Research Article: Game Theory of Social Distancing in Response to an Epidemic

Date Published: May 27, 2010

Publisher: Public Library of Science

Author(s): Timothy C. Reluga, Carl T. Bergstrom

Abstract: Social distancing practices are changes in behavior that prevent disease transmission by reducing contact rates between susceptible individuals and infected individuals who may transmit the disease. Social distancing practices can reduce the severity of an epidemic, but the benefits of social distancing depend on the extent to which it is used by individuals. Individuals are sometimes reluctant to pay the costs inherent in social distancing, and this can limit its effectiveness as a control measure. This paper formulates a differential-game to identify how individuals would best use social distancing and related self-protective behaviors during an epidemic. The epidemic is described by a simple, well-mixed ordinary differential equation model. We use the differential game to study potential value of social distancing as a mitigation measure by calculating the equilibrium behaviors under a variety of cost-functions. Numerical methods are used to calculate the total costs of an epidemic under equilibrium behaviors as a function of the time to mass vaccination, following epidemic identification. The key parameters in the analysis are the basic reproduction number and the baseline efficiency of social distancing. The results show that social distancing is most beneficial to individuals for basic reproduction numbers around 2. In the absence of vaccination or other intervention measures, optimal social distancing never recovers more than 30% of the cost of infection. We also show how the window of opportunity for vaccine development lengthens as the efficiency of social distancing and detection improve.

Partial Text: Epidemics of infectious diseases are a continuing threat to the health of human communities, and one brought to prominence in the public mind by the 2009 pandemic of H1N1 influenza [1]. One of the key questions of public health epidemiology is how individual and community actions can help mitigate and manage the costs of an epidemic. The basic problem I wish to address here is how rational social-distancing practices used by individuals during an epidemic will vary depending on the efficiency of the responses, and how these responses change the epidemic as a whole.

In this article, social distancing refers to the adoption of behaviors by individuals in a community that reduce those individuals’ risk of becoming infected by limiting their contact with other individuals or reducing the transmission risk during each contact. Typically, social distancing incurs some costs in terms of liberty, social capital, time, convenience, and money, so that people are only likely to adopt these measures when there is a specific incentive to do so. In addition to the personal consequences, the aggregate effects of social distancing form an economic externality, reducing the overall transmission of disease. This externality needs to be accounted for in the determination individuals optimal strategies, but, by definition, depends on the choice of strategy.

A problem with solving Eq. (12) under Eq. (14) is that it requires to be known from past time and to be known from future time. This is a common feature of boundary-value problems, and is resolved by considering all terminal conditions . Using standard numerical techniques, identifying an equilibrium in the described boundary-value problem reduces to scalar root finding for to match the given . The special form of the population game allows the solution manifold to be calculated directly by integrating backwards in time, rather than requiring iterative approaches like those used for optimal-control problems [23]. Code for these calculations is available from the author on request.

Here, I have described the calculations necessary to identify the equilibrium solution of the differential game for social distancing behaviors during an epidemic. The benefits associated with the equilibrium solution can be interpreted as the best outcome of a simple social-distancing policy. We find that the benefits of social distancing are constrained by fundamental properties of epidemic dynamics and the efficiency with which distancing can be accomplished. The efficiency results are most easily summarized in terms of the maximum efficiency , which is the percent reduction in contact rate per percent of infection cost invested per disease generation. As a rule-of-thumb, is an upper bound on the number of transmission generations individuals can isolate before the costs of social distancing outweigh the costs of infection. Social distancing is not practical if this efficiency is small compared to the number of generations in the fastest epidemics (). While social distancing can yield large reductions in transmission rate over short periods of time, optimal social-distancing strategies yield only moderate reductions in the cost of the epidemic.