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

TL;DR

We address computing the minimum eradication time and the associated optimal vaccination policy for a time-homogeneous SIR model using physics-informed neural networks. By formulating the problem as a Hamilton–Jacobi–Bellman PDE for the value function $u(x,y)$ and employing a variable-scaled PINN (VS-PINN), we solve the PDE in a mesh-free manner and extract the optimal bang-bang switching control via the dynamic programming principle. The work provides NTK-based theoretical support for the stability and efficiency of VS-PINN in solving the HJB equation, and validates the approach through extensive numerical experiments on learning $u$, $u^{r_0}$, the uncontrolled SIR flow, and the switching time $\tau$. The proposed framework offers a computationally efficient path to deriving vaccination strategies in epidemic models, with potential extensions to time-inhomogeneous settings and comparison to other data-driven control methods.

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.

Figures (9)

• Figure 1: Training $u$ and $u^{r_0}$ under the PINN framework.
• Figure 2: Procedure for solving $\tau$ using the DPP framework, with trained $u$, $u^{r_0}$, and the learned uncontrolled dynamics $(S(t), I(t))$.
• Figure 3: Training of uncontrolled dynamics via the PINN framework
• Figure 4: Feed-forward network with residual connections and normalization scheme.
• Figure 5: Collocation points and data points used in training $u$, $u^{r_0}$, $w$ and $\tau$. For training $u$ and $u^{r_0}$, collocation points (red dots) and data points (blue dots) are used (left). For training $w$, collocation points (red dots), data points (blue dots), and initial condition (green area) are used as shown in the middle. For training $\tau$, only collocation points are used (right).

Theorems & Definitions (5)

• Theorem 1: Theorem 1.2 of hynd2021eradication
• Theorem 2: Theorem 1.4 of hynd2021eradication
• Theorem 3
• Lemma 1
• proof : proof of Theorem \ref{['thm:ntk']}