Modeling the COVID-19 Pandemic: A Model-Driven and a Data-Driven Analyses Público

Abstract

The COVID-19 pandemic has sparked a plethora of mathematical models, each constructed with varying assumptions and methodologies. However, the nature of infectious disease dynamics, which are inherently complex and influenced by numerous factors, poses challenges to the development of accurate models. In this project, we propose a space-dependent compartmental model based on the classical Susceptible, Infected, Recovered (SIR) epidemiological model as well as a data-driven approach that combines the SIR model and data assimilation through physics-informed neural networks (PINN) using generated training data. The space-dependent model allows for control over each geographical unit (GU) in a global domain, making the problem scalable, while time-dependent parameters are included to simulate the effect of interventions like lockdowns and vaccination campaigns. Following the space-dependent model, we investigate the most influential parameters of the basic SIR model through a first-order local sensitivity analysis. The results show that the reproduction/death rate has the most impact on all compartments of the SIR model. This analysis provides insights into which parameters should be prioritized in future studies and can help in developing more effective interventions. Finally, the PINN approach based on SIR models exhibits satisfactory predictive capability for parameter estimation and dynamics simulation, even with limited and noisy data. This project is a promising start to the modeling of intricate and multifaceted dynamics pertaining to the transmission of infectious diseases.

Table of Contents

1 Introduction 1

2 Mono-region SIR-Like Models for the Outbreak 3

2.1 The basic SIR model........................... 3

2.2 Asymptotic behaviors of the SIR model ................ 5

3 Inter-Regional Mobility 6

3.1 The mobility matrix ........................... 6

3.2 The numerical solver........................... 8

3.3 Design of the objected-oriented solver................. 8

3.3.1 The local and global models ................... 9

3.4 Benchmarks and results ......................... 10

3.4.1 Consistency test with theorems for one-region models . . . . . 10

3.4.2 Cross-validation.......................... 11

3.4.3 Epidemic vs. endemic....................... 12

3.4.4 Effects of lockdown ....................... 13

4 Sensitivity Analysis for Mono-Regional Models 17

4.1 First-order local sensitivity analysis on the endemic model . . . . . 18

5 Parameter Estimation and Prediction with Physics-Informed Neural Networks 22

5.1 Introduction to physics-informed neural networks (PINN) . . . . . . 23

5.2 Using PINN for parameter estimation and prediction of the SIR endemic model................................ 27

5.2.1 Using PINN for noisy data ................... 28

5.2.2 Using PINN for limited data .................. 30

6 Conclusions: Limitations and Perspectives 33

Bibliography 37

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Mathematics
Palabra Clave	Physics-informed neural networks Compartmental models Space-explicit models Infectious disease modeling
Committee Chair / Thesis Advisor	Alessandro Veneziani, Emory University
Committee Members	Elizabeth Newman, Emory University Ruoxuan Xiong, Emory University

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