Synchronizing lockdown and vaccination policies for COVID-19: An optimal control approach based on piecewise constant strategies

We are still facing the devastating consequences of COVID-19. At the beginning of this pandemic, lockdown and non-pharmaceutical measures were the unique, effective strategy to overcome the ongoing outbreak. After almost a year, an exceptional effort gave the first efficient protective vaccines. Despite these significant advances, new challenges as its mass production and fair distribution emerge. Our work aims to address this balance by formulating an optimal epidemic control problem controlled by lockdown and vaccination but in a synchronized manner. In such a way that the sought-after solution optimizes the burden and economic implications of COVID-19 infections and deaths. Thus, we formulate an optimal control problem with a differential equation to describe the spread of COVID-19. Our formulation measures the efficiency of these controls by a functional cost involving the burden of COVID-19 quantified in DALYs and the costs regarding vaccination and lockdown. Then we minimize this cost subject to the controlled system and find optimal policies that are constant in time intervals of a given size. To this end, we apply the well-established heuristic method known as differential evolution. One of the advantages of these policies relies on their practical implementation since the health authority has to make only a finite number of different decisions. Our methodology to find optimal policies allows changes in the dynamics, the cost functional, or how frequently the policymaker changes actions. We show how a well-synchronized tradeoff between vaccination and lockdown could under-peak of the outbreak, with a delicate balance to overcome possible economic consequences.