Optimal control of a stochastic system related to the Kermack-McKendrick model

A stochastic optimal control problem for a two-dimensional system of differential equations related to the Kermack-McKendrick model for the spread of epidemics is considered. The aim is to maximize the expected value of the time during which the epidemic is under control, taking the quadratic control costs into account. An exact and explicit solution is found in a particular case.