Rational social distancing in epidemics with uncertain vaccination timing

Abstract

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.

Rational social distancing policy during epidemics with limited healthcare capacity

“…However, the small contribution of U f is always included in the figures and solutions we show here, for completeness. We note again that the arrival of a vaccine or treatment earlier during the course of the epidemic tends to enhance social distancing efforts [ 1 , 7 , 55 ]. If α ( i ) is not constant in that situation, the above approximation will not be accurate.…”

“…The individual at any given time is either susceptible, infectious, or recovered, and their fate can be modelled as a series of discrete transition events between these states. In order to make the situation tractable, we calculate the expected probability ψ j (t) that the individual is in compartment j at time t as a continuous where there is no behavioural modification (corresponding to equilibrium behaviour at an infection cost α = 0, grey lines); the Nash equilibrium for α = 400 (black lines), calculated numerically via forward-backward sweep, see section C in S1 Text (In order to demonstrate that the numerical solution is accurate, we also show the analytical solution of the same equations [62] as black dots); the utilitarian maximum for α = 400 (gold dashes); and finally the population behaviour for two optimal government policies, one being without cost to the government, γ g = 0 (gold lines), and one being costly, γ g = 0.5, with α = 400 (cyan lines). When government interventions are cost-free, they enable the population to reach the utilitarian maximum.…”

“…We ignore the situation where a vaccine becomes available during the epidemic. While the early arrival of a vaccine would have consequences for both equilibrium and globally optimal behaviour [ 1 , 7 , 29 , 55 ], this lies outside of the scope of this work. Judging from previous work, one would roughly expect that the earlier the vaccine is expected to arrive, the more incentivised both individuals and governments would be to increase their social distancing efforts.…”