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Optimal Bayesian Persuasion for Containing SIS Epidemics

Urmee Maitra, Ashish R. Hota, Philip E. Paré

TL;DR

A susceptible-infected-susceptible (SIS) epidemic model in which a large group of individuals decide whether to adopt partially effective protection without being aware of their individual infection status is considered, and the performance of the dynamic signaling scheme with the optimal static signaling scheme is compared.

Abstract

We consider a susceptible-infected-susceptible (SIS) epidemic model in which a large group of individuals decide whether to adopt partially effective protection without being aware of their individual infection status. Each individual receives a signal which conveys noisy information about its infection state, and then decides its action to maximize its expected utility computed using its posterior probability of being infected conditioned on the received signal. We first derive the static signal which minimizes the infection level at the stationary Nash equilibrium under suitable assumptions. We then formulate an optimal control problem to determine the optimal dynamic signal that minimizes the aggregate infection level along the solution trajectory. We compare the performance of the dynamic signaling scheme with the optimal static signaling scheme, and illustrate the advantage of the former through numerical simulations.

Optimal Bayesian Persuasion for Containing SIS Epidemics

TL;DR

A susceptible-infected-susceptible (SIS) epidemic model in which a large group of individuals decide whether to adopt partially effective protection without being aware of their individual infection status is considered, and the performance of the dynamic signaling scheme with the optimal static signaling scheme is compared.

Abstract

We consider a susceptible-infected-susceptible (SIS) epidemic model in which a large group of individuals decide whether to adopt partially effective protection without being aware of their individual infection status. Each individual receives a signal which conveys noisy information about its infection state, and then decides its action to maximize its expected utility computed using its posterior probability of being infected conditioned on the received signal. We first derive the static signal which minimizes the infection level at the stationary Nash equilibrium under suitable assumptions. We then formulate an optimal control problem to determine the optimal dynamic signal that minimizes the aggregate infection level along the solution trajectory. We compare the performance of the dynamic signaling scheme with the optimal static signaling scheme, and illustrate the advantage of the former through numerical simulations.

Paper Structure

This paper contains 10 sections, 3 theorems, 23 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. SIS Epidemic under Bayesian Persuasion
  3. Optimal Static Signal
  4. Optimal Dynamic Signal
  5. Numerical Results
  6. Static Signaling
  7. Dynamic Signaling
  8. Dynamic Signaling under Modified Cost Function
  9. Relaxing the Assumption $\mu_{\mathtt{I}} = 1$
  10. Conclusions

Key Result

Theorem 1

( hota2023bayesian) Under Assumption assumption:main, we have the following characterization of the SNE:

Figures (4)

  • Figure 1: Infected proportion at the SNE with respect to $\mu_{\texttt{S}}$ when Assumption \ref{['assump:cp_more']} is satisfied (left); and not satisfied (right).
  • Figure 2: Evolution of proportion of infected individuals (left), the proportions of individuals who remain unprotected (middle), and the signal $\mu_{\texttt{S}}$ (right) under both the static (dashed) and dynamic (solid) signaling schemes.
  • Figure 3: Comparison of the infection levels and dynamic control signals for the original and modified stage cost.
  • Figure 4: Variation of $\mu_{\texttt{S}}^{\mathtt{opt}}(\mu_{\mathtt{I}})$ (left), and the infected proportion at the SNE under $(\mu_{\texttt{S}}^{\mathtt{opt}}(\mu_{\mathtt{I}}),\mu_{\mathtt{I}})$ (right) with respect to $\mu_{\mathtt{I}}$.

Theorems & Definitions (7)

  • Theorem 1
  • Remark 1
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Remark 2