On the problem of minimizing the epidemic final size for SIR model via social distancing

📘 Introduction

Context and Motivation

We revisit the problem of minimizing the epidemic final size in the SIR model through social distancing interventions. Traditionally, this problem assumes a fixed interval structure for the timing of interventions. In contrast, we investigate a more flexible approach: L¹-constrained controls that limit total intervention effort, rather than when and how it's applied.

Key insight:

Even with this generalization, the optimal control still occurs over a single time interval when the transmission rate is constant, and over at most two (separate) time intervals if the transmission rate changes once.

The SIR Model with Control

System Dynamics

We consider the SIR model with a control variable u(t) (representing social distancing):

\[\begin{cases} \dot{S}(t) = -(1-u(t))\beta(t) S(t)I(t) \\ \dot{I}(t) = (1-u(t))\beta(t) S(t)I(t) - \gamma I(t) \end{cases}\]

where S, I are the susceptible and infected population fractions, u ∈ [0, ū] is the intervention intensity (lockdown), β is the transmission rate, and γ is the recovery rate.

Transmission Scenarios

We analyze two transmission cases:

1. Constant β:

\[\beta(t) = \beta_0 \quad \forall t \geq 0\]

2. Piecewise Constant β:

\[\beta(t) = \begin{cases} \beta_1 & \text{if } 0 \leq t < T_c \\ \beta_2 & \text{if } t \geq T_c \end{cases}\]

where Tc is the time when the transmission rate β changes.

Optimal Control Formulation

Objective

Our goal: Minimize the final epidemic size = maximize S(∞).

\[J(u) = S(\infty) = \lim_{t \to \infty} S(t)\]

Note: We deal with a non-standard cost function.

Constraints

We impose an L¹ budget constraint on the intervention:

\[\|u(\cdot)\|_{L^1} = \int_0^{\infty} u(t) \, dt \leq K\]

where K > 0 is the total budget and ū is the control upper bound.

This approach extends optimal control problems by requiring that each intervention occur within an interval of the form [t, t + δ], where t is a decision variable.

Precisely, we consider the following set of admissible controls:

\[\mathcal{U} = \left\{u : [0,\infty) \to [0,\bar{u}] \,\Big|\, u \text{ is measurable and } \|u\|_{L^1} \leq K \right\}\]