The goal of this paper is to prove the first and second orderoptimality conditions for some control problems governed bysemilinear elliptic equations with pointwise control constraintsand finitely many equality and inequality pointwise stateconstraints. To carry out the analysis we formulate a regularityassumption which is equivalent to the first order optimalityconditions. Though the presence of pointwise state constraintsleads to a discontinuous adjoint state, we prove that the optimalcontrol is Lipschitz in the whole domain. Necessary and sufficientsecond order conditions are proved with a minimal gap betweenthem.

A model is proposed for the control of a deterministic carrier-borne epidemic by immunising susceptibles. An optimal control is of bang-bang type and it is proved that it can have at most one switch.

A control scheme for the immunisation of susceptibles in the Kermack-McKendrick epidemic model for a closed population is proposed. The bounded control appears linearly in both dynamics and integral cost functionals and any optimal policies are of the “bang-bang” type. The approach uses Dynamic Programming and Pontryagin's Maximum Principle and allows one, for certain values of the cost and removal rates, to apply necessary and sufficient conditions for optimality and show that a one-switch candidate is the optimal control. In the remaining cases we are still able to show that an optimal control, if it exists, has at most one switch.