Global optimal vaccination in the SIR model: Properties of the value function and application to cost-effectiveness analysis

This work focuses on optimal vaccination policies for an Susceptible-Infected-Recovered (SIR) model; the impact of the disease is minimized with respect to the vaccination strategy. The problem is formulated as an optimal control problem and we show that the value function is the unique viscosity solution of an Hamilton-Jacobi-Bellman (HJB) equation. This allows to find the best 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 the global optimum solutions. Numerical applications illustrate the theoretical results. In addition the pertussis vaccination in adults is considered from two perspectives: first the maximization of DALY averted in presence of vaccine side-effects; then the impact of the herd immunity on the cost-effectiveness analysis is discussed on a concrete example.