Mean-Field Game Analysis of SIR Model with Social Distancing

As the current COVID-19 outbreak shows, disease epidemic is a complex problem which must be tackled with the right public policy. One of the main tools of the policymakers is to control the contact rate among the population, commonly referred to as social distancing, in order to reduce the spread of the disease. We pose a mean-field game model of individuals each choosing a dynamic strategy of making contacts, given the trade-off of utility from contacts but also risk of infection from those contacts. We compute and compare the mean-field equilibrium (MFE) strategy, which assumes individuals acting selfishly to maximize their own utility, to the socially optimal strategy, which is the strategy maximizing the total utility of the population. We prove that the infected always want to make more contacts than the level at which it would be socially optimal, which reinforces the need to reduce contacts of the infected (e.g. quarantining, sick paid leave). Additionally, we compute the socially optimal strategies, given the costs to incentivize people to change from their selfish strategies. We find that if we impose limited resources, curbing contacts of the infected is more important after the peak of the epidemic has passed. Lastly, we compute the price of anarchy of this system, to understand the conditions under which large discrepancies between the MFE and socially optimal strategies arise, which is when public policy would be most effective.