An optimal control theory approach to non-pharmaceutical interventions

Lin, Feng; Muthuraman, Kumar; Lawley, Mark

Sethi derived optimal closed-form results for isolation and immunization policies [37, 38] using an SI model. With this model, the population is partitioned into two parts, susceptible and infectious. The control is to either isolate and vaccinate at a maximum rate or do nothing. Infectious individuals who recover become susceptible once again, and thus immunity due to infection and subsequent recovery are not considered. Clancy [36] studied the properties of optimal policies for isolation and immunization assuming that all infectious individuals can be immediately isolated and all susceptible individuals can be immediately immunized. The policy takes no action when the number of infectious is below an optimal threshold and immediately isolates and/or immunizes when the number exceeds the threshold. However, they can only obtain optimal policies when the state space is small. Morton and Wickwire [40] developed optimal control policies for immunization assuming an infinite pandemic terminal time. However their switching curve derivation has an error in the derivatives (Eqs. 7a and 7b of [40] ) and thus their results are unclear. Behncke [41] derived mathematical properties of optimal vaccination programs under the following assumptions: 1) the time when vaccine becomes available is known; 2) infectious individuals can be immediately and completely isolated; and 3) the time horizon of the pandemic is infinite.