Optimal control of epidemic size and duration with limited resources

The total number of infections (epidemic size) and the time needed for the infection to go extinct (epidemic duration) represent two of the main indicators for the severity of infectious disease epidemics in human and livestock. However, few attempts have been made to address the problem of minimizing at the same time the epidemic size and duration from a theoretical point of view by using optimal control theory. Here, we investigate the multi–objective optimal control problem aiming to minimize, through either vaccination or isolation, a suitable combination of epidemic size and duration when both maximum control effort and total amount of resources available during the entire epidemic period are limited. Application of Pontryagin's Maximum Principle to a Susceptible–Infected–Removed epidemic model, shows that, when the resources are not sufficient to maintain the maximum control effort for the entire duration of the epidemic, the optimal vaccination control admits only bang–bang solutions with one or two switches, while the optimal isolation control admits only bang–bang solutions with one switch. We also find that, especially when the maximum control effort is low, there may exist a trade–off between the minimization of the two objectives. Consideration of this conflict among objectives can be crucial in successfully tackling real–world problems, where different stakeholders with potentially different objectives are involved. Finally, the particular case of the minimum time optimal control problem with limited resources is discussed.