Optimal protection and vaccination against epidemics with reinfection risk

We consider the problem of the optimal allocation of vaccination and protection measures for the Susceptible-Infected-Recovered-Infected (SIRI) epidemiological model, which generalizes the classical Susceptible-Infected-Recovered (SIR) and Susceptible-Infected-Susceptible (SIS) epidemiological models by allowing for reinfection. First, we introduce the controlled SIRI dynamical model, and discuss the existence and stability of the equilibrium points. Then, we formulate a finite-horizon optimal control problem where the cost of vaccination and protection is proportional to the mass of the population that adopts it. Our main contribution in this work arises from a detailed investigation into the existence/non-existence of singular control inputs, and establishing optimality of bang-bang controls. The optimality of bang-bang control is established by solving an optimal control problem with a running cost that is linear with respect to the input variables. The input variables are associated with actions including the vaccination and imposition of protective measures (e.g., masking or isolation). In contrast to most prior works, we rigorously establish the non-existence of singular controls (i.e., the optimality of bang-bang control for our SIRI model). Under the assumption that the reinfection rate exceeds the first-time infection rate, we characterize the structure of both the optimal control inputs, and establish that the vaccination control input admits a bang-bang structure. The numerical results provide valuable insights into the evolution of the disease spread under optimal control.