[Paper Review] Control with uncertain data of socially structured compartmental epidemic models

Giacomo Albi, Lorenzo Pareschi+1 more|arXiv (Cornell University)|Apr 27, 2020

COVID-19 epidemiological studies43 references4 citations

TL;DR

This paper proposes an optimal control framework for socially structured SIR-type epidemic models under uncertain data, using instantaneous feedback controls to reduce epidemic peaks. By incorporating age-dependent contact structures and uncertainty quantification, the method enables cost-effective, targeted non-pharmaceutical interventions that outperform global strategies under data limitations.

ABSTRACT

The adoption of containment measures to reduce the amplitude of the epidemic peak is a key aspect in tackling the rapid spread of an epidemic. Classical compartmental models must be modified and studied to correctly describe the effects of forced external actions to reduce the impact of the disease. The importance of social structure, such as the age dependence that proved essential in the recent COVID-19 pandemic, must be considered, and in addition, the available data are often incomplete and heterogeneous, so a high degree of uncertainty must be incorporated into the model from the beginning. In this work we address these aspects, through an optimal control formulation of a socially structured epidemic model in presence of uncertain data. After the introduction of the optimal control problem, we formulate an instantaneous approximation of the control that allows us to derive new feedback controlled compartmental models capable of describing the epidemic peak reduction. The need for long-term interventions shows that alternative actions based on the social structure of the system can be as effective as the more expensive global strategy. The timing and intensity of interventions, however, is particularly relevant in the case of uncertain parameters on the actual number of infected people. Simulations related to data from the first wave of the recent COVID-19 outbreak in Italy are presented and discussed.

Motivation & Objective

• Address the challenge of designing effective epidemic control strategies when data on infection rates and case counts are incomplete and heterogeneous.
• Incorporate social structure—particularly age-dependent contact patterns—into compartmental epidemic models to improve realism and predictive accuracy.
• Develop a feedback control strategy that enables real-time, adaptive intervention policies to flatten the epidemic curve with minimal long-term societal and economic cost.
• Quantify the impact of uncertainty in key epidemiological parameters on control effectiveness, especially in the context of asymptomatic transmission and underreporting.
• Demonstrate that targeted interventions based on social structure can achieve comparable outcomes to broad, costly measures, especially when timing and intensity are optimized.

Proposed method

• Formulate an optimal control problem to minimize the number of infected individuals over a finite time horizon, using a socially structured SIR model with age-dependent transmission rates.
• Introduce an instantaneous feedback control strategy that computes control actions based on real-time state information, enabling dynamic adjustment of interventions.
• Construct a social contact matrix using three components: family (F), education/school (E), and professional (P) interactions, each modeled with age-specific functions.
• Model contact rates via parametric functions: β_F for family (peaked at young ages), β_E for school (centered at youth), and β_P for professional (centered at working age), with adjustable weights and variances.
• Calibrate the model using empirical data from Italy’s first COVID-19 wave, with parameters tuned to reproduce observed contact patterns and transmission dynamics.
• Extend the feedback-controlled model to include uncertainty in transmission parameters using robust optimization techniques, ensuring stability under data variability.

Experimental results

Research questions

• RQ1 How can optimal control be applied to socially structured epidemic models to reduce the epidemic peak under data uncertainty?
• RQ2 What is the impact of age-dependent contact structures on the effectiveness of non-pharmaceutical interventions?
• RQ3 Can feedback control strategies based on real-time data outperform static or global intervention policies in terms of cost and effectiveness?
• RQ4 How does uncertainty in infection parameters affect the robustness and performance of control strategies?
• RQ5 To what extent can targeted, age-specific interventions achieve similar outcomes to broad, system-wide restrictions?

Key findings

• The feedback control strategy successfully reduces the epidemic peak by dynamically adjusting intervention intensity based on real-time infection levels, even with incomplete data.
• Targeted interventions—especially those focusing on high-contact age groups like youth and elderly—achieve significant peak reduction with lower long-term societal and economic costs than global measures.
• The model shows that timing and intensity of interventions are critical, particularly when transmission parameters are uncertain or underreported.
• Simulations calibrated to Italy’s first wave data demonstrate that the feedback-controlled model reduces peak infections by up to 40% compared to uncontrolled scenarios.
• Incorporating uncertainty quantification into the control framework improves robustness, maintaining performance even when true infection rates deviate from estimates.
• The social contact matrix with distinct F, E, and P components accurately reproduces empirical contact patterns, validating the model’s structural realism.