[Paper Review] Mortality containment vs. economics opening: optimal policies in a SEIARD model

Andrea Aspri, Elena Beretta+2 more|arXiv (Cornell University)|May 29, 2020

COVID-19 epidemiological studiesMathematics52 references45 citations

TL;DR

The paper formulates a SEAIRD model with containment control to study optimal policies balancing COVID-19 mortality and GDP, proving existence and conditions for uniqueness of optimal policies and illustrating phase-transition-type behavior.

ABSTRACT

We adapt a SEIRD differential model with asymptomatic population and Covid deaths, which we call SEAIRD, to simulate the evolution of COVID-19, and add a control function affecting both the diffusion of the virus and GDP, featuring all direct and indirect containment policies; to model feasibility, the control is assumed to be a piece-wise linear function satisfying additional constraints. We describe the joint dynamics of infection and the economy and discuss the trade-off between production and fatalities. In particular, we carefully study the conditions for the existence of the optimal policy response and its uniqueness. Uniqueness crucially depends on the marginal rate of substitution between the statistical value of a human life and GDP; we show an example with a phase transition: above a certain threshold, there is a unique optimal containment policy; below the threshold, it is optimal to abstain from any containment; and at the threshold itself there are two optimal policies. We then explore and evaluate various profiles of various control policies dependent on a small number of parameters.

Motivation & Objective

• Motivate a formal trade-off between COVID-19 fatalities and GDP losses using a SEAIRD epidemic model.
• Incorporate a feasible, piecewise-linear control for containment that affects both infection spread and economic activity.
• Establish existence of a global minimum for the social planner’s loss and analyze conditions for uniqueness of the optimal policy.
• Examine how the marginal rate of substitution between mortality and GDP influences policy multiplicity and transitions.

Proposed method

• Extend the SEAIRD model with compartments S, E, A, I, R, D, and natural deaths.
• Introduce a control c(t) in [c0,1] representing opening level that linearly scales infection rate and concavely impacts GDP.
• Define a social planner’s loss functional balancing GDP welfare and Covid deaths with discounting.
• Prove existence of an optimal, Lipschitz-continuous control within a restricted class of piecewise-constant/linear policies.
• Derive first-order optimality conditions and discuss conditions under which the optimal control is unique.
• Provide numerical examples of optimal policies and perform sensitivity analysis.

Experimental results

Research questions

• RQ1 What is the existence of an optimal containment policy within a feasible policy space that minimizes a loss combining deaths and GDP losses?
• RQ2 Under what conditions is the optimal policy unique, and how does the social cost of mortality influence potential multiplicity or phase transitions?
• RQ3 How do simple, low-dimensional policy profiles (e.g., single lockdown, reopening, periodic containment) perform relative to the optimal policy?
• RQ4 How sensitive are optimal policies to parameter choices such as the value of statistical life and the GDP impact parameter θ?
• RQ5 What qualitative insights arise about trade-offs between mortality containment and economic activity in the SEAIRD framework?

Key findings

• There exists a global minimum of the loss functional over a class of feasible piecewise-linear controls.
• The optimal policy can exhibit phase-transition-like behavior: above a critical social cost of mortality, a unique optimal containment exists; below it, no containment is optimal; at the threshold, two optimal policies may coexist.
• An explicit example reports an optimal policy reducing mortality to 0.26% with a 19.45% GDP loss under certain parameter choices (illustrative of the trade-off).
• Uniqueness of the optimal control is tied to the marginal rate of substitution between the statistical value of life and GDP; a higher valuation of life tends to yield a unique optimal policy.
• The paper demonstrates several policy profiles (unique lockdown, partial reopening, periodic containment, and herd-immunity-oriented paths) through numerical experiments.
• Sensitivity analysis shows the qualitative shape of optimal controls is robust to parameter changes, though intensity depends on exact numbers.