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Conceived and designed the experiments: TR. Performed the experiments: TR. Analyzed the data: TR. Wrote the paper: TR.

The author has declared that no competing interests exist.

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.

One of the easiest ways for people to lower their risk of infection during an epidemic is for them to reduce their rate of contact with infectious individuals. However, the value of such actions depends on how the epidemic progresses. Few analyses of behavior change to date have accounted for how changes in behavior change the epidemic wave. In this paper, I calculate the tradeoff between daily social distancing behavior and reductions in infection risk now and in the future. The subsequent analysis shows that, for the parameters and functional forms studied, social distancing is most useful for moderately transmissible diseases. Social distancing is particularly useful when it is inexpensive and can delay the epidemic until a vaccine becomes widely available. However, the benefits of social distancing are small for highly transmissible diseases when no vaccine is available.

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

Social distancing is an aspect of human behavior particularly important to epidemiology because of its universality; everybody can reduce their contact rates with other people by changing their behaviors, and reduced human contact reduces the transmission of many diseases. Theoretical work on social distancing has been stimulated by studies of agent-based influenza simulations indicating that small changes in behavior can have large effects on transmission patterns during an epidemic

Rather than treating behaviors as states, some models treat behaviors as parameters determined by simple functions of the available information. Reluga et al.

Building on the ground-breaking work of Fine and Clarkson

To study the best usage of social distancing, we apply differential-game theory at a population-scale. Differential games are games where strategies have a continuous time-dependence; at each point in time, a player can choose a different action. For instance, a pursuit-game between a target and a pursuer is a two-player differential game where each player's strategies consist of choosing how to move at each successive time until the target is caught by the pursuer or escapes. Geometrically, one might think of differential games as games where strategies are represented by curves instead of points. Two-player differential-game theory was systematically developed by Isaacs

In the

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.

To resolve this interdependence, we formulate our analysis as a population game where the payoff to each individual is determined by the individual's behavioral strategy and the average behavioral strategy used by the population as a whole. The model is related to that previously studied by Chen

The effectiveness of social distancing is represented by a function

Consider a Susceptible-Infected-Recovered (SIR) epidemic model with susceptible (

The total cost of the epidemic to the community,

To simplify our studies, we will work with the dimensionless version of the equations by taking:

Epidemics usually start with one or a few index cases, so we focus on scenarios where

For our further analysis, we will assume

We now formulate a differential game for individuals choosing their best social distancing practices relative to the aggregate behavior of the population as a whole. The following game-theoretic analysis combines the ideas of Isaacs

Since the events in the individual's life are stochastic, we can not predict the exact time spent in any one state or the precise payoff received at the end of the game. Instead, we calculate expected present values of each state at each time, conditional on the investment in social distancing. The expected present value is average value one expects after accounting for the probabilities of all future events, and discounting future costs relative to immediate costs. The expected present values

The adjoint equations governing the values of each state are derived from Markov decision process theory. They are

The dynamics are independent of

Solving a game refers to the problem of finding the best strategy to play, given that all the other players are also trying to find a best strategy for themselves. In some games, there is a single strategy that minimizes a player's costs no matter what their opponents do, so that strategy can very reasonably be referred to as a solution. In many games, no such strategy exists. Rather, the best strategy depends on the actions of the other players. Any strategy played by one player is potentially vulnerable to a lack of knowledge of the strategies of the other players. In such games, it is most useful to look for strategies that are equilibria, in the sense that every player's strategy is better than the alternatives, given knowledge of their opponent's strategies. A Nash equilibrium solution to a population game like that described by System (12) is a strategy that is a best response, even when everybody else is using the same strategy.

