Optimal lockdown and vaccination policies to contain the spread of a mutating infectious disease

We develop a piecewise deterministic control model to study optimal lockdown and vaccination policies to manage a pandemic. Lockdown is modeled as an impulse control that allows the decision maker to switch from one level of restrictions to another. Vaccination policy is a continuous control. Decisions are taken under the risk of mutations of the disease, with repercussions on the transmission rate. The decision maker follows a cost minimization objective. We first characterize the optimality conditions for impulse control and show how the prospect of a mutation affects the decision maker's choice by inducing her to anticipate the net benefit of operating under a different lockdown state once a mutation occurs. The problem admits infinitely many value functions. Under some parametric conditions, we show the existence of a minimum value function that is a natural candidate solution. Focusing on this specific value function, we finally study the features of the optimal policy, especially the timing of impulse control. We prove that uncertainty surrounding future "bad" versus "good" mutation of the disease expedites versus delays the adoption of lockdown measures.