Understanding Nash epidemics

Abstract

Faced with a dangerous epidemic humans will spontaneously social distance to reduce their risk of infection at a socioeconomic cost. Compartmentalized epidemic models have been extended to include this endogenous decision making: Individuals choose their behavior to optimize a utility function, self-consistently giving rise to population behavior. Here, we study the properties of the resulting Nash equilibria, in which no member of the population can gain an advantage by unilaterally adopting different behavior. We leverage an analytic solution that yields fully time-dependent rational population behavior to obtain, 1) a simple relationship between rational social distancing behavior and the current number of infections; 2) scaling results for how the infection peak and number of total cases depend on the cost of contracting the disease; 3) characteristic infection costs that divide regimes of strong and weak behavioral response; 4) a closed form expression for the value of the utility. We discuss how these analytic results provide a deep and intuitive understanding of the disease dynamics, useful for both individuals and policymakers. In particular, the relationship between social distancing and infections represents a heuristic that could be communicated to the population to encourage, or “bootstrap,” rational behavior.