SIR Epidemics With State-Dependent Costs and ICU Constraints: A Hamilton-Jacobi Verification Argument and Dual LP Algorithms

TL;DR

This paper introduces a simplified proof of optimal control in SIR epidemic models with state-dependent costs and ICU constraints, using Hamilton-Jacobi methods and dual linear programming algorithms.

Contribution

It provides a new verification approach for optimal controls and extends linear programming methods to handle complex, possibly discontinuous costs in epidemic modeling.

Findings

Simplified proof of control optimality using Hamilton-Jacobi theory.

Extension of linear programming algorithms to discontinuous costs.

Framework for analyzing reachable sets in epidemic control scenarios.

Abstract

The aim of this paper is twofold. On one hand, we strive to give a simpler proof of the optimality of greedy controls when the cost of interventions is control-affine and the dynamics follow a state-constrained controlled SIR model. This is achieved using the Hamilton-Jacobi characterization of the value function, via the verification argument and explicit trajectorybased computations. Aside from providing an alternative to the Pontryagin complex arguments in [5], this method allows one to consider more general classes of costs; in particular statedependent ones. On the other hand, the paper is completed by linear programming methods allowing to deal with possibly discontinuous costs. In particular, we propose a brief exposition of classes of linearized dynamic programming principles based on our previous work and ensuing dual linear programming algorithms. We emphasize the particularities of our state space and possible generations of forward scenarios using the description of reachable sets.