analyze how the epidemic's spread rate is influenced by the time scale of the payoff mechanism's dynamics. Additionally, we derive anytime upper bounds on both the infectious fraction of the population and the instantaneous cost a social planner must incur to control the spread, allowing us to quantify the trade-off between peak infection prevalence and the corresponding cost. For a class of one-parameter perturbed best response models, we propose a method to learn the model's parameter from data.

Epidemic Population Games And Perturbed Best Response Dynamics

TL;DR

The paper addresses how to design a dynamic payoff mechanism to mitigate epidemic spread by steering a population of strategic agents toward safer behaviors under a budget. It couples a modified epidemic model with a perturbed best-response (PBR) evolutionary dynamic and uses system-theoretic, passivity-based Lyapunov analysis to guarantee global convergence to an endemic equilibrium that minimizes infection prevalence under budget constraints. A key contribution is the incorporation of noise in agent decision-making via a one-parameter logit-type choice function , together with learning methods to estimate from data and to compute optimal stationary rewards and transmission rate under budget limits; the framework also yields anytime upper bounds on the infectious fraction and on the instantaneous cost. The results demonstrate that the time-scale parameter of the payoff dynamics and the learning-parameter estimation combine to enable practical, data-driven control of epidemics with provable stability, offering a principled approach for adaptive incentives in public health. The work has implications for designing affordable, robust incentive schemes that account for bounded rationality and noisy reward perception in large populations.

Abstract

This paper proposes an approach to mitigate epidemic spread in a population of strategic agents by encouraging safer behaviors through carefully designed rewards. These rewards, which adapt to the evolving state of the epidemic, are ascribed by a dynamic payoff mechanism we seek to design. We use a modified SIRS model to track how the epidemic progresses in response to the agents' strategic choices. By employing perturbed best response evolutionary dynamics to model the population's strategic behavior, we extend previous related work so as to allow for noise in the agents' perceptions of the rewards and intrinsic costs of the available strategies. Central to our approach is the use of system-theoretic methods and passivity concepts to obtain a Lyapunov function, ensuring the global asymptotic stability of an endemic equilibrium with minimized infection prevalence under budget constraints. We leverage the Lyapunov function to analyze how the epidemic's spread rate is influenced by the time scale of the payoff mechanism's dynamics. Additionally, we derive anytime upper bounds on both the infectious fraction of the population and the instantaneous cost a social planner must incur to control the spread, allowing us to quantify the trade-off between peak infection prevalence and the corresponding cost. For a class of one-parameter perturbed best response models, we propose a method to learn the model's parameter from data.
Paper Structure (17 sections, 7 theorems, 72 equations, 4 figures)

This paper contains 17 sections, 7 theorems, 72 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Evolutionary Dynamics Model
  3. PBR Learning Rule and Epidemic Population Game
  4. Problem Formulation and Paper Structure
  5. Motivation and Comparative Literature Review
  6. A Solution to Main Problem
  7. Optimal Design of $\bar{r}, \bar{\beta}$
  8. Anytime Bound
  9. Learning One-Parameter Choice Function $C^\mu$
  10. Simulations
  11. Conclusions and Future Research Plans
  12. Evaluation of (EPG) under Different Learning Rules for (PBR EDM)
  13. Proof of Theorem \ref{['theorem:stability']}
  14. Proof of Proposition \ref{['prop:local_exponential_stability']}
  15. Proof of Lemma \ref{['proposition:estimated_perturbation_parameter']}
  16. ...and 2 more sections

Key Result

Theorem 1

Consider the closed-loop model consisting of (EPG) and (PBR EDM), with the parameters $\bar{\beta} \in (\sigma, \vec{\beta}_n)$ and $\bar{r} \in \mathbb R_{\geq 0}^n$ given. Suppose $\mathcal{B}\text{\footnotesize $(t)$} \geq \mathcal{B}_{\text{min}}, ~\forall t \geq 0$ holds for some $\bar{\beta} >

Figures (4)

  • Figure 1: Simulation results for Example \ref{['ex:exampleLogit']} illustrating (a) the infectious fraction $I\text{\footnotesize $(t)$}$ of the population with respect to $I^\ast$ and (b) the average cost $r'\text{\footnotesize $(t)$} x\text{\footnotesize $(t)$}$, evaluated with different values of $\kappa$ for \ref{['eq:G']}.
  • Figure 2: Plots (a)--(c) illustrate how the cost upper bound \ref{['eq:cost_upper_bound']} varies depending on $\bar{\beta}$ for each fixed $\mu_U$: (a) $\mu_U=1$, (b) $\mu_U=2$, and (c) $\mu_U=5$. The red circle indicates the optimal endemic transmission rate $\beta^\ast =0.1691$ and the budget $c^\ast = 0.15$. Plot (d) depicts the smallest $\bar{\beta}_{\text{min}}$ under which the cost upper bound does not exceed the budget $c^\ast = 0.15$ for $\mu_U \in [1, 5]$, i.e., $\bar{\beta}_{\text{min}} = \min \{\bar{\beta} > 0 \,|\, \mu_U \lambda (\bar{\beta} \!-\! \vec{\beta}_n) + \tilde{c}' (\bar{I}) \mathop{\mathrm{arg\,max}}\limits_{z \in \mathrm{int} (\mathbb X)} ( z' (\lambda \vec{\beta}) \!-\! \bar{Q} (z) ) \leq c^\ast \}$.
  • Figure 3: Simulation results for Scenario 1 and Scenario 2 illustrating (a) the infectious fraction $I\text{\footnotesize $(t)$}$ of the population with respect to $I^\ast$ and (b) the average cost $r'\text{\footnotesize $(t)$} x\text{\footnotesize $(t)$}$, where the parameters $\bar{r}, \bar{\beta}$ of (EPG) are updated using the optimal $r^\ast, \beta^\ast$ beginning from the day $t_0 = 240$, as indicated by the green dotted vertical line.
  • Figure 4: The ratio $I\text{\footnotesize $(t)$}/I^\ast$ for (EPG) evaluated with different noise distributions for the choice function \ref{['eq:perturbed_revision_protocol']}. The square area highlighted in (a) is presented at an enlarged scale in (b).

Theorems & Definitions (14)

  • Example 1
  • Definition 1
  • Remark 1
  • Theorem 1
  • Proposition 1
  • Remark 2
  • Corollary 1
  • Remark 3
  • Example 2
  • Lemma 1
  • ...and 4 more