Abstract
The classical Kermack-McKendrick model for the spread of an epidemic through a closed population has recently been extended by Billard to allow for the recovery and possible reinfection of infective cases. In this paper, we study the optimal control of such an epidemic through immunization of susceptibles when costs are proportional to the area under the infectives trajectory plus the total number of immunizations. When the immunization rate is bounded, optimal controls are of bang-bang type and are characterized by switching curves in the epidemic state space. Explicit expressions for these curves are obtained in the case of deterministic dynamics. When the epidemic is described by a Markov chain, numerical solutions for the switching curve are easy to obtain by dynamic programming, and useful analytic approximations to them are described. The results include those for the so-called general epidemic in which no recovery is allowed.
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