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

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

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

Total costs and savings.

<p>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 .</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