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Spatiotemporal Monitoring of Epidemics via Solution of a Coefficient Inverse Problem

Michael V. Klibanov, Jingzhi Li, Zhipeng Yang

TL;DR

This work tackles a coefficient inverse problem for a spatiotemporal SIR PDE system by recovering $\beta(\mathbf{x})$, $\gamma(\mathbf{x})$, and the state $S,I,R$ from minimal interior and boundary data. It develops a convexification framework using a Carleman weight $\varphi_{\lambda}$ to build a strongly convex Tikhonov-like functional $J_{\lambda,\xi}(W)$, enabling global convergence of gradient descent to the true solution as data noise vanishes. Theoretical results include strong convexity (Theorem 5.1), accuracy estimates (Theorem 5.2), and uniqueness (Theorem 5.3), together with global convergence of an iterative method (Theorem 5.4). Numerical experiments with letter-shaped inclusions for the unknown coefficients demonstrate accurate reconstruction of $\beta$, $\gamma$, and the SIR fields, even under noisy measurements, highlighting potential for substantial cost savings in spatiotemporal epidemic monitoring.

Abstract

Let S,I and R be susceptible, infected and recovered populations in a city affected by an epidemic. The SIR model of Lee, Liu, Tembine, Li and Osher, \emph{SIAM J. Appl. Math.},~81, 190--207, 2021 of the spatiotemoral spread of epidemics is considered. This model consists of a system of three nonlinear coupled parabolic Partial Differential Equations with respect to the space and time dependent functions S,I and R. For the first time, a Coefficient Inverse Problem (CIP) for this system is posed. The so-called \textquotedblleft convexification" numerical method for this inverse problem is constructed. The presence of the Carleman Weight Function (CWF) in the resulting regularization functional ensures the global convergence of the gradient descent method of the minimization of this functional to the true solution of the CIP, as long as the noise level tends to zero. The CWF is the function, which is used as the weight in the Carleman estimate for the corresponding Partial Differential Operator. Numerical studies demonstrate an accurate reconstruction of unknown coefficients as well as S,I,R functions inside of that city. As a by-product, uniqueness theorem for this CIP is proven. Since the minimal measured input data are required, then the proposed methodology has a potential of a significant decrease of the cost of monitoring of epidemics.

Spatiotemporal Monitoring of Epidemics via Solution of a Coefficient Inverse Problem

TL;DR

This work tackles a coefficient inverse problem for a spatiotemporal SIR PDE system by recovering , , and the state from minimal interior and boundary data. It develops a convexification framework using a Carleman weight to build a strongly convex Tikhonov-like functional , enabling global convergence of gradient descent to the true solution as data noise vanishes. Theoretical results include strong convexity (Theorem 5.1), accuracy estimates (Theorem 5.2), and uniqueness (Theorem 5.3), together with global convergence of an iterative method (Theorem 5.4). Numerical experiments with letter-shaped inclusions for the unknown coefficients demonstrate accurate reconstruction of , , and the SIR fields, even under noisy measurements, highlighting potential for substantial cost savings in spatiotemporal epidemic monitoring.

Abstract

Let S,I and R be susceptible, infected and recovered populations in a city affected by an epidemic. The SIR model of Lee, Liu, Tembine, Li and Osher, \emph{SIAM J. Appl. Math.},~81, 190--207, 2021 of the spatiotemoral spread of epidemics is considered. This model consists of a system of three nonlinear coupled parabolic Partial Differential Equations with respect to the space and time dependent functions S,I and R. For the first time, a Coefficient Inverse Problem (CIP) for this system is posed. The so-called \textquotedblleft convexification" numerical method for this inverse problem is constructed. The presence of the Carleman Weight Function (CWF) in the resulting regularization functional ensures the global convergence of the gradient descent method of the minimization of this functional to the true solution of the CIP, as long as the noise level tends to zero. The CWF is the function, which is used as the weight in the Carleman estimate for the corresponding Partial Differential Operator. Numerical studies demonstrate an accurate reconstruction of unknown coefficients as well as S,I,R functions inside of that city. As a by-product, uniqueness theorem for this CIP is proven. Since the minimal measured input data are required, then the proposed methodology has a potential of a significant decrease of the cost of monitoring of epidemics.
Paper Structure (13 sections, 130 equations, 5 figures)

This paper contains 13 sections, 130 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Statements of Forward and Inverse Problems
  3. Transformation
  4. Carleman Estimate
  5. The Convexification Functional
  6. The functional
  7. Strong convexity of the functional $J_{\lambda , \xi }\left( W\right)$ on the set $\overline{B\left( R\right) }$
  8. The accuracy of the minimizer
  9. Global convergence of the gradient descent method
  10. Proof of Theorem 5.1
  11. Proof of Theorem 5.2
  12. Numerical Studies
  13. Summary

Figures (5)

  • Figure 1: Test 1. The reconstructed results of function $\beta( \mathbf{x})$ with different $\lambda$ in \ref{['5.4']}. The function $\beta(\mathbf{x})$ is given in \ref{['8.7']} with $c_{\beta }=0.6$ inside of the letter 'A' and the function $\gamma(\mathbf{x})$ is given in \ref{['8.8']} with $c_{\gamma}=0.4$ inside of the letter '$\Omega$'.
  • Figure 2: Test 1. The reconstructed results of function $\gamma( \mathbf{x})$ with different $\lambda$ in \ref{['5.4']}. The function $\beta(\mathbf{x})$ is given in \ref{['8.7']} with $c_{\beta }=0.6$ inside of the letter 'A' and the function $\gamma(\mathbf{x})$ is given in \ref{['8.8']} with $c_{\gamma}=0.4$ inside of the letter '$\Omega$'. Thus, it is clear from Figure \ref{['plot_diff_lambda_beta']} and Figure \ref{['plot_diff_lambda_gamma']} that the presence of the CWF in the functional $J_{\lambda ,\xi }\left( W\right)$ is necessary since the images for $\lambda =0$ are poor. It is also clear that $\lambda =3$ is the optimal value of the parameter $\lambda$.
  • Figure 3: Test 1. The exact (top line) and reconstructed (bottom line) functions $\rho_{S}$ (left column), $\rho_{I}$ (middle line) and $\rho_{R}$ (right line) at $t=0.75$, where the function $\beta(\mathbf{x})$ is given in \ref{['8.7']} with $c_{\beta}=0.6$ inside of the letter 'A' and the function $\gamma(\mathbf{x})$ is given in \ref{['8.8']} with $c_{\gamma}=0.4$ inside of the letter '$\Omega$'.
  • Figure 4: Test 2. The exact (left) and reconstructed (right) functions $\beta (\mathbf{x})$ (top line) and $\gamma (\mathbf{x})$ (bottom line), where the function $\beta (\mathbf{x})$ is given in \ref{['8.7']} with $c_{\beta }=0.4$ inside of the letter 'B' and the function $\gamma (\mathbf{x})$ is given in \ref{['8.8']} with $c_{ \gamma }=0.6$ inside of the letter 'D'.
  • Figure 5: Test 3. The reconstructed functions $\beta (\mathbf{x})$ (top line) and $\gamma (\mathbf{x})$ (bottom line) with $\sigma =0.01$ (1st column), $\sigma =0.02$ (2ed column), $\sigma =0.03$ (3rd column) and $\sigma =0.05$ (4th column).