  We consider a SIR model with temporary immunity and time dependent
transmission rate. We assume time dependent vaccination which confers the same
immunity as natural infection. We study two types of vaccination strategies: i)
optimal vaccination, in the sense that it minimizes the effort of vaccination
in the set of vaccination strategies for which, for any sufficiently small
perturbation of the disease free state, the number of infectious individuals is
monotonically decreasing; ii) Nash-equilibria strategies where all individuals
simultaneously minimize the joint risk of vaccination versus the risk of the
disease. The former case corresponds to an optimal solution for mandatory
vaccinations, while the second correspond to the equilibrium to be expected if
vaccination is fully voluntary. We are able to show the existence of both an
optimal and Nash strategies in a general setting. In general, these strategies
will not be functions but Radon measures. For specific forms of the
transmission rate, we provide explicit formulas for the optimal and the Nash
vaccination strategies.
