On the Optimal Vaccination Control of SIR Model with Erlang-Distributed Infectious Period

In this work, we investigate optimal vaccination strategies in SIR models with Erlang-distributed infectious period (the Erlang distribution is a gamma distribution with integer shape parameter). We consider the problem of minimizing the epidemic burden under either limited vaccination resources or linear costs of vaccination. By applying the Pontryagin’s minimum principle, we prove that both the optimal control problems admit only bang–bang solutions with at most one switch, the one-switch control being the case of reactive vaccination. We show that different assumptions on the infectious period distribution may lead to qualitatively and quantitatively different shapes of the optimal control. The main results are the following. We prove that in the absence of control the epidemic burden in SIR models is the same irrespectively of the infectious period distribution. However, when the vaccination resources are limiting, the optimal control problem leads to solutions where the epidemic burden is larger (and the time needed to complete vaccination campaigns is longer) in models with Erlang-distributed infectious periods with respect to exponential ones.