[Paper Review] Optimal Lockdown Policies driven by Socioeconomic Costs

Elena Gubar, Laura Policardo+2 more|arXiv (Cornell University)|May 18, 2021

COVID-19 epidemiological studies4 citations

TL;DR

This paper proposes a heterogeneous SQAIRD model integrating age-specific demographics (young, adults, elderly) to optimize lockdown policies under convex socioeconomic costs. Using optimal control theory, it demonstrates that targeted lockdowns—prioritizing less costly groups—reduce pandemic costs more effectively than uniform or laissez-faire approaches, with simulations showing up to 43% lower aggregate costs under targeted policies.

ABSTRACT

In this research paper we modify a classical SIR model to better adapt to the dynamics of COVID-19, that is we propose the heterogeneous SQAIRD model where COVID-19 spreads over a population of economic agents, namely: the elderly, adults and young people. We then compute and simulate an optimal control problem faced by a Government, where its objective is to minimize the costs generated by the pandemics using as control a compulsory quarantine measure (that is, a lockdown). We first analyze the problem from a theoretical perspective, claiming that different lockdown policies (total lockdown, no lockdown or partial lockdown) may justified by different cost (concave or convex) structures of the economies. We then focus on a particular cost structure (convex costs) and we simulate a targeted optimal policy vs. a uniform optimal policy, by dividing the whole population in three demographic groups (young, adults and old). We also simulate the dynamic of the pandemic with no policy implemented. Simulations highlighted the fact that: a) a policy of lockdown is always better than the \emph{laissez faire} policy, because it limits the costs that the pandemic generates in an uncontrolled situation; b) a targeted policy based on age of the individuals outperforms a uniform policy in terms of costs that it generates, being a targeted policy less costly and equally effective in the control of the pandemic.

Motivation & Objective

• To develop a dynamic, age-heterogeneous epidemic model (SQAIRD) that captures differential transmission and impact of COVID-19 across young, adult, and elderly populations.
• To analyze optimal lockdown policies using a social cost function that incorporates treatment, quarantine, and death costs, under both convex and concave cost structures.
• To compare the efficiency of uniform versus targeted lockdown policies in minimizing total socioeconomic costs during a pandemic.
• To evaluate the long-term economic and health trade-offs of lockdown timing and intensity, particularly under convex cost functions.
• To provide a policy-relevant framework for governments to design cost-effective, age-targeted interventions during infectious disease outbreaks.

Proposed method

• Extends the classical SIR model into a three-group SQAIRD framework (Susceptible, Quarantined, Asymptomatic, Infected, Recovered, Dead) with age-specific compartments.
• Applies optimal control theory to a dynamic system of ODEs, where the control variable is the rate of quarantine (lockdown intensity), subject to state dynamics and cost minimization.
• Models social costs as a convex function of the quarantine rate, reflecting increasing marginal costs of imposing restrictions.
• Calibrates model parameters using real-world data on infection rates, recovery, mortality, and treatment costs across age groups.
• Conducts numerical simulations comparing three scenarios: no policy (laissez-faire), uniform lockdown, and targeted lockdown based on age-specific cost-efficiency.
• Uses parameter sets (Exp. 1 and Exp. 2) to test robustness and quantify cost differentials under varying assumptions on transmission and mortality.

Experimental results

Research questions

• RQ1 Under what conditions does a targeted lockdown outperform a uniform lockdown in minimizing total socioeconomic costs during a pandemic?
• RQ2 How does the curvature of the cost function (convex vs. concave) influence the optimal lockdown policy (e.g., partial, full, or no lockdown)?
• RQ3 What is the quantitative impact of implementing a lockdown versus no policy on peak infection, death toll, and aggregate costs?
• RQ4 How do age-specific differences in infection risk, mortality, and quarantine cost affect the efficiency of targeted interventions?
• RQ5 To what extent can age-based targeting reduce total economic and health costs compared to uniform restrictions?

Key findings

• A lockdown policy is always superior to a laissez-faire approach, reducing aggregate costs by approximately 43% in the most favorable simulation (Exp. 1: $3.58×10¹² vs. $2.04×10¹²).
• Targeted lockdowns—focusing on the age group with the lowest quarantine cost—achieve the same pandemic control with 28% lower total costs than uniform policies in Exp. 1.
• Under convex cost structures, the optimal policy features an endogenous lockdown with a defined start and end time, rather than a constant or boundary solution.
• In Exp. 2, the peak number of infected individuals reaches 6.57 million under no policy, compared to 2.65 million under optimal targeted lockdown, demonstrating significant mitigation.
• The model shows that excluding asymptomatic individuals from quarantine leads to higher transmission, but the cost of quarantining them is lower than the cost of uncontrolled spread.
• The cost of death is modeled as a major driver of total social cost, with $2.8 million per death in Exp. 1, significantly influencing the optimal policy structure.