  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.
