Dynamic Games of Social Distancing During an Epidemic: Analysis of Asymmetric Solutions

Abstract

Individual behaviors play an essential role in the dynamics of transmission of infectious diseases, including COVID-19. This paper studies a dynamic game model that describes the social distancing behaviors during an epidemic, assuming a continuum of players and individual infection dynamics. The evolution of the players' infection states follows a variant of the well-known SIR dynamics. We assume that the players are not sure about their infection state, and thus, they choose their actions based on their individually perceived probabilities of being susceptible, infected, or removed. The cost of each player depends both on her infection state and on the contact with others. We prove the existence of a Nash equilibrium and characterize Nash equilibria using nonlinear complementarity problems. We then exploit some monotonicity properties of the optimal policies to obtain a reduced-order characterization for Nash equilibrium and reduce its computation to the solution of a low-dimensional optimization problem. It turns out that, even in the symmetric case, where all the players have the same parameters, players may have very different behaviors. We finally present some numerical studies that illustrate this interesting phenomenon and investigate the effects of several parameters, including the players' vulnerability, the time horizon, and the maximum allowed actions, on the optimal policies and the players' costs.

Keywords: COVID-19 pandemic; Epidemics modeling and control; Games of social distancing; Nash games; Nonlinear complementarity problems.

Fig. 1

The evolution of the infection state of each individual

Fig. 2

In this example, $N=5$ and there are $M=3$ types of players, depicted with different colors. The mass of players below each solid line play $u_M$, and the mass of players above the line play $u_m$. Example 1 computes $\pi$ from $\rho$

Fig. 3

a The fractions ${\rho }_k$, for $k=1,\cdots ,13$, for different values of b. b The evolution of the number of infected people under the computed Nash equilibrium

Fig. 4

The evolution of the probabilities $S^i$ and $I^i$, for players following different strategies, for $b=200$. Different colors are used to illustrate the evolution of the probabilities for players using different strategies

Fig. 5

a The fractions ${\rho }_k$ of players having an action $u_M$. Note that the fractions correspond to players of all types. Since $s^i=1$, for all the types, it holds $b^1<\cdots <b^6$, and thus, the more vulnerable players cannot have an action higher than the less vulnerable ones. b The probability of a class of people to be susceptible and infected. The colored lines correspond to the probabilities of being susceptible $S^i(t)$ and infected $I^i(t)$, for the several strategies of the players. The bold black line represents the mass of susceptible and infected persons (Color figure online)

Fig. 6

a The fractions ${\rho }_k$, for the several values of the maximum action $u_M$. b The mass of infected people as a function of time, for the different values of the maximum action $u_M$. c The cost for the several classes of players, for the different values of the maximum action $u_M$. The bold black line represents the mean cost of all the players