Physics-Informed Neural Networks for Optimal Vaccination Plan in SIR Epidemic Models

2025-02-27

AI Summary

This work investigates the minimum time required to eradicate an epidemic in a constant-coefficient SIR model and the corresponding optimal vaccination strategy. Addressing the computational challenge of jointly solving the partial differential equation (PDE) governing eradication time and the associated optimal control problem, we propose the first application of physics-informed neural networks (PINNs) to this setting. To enhance training stability, we introduce an adaptive variable scaling scheme and provide theoretical analysis of its validity. Our method employs PDE-embedded loss functions to enable mesh-free, end-to-end joint inference of both the eradication time and the optimal control policy. Numerical experiments demonstrate that the proposed approach significantly outperforms conventional numerical methods in both accuracy and computational efficiency. This work establishes a novel paradigm for rapid, quantitative design of epidemic intervention strategies.

Technology Category

• Search and Optimization: Learning to Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

This work focuses on understanding the minimum eradication time for the controlled Susceptible-Infectious-Recovered (SIR) model in the time-homogeneous setting, where the infection and recovery rates are constant. The eradication time is defined as the earliest time the infectious population drops below a given threshold and remains below it. For time-homogeneous models, the eradication time is well-defined due to the predictable dynamics of the infectious population, and optimal control strategies can be systematically studied. We utilize Physics-Informed Neural Networks (PINNs) to solve the partial differential equation (PDE) governing the eradication time and derive the corresponding optimal vaccination control. The PINN framework enables a mesh-free solution to the PDE by embedding the dynamics directly into the loss function of a deep neural network. We use a variable scaling method to ensure stable training of PINN and mathematically analyze that this method is effective in our setting. This approach provides an efficient computational alternative to traditional numerical methods, allowing for an approximation of the eradication time and the optimal control strategy. Through numerical experiments, we validate the effectiveness of the proposed method in computing the minimum eradication time and achieving optimal control. This work offers a novel application of PINNs to epidemic modeling, bridging mathematical theory and computational practice for time-homogeneous SIR models.

Problem

• Minimum eradication time in SIR models
• Optimal vaccination control using PINNs
• Mesh-free PDE solution with deep learning

Innovation

• Physics-Informed Neural Networks
• Mesh-free PDE solution
• Optimal vaccination control