Game theory of social distancing in response to an epidemic

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.

Figure 1. Contour plots of relative risk surface for equilibrium strategies.

The relative risk is presented in feedback form with implicit coordinates (left) and transformed to explicit coordinates (right) for the infinite-horizon problem with maximum efficiency . The greater the value of the susceptible state (), the greater the instantaneous social distancing. We find that increasing the number of susceptible individuals always decreases the investment in social distancing, and the greatest investments in social distancing occur when the smallest part of the population is susceptible. Note that in the dimensionless model, the value of the infection state .

Figure 2. Epidemic solutions with equilibrium social distancing and without social distancing.

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 , ). In the phase plane (right), we see that both epidemics track each other perfectly until , when individuals begin to use social distancing to reduce transmission. Eventually, social distancing leads to a smaller epidemic. The convexity change appearing at the bottom the phaseplane orbit with social distancing corresponds to the cessation of social distancing.

Figure 3. Social distancing threshold.

This is the threshold that dictates whether or not equilibrium behavior involves some social distancing. It depends on both the basic reproduction number and the maximum efficiency , and is independent of the exact form of . As rough rules of thumb, if or , then equilibrium behavior involves no social distancing.

Figure 4. Total costs and savings.

Plots of the total per-capita cost of an epidemic (left) under equilibrium social distancing for the infinite-horizon problem with several efficiencies under Eq. (6), and the corresponding per-capita savings (right). Savings in expected cost compared to universal abstention from social distancing are largest for moderate basic reproduction numbers, but are relatively small, even in the limit of infinitely efficient social distancing. The case corresponds to infection of the minimum number of people necessary to reduce the reproduction ratio below .

Figure 5. Solutions when vaccine becomes available after a fixed time.

These are time series of an equilibrium solution for social distancing when mass vaccination occurs generations (left) and generations (right) after the start of the epidemic. Investments in social distancing begin well after the start of the epidemic but continue right up to the time of vaccination. Social distancing begins sooner when vaccine development is faster. For these parameter values (), individuals save % of the cost of infection per capita (left) and % of the cost of infection (right).

Figure 6. Windows of Opportunity for Vaccination.

Plots of how the net expected losses per individual () depend on the delay between the start of social-distancing practices and the date when mass-vaccination becomes universally available if individuals use a Nash equilibrium strategy. The more efficient social distancing, the less individuals invest prior to vaccine introduction. The blue lines () do not use social distancing, as the efficiency is below the threshold. The dotted lines represent the minimal asymptotic epidemic costs necessary to stop an epidemic.