We investigate the time-optimal control problem in SIR(Susceptible-Infected-Recovered) epidemic models, focusing on different controlpolicies: vaccination, isolation, culling, and reduction of transmission.Applying the Pontryagin's Minimum Principle (PMP) to the unconstrained controlproblems (i.e. without costs of control or resource limitations), we provethat, for all the policies investigated, only bang-bang controls with at mostone switch are admitted. When a switch occurs, the optimal strategy is to delaythe control action some amount of time and then apply the control at themaximum rate for the remainder of the outbreak. This result is in contrast withprevious findings on the unconstrained problems of minimizing the totalinfectious burden over an outbreak, where the optimal strategy is to use themaximal control for the entire epidemic. Then, the critical consequence of ourresults is that, in a wide range of epidemiological circumstances, it may beimpossible to minimize the total infectious burden while minimizing theepidemic duration, and vice versa. Moreover, numerical simulations highlightedadditional unexpected results, showing that the optimal control can be delayedalso when the control reproduction number is lower than one and that theswitching time from no control to maximum control can even occur after the peakof infection has been reached. Our results are especially important forlivestock diseases where the minimization of outbreaks duration is a prioritydue to sanitary restrictions imposed to farms during ongoing epidemics, such asanimal movements and export bans.
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