A comparative optimal control study of protecting susceptibles and isolating infecteds in SIR models

There are several approaches to controlling the spread of infectious diseases. Among the most widely used measures are the protection of susceptible individuals and the isolation of infected ones. This raises a critical question: which strategy, protecting susceptible individuals, isolating infected ones, or combining both, most effectively mitigates disease transmission? The work proposed here, address this question by analyzing protection and isolation strategies within the framework of the classical SIR (Susceptible-Infected-Recovered) epidemic model. Three scenarios are considered: (i) combined protection and isolation, (ii) protection alone, and (iii) isolation alone. Using Optimal Control Theory, we show that, in the short-term context of epidemic control, combining protection and isolation generally provides the most effective strategy under the considered conditions. However, when the combination is unavailable or unnecessary, isolation tends to outperform protection, particularly under challenging conditions such as high transmission rates or low recovery rates. Protection alone can still be effective, but primarily in settings where recovery rates are sufficiently high. These results contrast with those obtained from autonomous models, in which, in the long-term dynamics, protecting susceptible individuals proved to be more effective than isolating the infected.