Game Theory of Social Distancing in Response to an Epidemic

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

Erratic Flu Vaccination Emerges from Short-Sighted Behavior in Contact Networks

The effectiveness of seasonal influenza vaccination programs depends on individual-level compliance. Perceptions about risks associated with infection and vaccination can strongly influence vaccination decisions and thus the ultimate course of an epidemic. Here we investigate the interplay between contact patterns, influenza-related behavior, and disease dynamics by incorporating game theory into network models. When individuals make decisions based on past epidemics, we find that individuals with many contacts vaccinate, whereas individuals with few contacts do not. However, the threshold number of contacts above which to vaccinate is highly dependent on the overall network structure of the population and has the potential to oscillate more wildly than has been observed empirically. When we increase the number of prior seasons that individuals recall when making vaccination decisions, behavior and thus disease dynamics become less variable. For some networks, we also find that higher flu transmission rates may, counterintuitively, lead to lower (vaccine-mediated) disease prevalence. Our work demonstrates that rich and complex dynamics can result from the interaction between infectious diseases, human contact patterns, and behavior

Contour plots of relative risk surface for equilibrium strategies.

<p>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 .</p

Solutions when vaccine becomes available after a fixed time.

<p>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).</p

Equilibria of an Epidemic Game with Piecewise Linear Social Distancing Cost

Around the world, infectious disease epidemics continue to threaten people’s health. When epidemics strike, we often respond by changing our behaviors to reduce our risk of infection. This response is sometimes called “social distancing”. Since behavior changes can be costly, we would like to know the optimal social distancing behavior. But the benefits of changes in behavior depend on the course of the epidemic, which itself depends on our behaviors. Differential population game theory provides a method for resolving this circular dependence. Here, I present the analysis of a special case of the differential SIR epidemic population game with social distancing when the relative infection rate is linear but bounded below by zero. Equilibrium solutions are constructed in closed-form for an open-ended epidemic. Constructions are also provided for epidemics that are stopped by the deployment of a vaccination that becomes available a fixed-time after the start of the epidemic. This can be used to anticipate a window of opportunity during which mass vaccination can significantly reduce the cost of an epidemic. c ○ 2013 Tim Reluga2 Reluga 1

Social distancing threshold.

<p>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.</p

Windows of Opportunity for Vaccination.

<p>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.</p

Epidemic solutions with equilibrium social distancing and without social distancing.

<p>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.</p