  During epidemics people may reduce their social and economic activity to
lower their risk of infection. Such social distancing strategies will depend on
information about the course of the epidemic but also on when they expect the
epidemic to end, for instance due to vaccination. Typically it is difficult to
make optimal decisions, because the available information is incomplete and
uncertain. Here, we show how optimal decision-making depends on information
about vaccination timing in a differential game in which individual
decision-making gives rise to Nash equilibria, and the arrival of the vaccine
is described by a probability distribution. We predict stronger social
distancing the earlier the vaccination is expected and also the more sharply
peaked its probability distribution. In particular, equilibrium social
distancing only meaningfully deviates from the no-vaccination equilibrium
course if the vaccine is expected to arrive before the epidemic would have run
its course. We demonstrate how the probability distribution of the vaccination
time acts as a generalised form of discounting, with the special case of an
exponential vaccination time distribution directly corresponding to regular
exponential discounting.
