Optimal policy for control of epidemics with constrained time intervals and region-based interactions

TL;DR

This paper develops a policy model integrated with the SIR epidemic framework to determine optimal, constrained-time policies for controlling epidemics across multiple regions, considering economic and compliance costs.

Contribution

It introduces a hierarchical, multi-layer game-theoretic approach with piece-wise constant policies and analyzes their effects on epidemic control, including case studies on COVID-19.

Findings

Optimal policies depend on the minimum time interval, with larger intervals leading to step-down strategies.

The model's solutions approximate existing optimal control results for small policy intervals.

Counterfactual analysis shows different epidemic trajectories if herd immunity was achieved earlier.

Abstract

We introduce a policy model coupled with the susceptible-infected-recovered (SIR) epidemic model to study interactions between policy-making and the dynamics of epidemics. We consider both single-region policies, as well as game-theoretic models involving interactions among several regions, and hierarchical interactions among policy-makers modeled as multi-layer games. We assume that the policy functions are piece-wise constant with a minimum time interval for each policy stage, considering policies cannot change frequently in time or they cannot be easily followed. The optimal policy is obtained by minimizing a cost function which consists of an implementation cost, an impact cost, and, in the case of multi-layer games, a non-compliance cost. We show in a case study of COVID-19 in France that when the cost function is reduced to the impact cost and is parameterized as the final epidemic size, the solution approximates that of the optimal control in Bliman et al, J. Optim. Theory Appl., 189, 2021, for sufficiently small minimum policy time interval. For a larger time interval however the optimal policy is a step down function, quite different from the step up structure typically deployed during the COVID-19 pandemic. In addition, we present a counterfactual study of how the pandemic would have evolved if herd immunity was reached during the second wave in the county of Los Angeles, California. Lastly, we study a case of three interacting counties with and without a governing state.