This presentation considers optimal vaccination policies for a SIR (Susceptible - Infected - Recovered) model. The cost is optimized with respect to the vaccination strategies. We show that the value function is the unique viscosity solution of a HJB equation. This allows to find the global optimal vaccination policy. At odds with existing literature, it is seen that the value function is not always smooth (sometimes only Lipschitz) and the optimal vaccination policies are not unique. Moreover we rigorously analyze the situation when vaccination can be modeled as instantaneous (with respect to the time evolution of the epidemic) and identify for the first time the global optimum solutions. We show that the optimal solution divides the space into two regions : into one it is best to vaccinate with maximum effort and outside it is optimal to do nothing.

When the vaccination is not compulsory and moreover has some secondary effects the individual decision (to vaccinate or not) may be different from the optimal strategy of the society as a whole. Indeed, in some cases, vaccination would be optimal for the society but not for an individual. We introduce mathematical models that study this difference and propose a structural explanation using a "Mean Field Games" model (à la Lions and Lasry).

Optimal vaccination, SIR model, HJB equation, Mean Field Games