  We investigate the time-optimal control problem in SIR
(Susceptible-Infected-Recovered) epidemic models, focusing on different control
policies: vaccination, isolation, culling, and reduction of transmission.
Applying the Pontryagin's Minimum Principle (PMP) to the unconstrained control
problems (i.e. without costs of control or resource limitations), we prove
that, for all the policies investigated, only bang-bang controls with at most
one switch are admitted. When a switch occurs, the optimal strategy is to delay
the control action some amount of time and then apply the control at the
maximum rate for the remainder of the outbreak. This result is in contrast with
previous findings on the unconstrained problems of minimizing the total
infectious burden over an outbreak, where the optimal strategy is to use the
maximal control for the entire epidemic. Then, the critical consequence of our
results is that, in a wide range of epidemiological circumstances, it may be
impossible to minimize the total infectious burden while minimizing the
epidemic duration, and vice versa. Moreover, numerical simulations highlighted
additional unexpected results, showing that the optimal control can be delayed
also when the control reproduction number is lower than one and that the
switching time from no control to maximum control can even occur after the peak
of infection has been reached. Our results are especially important for
livestock diseases where the minimization of outbreaks duration is a priority
due to sanitary restrictions imposed to farms during ongoing epidemics, such as
animal movements and export bans.
