Optimal Isolation Strategies in an SIR Model with Erlang–Distributed Infectious Period

Abstract

In classical epidemic models, a major simplification consists in assuming that the infectious period is exponentially distributed. Here, we first attempt to investigate the consequences of relaxing this assumption on the performances of time–variant disease control strategies by using optimal control theory. In the framework of a basic susceptible–infected– removed (SIR) model, an Erlang distribution of the infectious period is considered and optimal isolation strategies are searched for. The objective functional takes into account the cost of the isolation efforts and the sanitary costs due to the incidence of the epidemic outbreak. Applying the Pontryagin's minimum principle, we prove that the problem admits only bang–bang solutions with at most two switches. Finally, by means of numerical simulations, we show how the shape of the optimal solutions is affected by the different distributions of the infectious period, and by the relative weight of the two cost components.