Physics-informed neural networks for optimal vaccination plan in SIR epidemic models

This work investigates the minimum eradication time in a controlled susceptible-infectious-recovered model with constant infection and recovery rates. The eradication time is defined as the earliest time the infectious population falls below a prescribed threshold and remains below it. Leveraging the fact that this problem reduces to solving a Hamilton-Jacobi-Bellman (HJB) equation, we propose a mesh-free framework based on a physics-informed neural network to approximate the solution. Moreover, leveraging the well-known structure of the optimal control of the problem, we efficiently obtain the optimal vaccination control from the minimum eradication time using the dynamic programming principle. To improve training stability and accuracy, we incorporate a variable scaling method and provide theoretical justification through a neural tangent kernel analysis. Numerical experiments show that this technique significantly enhances convergence, reducing the mean squared residual error by approximately 80% compared with standard physics-informed approaches. Furthermore, the method accurately identifies the optimal switching time. These results demonstrate the effectiveness of the proposed deep learning framework as a computational tool for solving optimal control problems in epidemic modeling as well as the corresponding HJB equations.