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A Metric Hybrid Planning Approach to Solving Pandemic Planning Problems with Simple SIR Models

Ari Gestetner, Buser Say

TL;DR

Problem: design continuous-time lockdown schedules within an SIR-based pandemic model to keep infections and removals under thresholds. Approach: formalize as a metric hybrid planning problem $\Pi$ and solve with SCIPPlan, incorporating a lockdown-dependent infection rate $b(a_{1})$ and domain-specific valid inequalities. Contributions: finiteness and correctness guarantees for SCIPPlan in this setting and extensive experiments showing substantial runtime reductions from valid inequalities and improvements from variable step durations. Significance: demonstrates an exact, continuous-time planning framework for pandemic mitigation that leverages closed-form SIR solutions and constraint-generation, with potential applicability to related social-physics planning problems.

Abstract

A pandemic is the spread of a disease across large regions, and can have devastating costs to the society in terms of health, economic and social. As such, the study of effective pandemic mitigation strategies can yield significant positive impact on the society. A pandemic can be mathematically described using a compartmental model, such as the Susceptible Infected Removed (SIR) model. In this paper, we extend the solution equations of the SIR model to a state transition model with lockdowns. We formalize a metric hybrid planning problem based on this state transition model, and solve it using a metric hybrid planner. We improve the runtime effectiveness of the metric hybrid planner with the addition of valid inequalities, and demonstrate the success of our approach both theoretically and experimentally under various challenging settings.

A Metric Hybrid Planning Approach to Solving Pandemic Planning Problems with Simple SIR Models

TL;DR

Problem: design continuous-time lockdown schedules within an SIR-based pandemic model to keep infections and removals under thresholds. Approach: formalize as a metric hybrid planning problem and solve with SCIPPlan, incorporating a lockdown-dependent infection rate and domain-specific valid inequalities. Contributions: finiteness and correctness guarantees for SCIPPlan in this setting and extensive experiments showing substantial runtime reductions from valid inequalities and improvements from variable step durations. Significance: demonstrates an exact, continuous-time planning framework for pandemic mitigation that leverages closed-form SIR solutions and constraint-generation, with potential applicability to related social-physics planning problems.

Abstract

A pandemic is the spread of a disease across large regions, and can have devastating costs to the society in terms of health, economic and social. As such, the study of effective pandemic mitigation strategies can yield significant positive impact on the society. A pandemic can be mathematically described using a compartmental model, such as the Susceptible Infected Removed (SIR) model. In this paper, we extend the solution equations of the SIR model to a state transition model with lockdowns. We formalize a metric hybrid planning problem based on this state transition model, and solve it using a metric hybrid planner. We improve the runtime effectiveness of the metric hybrid planner with the addition of valid inequalities, and demonstrate the success of our approach both theoretically and experimentally under various challenging settings.
Paper Structure (24 sections, 1 theorem, 12 equations, 6 figures, 2 tables)

This paper contains 24 sections, 1 theorem, 12 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. The Metric Hybrid Planning Problem
  4. Solving the Metric Hybrid Planning Problem
  5. The SIR Model with Constant Infection and Recovery Rates
  6. Pandemic Planning Using a Metric Hybrid Planner
  7. The State Transition Model with Lockdowns
  8. The Pandemic Planning Problem
  9. Valid Inequalities
  10. The Mathematical Optimization Model
  11. Theoretical Results
  12. Related Problems and Extensions
  13. Experimental Results
  14. Design of Experiments
  15. Experimental Setup
  16. ...and 9 more sections

Key Result

Lemma 1

SCIPPlan finds a solution to the pandemic planning problem within some constant $\gamma > 0$ constraint violation tolerance or proves its infeasibility in finite number of constraint generation iterations.

Figures (6)

  • Figure 1: Visualization of two pandemic curves based on the SIR model. The purple curves represent susceptible individuals, blue curves represent the infected individuals when there are no lockdowns and the green curves represent the removed individuals. The goal of pandemic planning problem is to keep the number of removed individuals under a certain threshold (i.e., orange dashed line) while also keeping the number of infected individuals under a certain threshold (i.e., black dashed line). On the left is the visualization of the natural spread of a disease without any lockdowns and on the right is the spread of the same disease under the plan that is produced by SCIPPlan using lockdowns (i.e., red lines).
  • Figure 2: Visualization of the effect of valid inequalities on the logarithmic runtime performance of SCIPPlan in seconds where each data point corresponds to an instance. The addition of the valid inequalities improves the runtime performance of SCIPlan by around one to two orders of magnitude on average.
  • Figure 3: Visualization of logarithmic runtime performance in seconds for both problem settings. Overall, decreasing the value of parameter $\delta$ typically decreases the runtime performance of SCIPPlan, since the lower values of $\delta$ correspond higher values of horizon $H$.
  • Figure 4: Visualization of total action durations in days for both problem settings. Overall, using smaller values of parameter $\delta$ increases the solution quality and allows for a more granular control of the population.
  • Figure 5: Visualization of total action durations in days over different values of initial infected. The use of the variable step duration over fixed step duration results in higher solution quality (i.e., lower total action duration) in all instances.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Lemma 1: Finiteness and correctness of SCIPPlan
  • proof