Optimal control for a SIR model with limited hospitalised patients

TL;DR

This paper develops an optimal control strategy for managing infectious disease spread using an SIR model, balancing health outcomes and social costs under capacity constraints, with a three-phase intervention approach.

Contribution

It introduces a rigorous derivation of a boundary-bang optimal quarantine strategy with three distinct phases, considering capacity and economic constraints.

Findings

Optimal strategy involves no intervention, then maximum infection maintenance, followed by partial lockdown.

The boundary-bang control policy is optimal and matches numerical solutions.

Refined transition timings improve intervention effectiveness.

Abstract

This paper analyses the optimal control of infectious disease propagation using a classic susceptible-infected-recovered (SIR) model characterised by permanent immunity and the absence of available vaccines. The control is performed over a time-dependent mean reproduction number, in order to minimise the cumulative number of ever-infected individuals (recovered), under different constraints. We consider constraints on isolation measures ranging from partial lockdown to non-intervention, as well as the social and economic costs associated with such isolation, and the capacity limitations of intensive care units that limits the number of infected individuals to a maximum allowed value. We rigorously derive an optimal quarantine strategy based on necessary optimality conditions. The obtained optimal strategy is of a boundary-bang type, comprising three phases: an initial phase with no intervention, a second phase maintaining the infected population at its maximum possible value, and a final phase of partial lockdown applied over a single interval. The optimal policy is further refined by optimising the transition times between these phases. We show that these results are in excellent agreement with the numerical solution of the problem.