Optimal policies involving the isolation of infectives are derived for a deterministic epidemic model with non-standard infection rate function. Denoting by y the number of infective individuals and x the number of susceptible individuals in the population, we replace the classical Kermack–McKendrick infection rate function βxy (for some constant β) with βxy/(x + y). This modified model has been studied by various previous authors, but not in the context of control policies. We show that the optimal isolation policy is to intervene with maximal effort when y ⩽ Ax, and not to intervene otherwise, for some constant A whose value can be given explicitly in terms of the parameters of the model.
Clancy, D., & Piunovskiy, A. (2005). An explicit optimal isolation policy for a deterministic epidemic model. Applied Mathematics and Computation, 163(3), 1109-1121. https://doi.org/10.1016/j.amc.2004.06.028