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On the Estimation of the Time-Dependent Transmission Rate in Epidemiological Models

Jorge P. Zubelli, Jennifer Loria, Vinicius V. L. Albani

TL;DR

The paper addresses the estimation of a time-varying transmission rate in an SEIR-like epidemiological model featuring nine compartments and COVID-19–relevant dynamics. It develops a nonparametric inverse-problem framework using a parameter-to-solution map $\mathcal{J}$ and applies Tikhonov regularization to recover $\beta$ from incidence data, proving well-posedness, continuity, and Fréchet differentiability of $\mathcal{J}$. The authors establish existence, stability, and convergence of regularized solutions and validate the approach with synthetic data and real-world Chicago 2020 and Canada 2021 datasets, including vaccination effects. The work provides a rigorous pathway for stable, data-consistent inference of time-dependent transmission rates to inform public health decision-making.

Abstract

The COVID-19 pandemic highlighted the need to improve the modeling, estimation, and prediction of how infectious diseases spread. SEIR-like models have been particularly successful in providing accurate short-term predictions. This study fills a notable literature gap by exploring the following question: Is it possible to incorporate a nonparametric susceptible-exposed-infected-removed (SEIR) COVID-19 model into the inverse-problem regularization framework when the transmission coefficient varies over time? Our positive response considers varying degrees of disease severity, vaccination, and other time-dependent parameters. In addition, we demonstrate the continuity, differentiability, and injectivity of the operator that link the transmission parameter to the observed infection numbers. By employing Tikhonov-type regularization to the corresponding inverse problem, we establish the existence and stability of regularized solutions. Numerical examples using both synthetic and real data illustrate the model's estimation accuracy and its ability to fit the data effectively.

On the Estimation of the Time-Dependent Transmission Rate in Epidemiological Models

TL;DR

The paper addresses the estimation of a time-varying transmission rate in an SEIR-like epidemiological model featuring nine compartments and COVID-19–relevant dynamics. It develops a nonparametric inverse-problem framework using a parameter-to-solution map and applies Tikhonov regularization to recover from incidence data, proving well-posedness, continuity, and Fréchet differentiability of . The authors establish existence, stability, and convergence of regularized solutions and validate the approach with synthetic data and real-world Chicago 2020 and Canada 2021 datasets, including vaccination effects. The work provides a rigorous pathway for stable, data-consistent inference of time-dependent transmission rates to inform public health decision-making.

Abstract

The COVID-19 pandemic highlighted the need to improve the modeling, estimation, and prediction of how infectious diseases spread. SEIR-like models have been particularly successful in providing accurate short-term predictions. This study fills a notable literature gap by exploring the following question: Is it possible to incorporate a nonparametric susceptible-exposed-infected-removed (SEIR) COVID-19 model into the inverse-problem regularization framework when the transmission coefficient varies over time? Our positive response considers varying degrees of disease severity, vaccination, and other time-dependent parameters. In addition, we demonstrate the continuity, differentiability, and injectivity of the operator that link the transmission parameter to the observed infection numbers. By employing Tikhonov-type regularization to the corresponding inverse problem, we establish the existence and stability of regularized solutions. Numerical examples using both synthetic and real data illustrate the model's estimation accuracy and its ability to fit the data effectively.
Paper Structure (9 sections, 23 theorems, 57 equations, 6 figures, 2 tables)

This paper contains 9 sections, 23 theorems, 57 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The Proposed Model
  3. Model Well-Posedness
  4. The Parameter-to-Solution Map
  5. The Regularized Inverse Problem
  6. Numerical Examples
  7. Synthetic Data
  8. Real Data
  9. Conclusions

Key Result

Proposition 2.1

Let $\boldsymbol{u}$ be a continuous solution of the ODE problem in Eq. re_edo and defined in the interval $J=[0,T_{0})$, with $0<T<T_{0}\leq \infty$. Let also Assumption hip_solutions hold. Then, $u_{1}(t) > 0$ , $\forall t \in J$.

Figures (6)

  • Figure 1: Schematic representation of the epidemiological model of Eqs. (1)-(9).
  • Figure 2: Top left: Reconstruction normalized error evolution considering different regularization parameter values and noise levels. Top right: Corresponding evolution of the normalized distance of model predictions to the observed data. Bottom left: Comparison between the reconstructed transmission parameters and the true one (solid black line) for the regularization parameter $\alpha = 10^{-3}$. Bottom right: Comparison of the corresponding model predicted infections with the true number of cases (no noise). No vaccination.
  • Figure 3: Top Left: Reconstruction normalized error evolution considering different regularization parameter values and noise levels. Top right: Corresponding evolution of the normalized distance of model predictions to the observed data. Bottom Left: Comparison between the reconstructed transmission parameters and the true one (solid black line) for the regularization parameter $\alpha = 10^{-3}$. Bottom right: Comparison of the corresponding model predicted infections with the true number of cases (no noise). Vaccination.
  • Figure 4: Left: Comparison between the reconstructed and transmission parameters and the true transmission (solid black line) for the regularization parameter $\alpha = 10^{-3}$. Right: Comparison of the corresponding model predicted infections with the true number of cases (no noise). Wrong vaccination parameter.
  • Figure 5: Left: Reconstructed transmission parameters for different regularization parameter values. Right: Comparison of the corresponding model predictions of cases with the observed daily infections. COVID-19 infections in Chicago, during 2020.
  • ...and 1 more figures

Theorems & Definitions (43)

  • Definition 2.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Corollary 2.1
  • Proposition 2.4
  • proof
  • ...and 33 more