A parsimonious model of optimal social distancing and vaccination during an outbreak

Motivated by the complicated control issues of COVID-19, this article aims at investigating the optimal control of an epidemic of a Susceptible-Infective-Removed-Susceptible (SIRS) infection, where social distancing is the only control action in a first stage, whereas a combination of social distancing and vaccination is available in a second stage. The resulting two-control optimal problem is set within a parsimonious economic framework in which a social planner minimises an objective function weighting epidemiological and economic costs by choosing the strength of social distancing in the first stage and both social distancing and the extent of an income tax to finance vaccination in the second stage. The article shows (i) how to mix social distancing and vaccination depending on the planner's degree of rationality; (ii) the importance of the planner's expectation about the date of vaccine arrival, and how the actual effi cacy of the vaccine against the infection can affect the optimal social distancing policy in the pre-vaccination period, and (iii) the use of the social distancing instrument as the only optimal control under vaccine rationing.