The equilibria of System (12) can be calculated using the general methods of Isaacs

The relative risk is presented in feedback form with implicit coordinates

Two cases are immediately interesting. The first is the infinite-horizon problem – what is the equilibrium behavior when there is never a vaccine and the epidemic continues on until its natural end? The second is the finite-horizon problem – if a vaccine is introduced at time

The infinite-horizon and finite-horizon problems are distinguished by their boundary conditions. In the finite-horizon case, we assume all susceptible individuals are vaccinated at final time

Most of the equilibria we calculate are obtained numerically. Some exceptions are the special cases where

A problem with solving Eq. (12) under Eq. (14) is that it requires

Before presenting the results, it is helpful to develop some intuition for the importance of the maximum efficiency

For the infinite-horizon problem, an example equilibrium strategy and the corresponding dynamics in the absence of social distancing are shown in

Social distancing reduces the epidemic peak and prolongs the epidemic, as we can see by comparing a time series with subgame-perfect social distancing (top left) and a time series with the same initial condition but no social distancing (bottom left) (parameters

This is the threshold that dictates whether or not equilibrium behavior involves some social distancing. It depends on both the basic reproduction number

The exact window over which social distancing is used depends on the basic reproduction number, the initial and terminal conditions, and the efficiency of distancing measures. The feedback form of equilibrium strategies, transformed from

The consequences of social distancing are shown in

Plots of the total per-capita cost of an epidemic

We can also calculate solutions of the finite-time horizon problem where a vaccine becomes universally available at a fixed time after the detection of disease (

These are time series of an equilibrium solution for social distancing when mass vaccination occurs

Plots of how the net expected losses per individual (

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

Our calculations have determined the equilibrium strategies from the perspective of individuals. Alternatively, we could ask what the optimal social distancing practices are from the perspective of minimizing the total cost of the epidemic to the community. Determination of the optimal community strategy leads to a nonlinear optimal control problem that can be studied using standard procedures

The results presented require a number of caveats. I have, for instance, only considered one particular form for the relative risk function. Most of the analysis has been undertaken in the absence of discounting (

The simple epidemic model is particularly weak in its prediction of the growths of epidemics because it assumes the population is randomly mixed at all times. We know, however, that the contact patterns among individuals are highly structured, with regular temporal, spatial, and social correlations. One consequence of heterogeneous contact structure is that epidemics proceed more slowly than the simple epidemic model naively predicts. Thus, the simple epidemic model is often considered as a worst-case-scenario, when compared with more complex network models

One of the fundamental assumptions in our analysis is that there are no cost-neutral behavior changes that can reduce contact rates. In fact, life-experience provides good evidence that many conventional aspects of human behavior are conditional on cultural norms, and that different cultures may adopt alternative conventions. The introduction of a new infectious disease may alter the motivational pressures so that behavioral norms that were previously equivalent are no longer, and that one norm is now preferred to the others. In such cases, there are likely to be switching costs that retard the rapid adoption of the better behaviors that conflict with cultural norms. The rate of behavior change, then, would be limited by the rate of adoption of compensatory changes in cultural norms that reduce the cost of social distancing.

Another deep issue is that behavior changes have externalities beyond influencing disease incidence, but we have not accounted for these externalities. People's daily activities contribute not just to their own well-being but also to the maintenance of our economy and infrastructure. Social distancing behaviors may have serious negative consequences for economic productivity, which might feed back into slowing the distribution of vaccines and increasing daily cost-of-living expenses.

We can extend our analysis to include economic feedbacks by incorporating capital dynamics explicitly. Individuals may accumulate capital resources like food, water, fuel, and prophylactic medicine prior to an epidemic, but these resources will gradually be depleted and might be difficult to replace if social distancing interferes with the economy flow of goods and services. Further capital costs at the community and state scales may augment epidemic valuations. These factors appear to have been instrumental in the recent US debate of school-closure policies. One feature of a model with explicit capital dynamics is the possibility of large economic shocks. This and related topics will be explored in future work.

These calculations raise two important mathematical conjectures which I have not attempted to address. The first is that the social distancing game possesses a unique subgame-perfect Nash equilibrium. There is reasonable numerical evidence of this in cases where the relative risk function

As with all game-theoretic models, human behavior is unlikely to completely agree with our equilibria for many reasons, including incomplete information about the epidemic and vaccine and strong prior beliefs that impede rational responses. On the other hand, our approach is applicable to a large set of related models. We can analyze many more realistic representations of pathogen life-cycles. For instance, arbitrary infection-period distributions and infection rates can be approximated using a linear chain of states or delay-equations

The author thanks A. Bressan, A. Galvani, and E. Shim for helpful discussion, and two anonymous referees for their valuable criticisms.