Optimal vaccination strategies for a heterogenous population using multiple objectives: The case of L1 and L2-formulations

Abstract

The choice of the objective functional in optimization problems coming from biomedical and epidemiological applications plays a key role in optimal control outcomes. In this study, we investigate the role of the objective functional on the structure of the optimal control solution for an epidemic model for sexually transmitted infections that includes a core group with higher sexual activity levels than the rest of the population. An optimal control problem is formulated to find a targeted vaccination program able to control the spread of the infection with minimum vaccine deployment. Both $L_{1}-$ and $L_{2}-$objectives are considered as an attempt to explore the trade-offs between control dynamics and the functional form characterizing optimality. The results show that the optimal vaccination policies for both the $L_{1}-$ and the $L_{2}-$formulation share one important qualitative property, that is, immunization of the core group should be prioritized by policymakers to achieve a fast reduction of the epidemic. However, quantitative aspects of this result can be significantly affected depending on the choice of the control weights between formulations. Overall, the results suggest that with appropriate weight constants, the optimal control outcomes are reasonably robust with respect to the $L_{1}-$ or $L_{2}-$formulation. This is particularly true when the monetary cost of the control policy is substantially lower than the cost associated with the disease burden. Under these conditions, even if the $L_{1}-$formulation is more realistic from a modeling perspective, the $L_{2}-$formulation can be used as an approximation and yield qualitatively comparable outcomes.