[Paper Review] Mean-Field Game Analysis of SIR Model with Social Distancing

Cho S|arXiv (Cornell University)|May 14, 2020

COVID-19 epidemiological studies19 references19 citations

TL;DR

This paper formulates a mean-field game model to analyze social distancing in an SIR epidemic framework, where individuals balance contact utility against infection risk. It shows that selfish behavior leads infected individuals to make more contacts than socially optimal, necessitating public policies like quarantine or paid sick leave to reduce transmission, especially post-peak, and quantifies policy effectiveness via the price of anarchy.

ABSTRACT

The current COVID-19 pandemic has proven that proper control and prevention of infectious disease require creating and enforcing the appropriate public policies. One critical policy imposed by the policymakers is encouraging the population to practice social distancing (i.e. controlling the contact rate among the population). Here we pose a mean-field game model of individuals each choosing a dynamic strategy of making contacts, given the trade-off of gaining utility but also risking infection from additional contacts. We compute and compare the mean-field equilibrium (MFE) strategy, which assumes each individual acting selfishly to maximize its own utility, to the socially optimal strategy, which maximizes the total utility of the population. We prove that the optimal decision of the infected is always to make more contacts than the level at which it would be socially optimal, which reinforces the important role of public policy to reduce contacts of the infected (e.g. quarantining, sick paid leave). Additionally, we include cost to incentivize people to change strategies, when computing the socially optimal strategies. We find that with this cost, policies reducing contacts of the infected should be further enforced after the peak of the epidemic has passed. Lastly, we compute the price of anarchy (PoA) of this system, to understand the conditions under which large discrepancies between the MFE and socially optimal strategies arise, which is when intervening public policy would be most effective.

Motivation & Objective

• To model individual contact decisions in an epidemic as a dynamic trade-off between utility and infection risk.
• To compare mean-field equilibrium (selfish) strategies with socially optimal strategies that maximize total population utility.
• To evaluate the effectiveness of public policies in reducing contact rates, especially among infected individuals.
• To quantify the degradation of system performance due to selfish behavior using the price of anarchy.
• To assess how disease characteristics (e.g., incubation period, recovery rate) affect the need for centralized policy intervention.

Proposed method

• Models a continuous-time mean-field game with three compartments: susceptible (S), infected (I), and recovered (R), assuming large, well-mixed populations.
• Uses a contact rate function $ C(\cdot) $ that depends on individual contact strategies $ c_z(t) $, with frequency- and density-dependent transmission formulations.
• Derives Hamilton-Jacobi-Bellman equations for individual optimal control under mean-field assumptions, solving for equilibrium and socially optimal strategies.
• Incorporates a cost $ k $ per unit change in contact rate to model central planner’s cost in enforcing behavioral changes.
• Computes the price of anarchy (PoA) as the ratio of system performance under selfish vs. optimal strategies to assess policy necessity.
• Extends the SIR model to include an exposed (E) compartment to study the impact of incubation periods on social distancing sustainability.

Experimental results

Research questions

• RQ1 How do selfish individuals' contact strategies in an epidemic differ from the socially optimal contact rates?
• RQ2 What is the impact of public policies—such as quarantining or paid sick leave—on reducing contact rates among infected individuals?
• RQ3 When is the price of anarchy highest, indicating the greatest need for centralized policy intervention?
• RQ4 How does the presence of an incubation period affect the sustainability of social distancing behaviors?
• RQ5 Under what disease parameters is it most beneficial to enforce contact reductions after the epidemic peak?

Key findings

• The mean-field equilibrium (MFE) strategy for infected individuals always results in higher contact rates than the socially optimal strategy, indicating a systemic failure of self-regulation.
• The socially optimal strategy for infected individuals is always lower than the selfish equilibrium, reinforcing the need for policies like quarantine or paid sick leave to reduce their contact rates.
• When a cost $ k $ is imposed on changing contact rates, the optimal strategy delays contact reduction until after the epidemic peak, especially when $ k $ is high.
• The price of anarchy is highest for diseases with low recovery rate $ \mu $, high infection utility $ a_I $, high transmission probability $ \beta $, and intermediate contact utility $ b_I $, indicating strong need for policy intervention.
• Diseases with long incubation periods (e.g., COVID-19) make sustained social distancing more difficult due to uncertainty about infection status and presymptomatic transmission.
• Even with only susceptible individuals practicing social distancing, the system-wide peak of infection is flattened, showing that behavioral adaptation can significantly reduce epidemic burden.