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% Author:   Phillip Rothenbeck
% Title:    Investigating the Evolution of the COVID-19 Pandemic in Germany Using Physics-Informed Neural Networks
% File:     chap01-introduction/chap01-introduction.tex
% Part:     introduction
% Description:
%         summary of the content in this chapter
% Version:  01.01.2012
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\chapter{Introduction}
\label{chap:introduction}

In the early months of 2020, Germany, like many other countries, was struck by
the novel \emph{Coronavirus Disease} (COVID-19)~\cite{WHO}. The pandemic, which
originates in Wuhan, China, had a profound impact on the global community,
paralyzing it for over two years. In response to the pandemic, the German
government employed a multifaceted approach~\cite{RKI}, encompassing the
introduction of vaccines and non-pharmaceutical mitigation policies such as
lockdowns. Between mitigation policies and varying strains of COVID-19, which
have exhibited varying degrees of infectiousness and lethality~\cite{RKIa},
Germany had recorded over 38,400,000 infection cases and 174,000 deaths, by
the end of June in 2023~\cite{SRD}. In light of these figures the need for an
analysis arises.\\

The dynamics of disease transmission in the real-world are complex. A multitude
of factors influence the course of a disease, and it is
challenging to gain a comprehensive understanding of these factors and develop
tools that allow for the comparison of disease courses across different
diseases and time points. The common approach in epidemiology to address this is
the utilization of epidemiological models that approximate the dynamics by
focusing on specific factors and modeling these using mathematical tools. These
models provide epidemiological parameters that determine the behavior of a
disease within the boundaries of the model. A seminal epidemiological model is
the \emph{SIR model}, which was first proposed by Kermack and McKendrick~\cite{1927}
in 1927. The SIR model is a compartmental model that divides the entire
population into three distinct groups: the \emph{susceptible} compartment, $S$;
the \emph{infectious} compartment, $I$; and the \emph{removed} compartment, $R$.
In the context of the SIR model, the constant parameters of the transmission
rate $\beta$ and the recovery rate $\alpha$ serve to quantify and determine the
course of a pandemic. However, a pandemic is not a static entity, therefore Liu
and Stechlinski~\cite{Liu2012}, and Setianto and Hidayat~\cite{Setianto2023}
propose an SIR model with time-dependent epidemiological parameters and
reproduction numbers $\Rt$. The SIR model is defined by a system of differential
equations, that incorporate the parameters $\alpha$ and $\beta$, thereby
depicting the fluctuation between the three compartments. For a given set of
data, the epidemiological parameters can be identified by solving the set of
differential systems. Recently, the data-driven approach of \emph{Physics-Informed Neural Networks}
(PINN) has gained attention due to its capability of finding solutions to
differential equations by fitting its predictions to both given data and the
governing system of differential equations. By employing this methodology,
Shaier \etal~\cite{Shaier2021} were able to find the epidemiological parameters
on data for different diseases. Additionally, Millevoi \etal~\cite{Millevoi2023}
were able to identify the reproduction number $\Rt$ for both synthetic and
Italian COVID-19 data using an approach based on a reduced version of the SIR
model.\\

The objective of this thesis is to identify the epidemiological parameters
$\alpha$ and $\beta$, as well as the reproduction number $\Rt$ of COVID-19 over
the first 1200 days of recorded data in Germany and its federal states. The
Robert Koch Institute (RKI)\footnote{\url{https://www.rki.de/EN/Home/homepage_node.html}} has compiled data on both reported cases and
associated moralities from the beginning of the outbreak in Germany to the
present. We utilize and preprocess this data according to the required format of
our approaches. As the raw data lacks information on recovery incidence, we
introduce the recovery queue that simulates a recovery period. To estimate the
epidemiological parameters we adopt the approach of Shaier
\etal~\cite{Shaier2021}, which utilizes a PINN learning the data, that consists
of time points with their respective sizes of  the $S, I$ and $R$ compartments,
to predict the epidemiological parameters based on the data and the governing
system of differential equations. Additionally, we apply the methodology by
Millevoi \etal~\cite{Millevoi2023} to estimate the time-dependent reproduction
number, $\Rt$, over a 1200-day period for each German federal state and Germany
as a whole in the reduced SIR model. Thus needing only the size of the $I$
group for each time step. To validate the effectiveness of these methods, we
first conduct experiments on a small synthetic dataset before applying the
techniques to real-world data. We then analyze the plausibility of our results
by comparing them to real-world events and data such as vaccination ratios of
each region or the peaks of impactful variants. This analysis demonstrates the
relevance of our findings and reveals a correlation between our results and
real-world developments, thus supporting the effectiveness of our approach.\\


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\section{Related work}
\label{sec:relatedWork}
In this section, we categorize our work into the context of existing literature
on the topic of solving the epidemiological models for real-world data. The
first work, by Smirnova \etal~\cite{Smirnova2017}, endeavors to identify a
stochastic methodology for estimating the time-dependent transmission rate
$\beta(t)$. They achieve this by projecting the time-dependent transmission
rate onto a finite subspace, that is defined by Legendre polynomials.
Subsequently, they compare the three regularization techniques of variational
(Tikhonov's) regularization, truncated singular value decomposition (TSVD), and
modified TSVD to ascertain the most reliable method for forecasting with
limited data. Their findings indicate that modified TSVD provides the most
stable forecasts on, as demonstrated on both simulated data and real-world data
from the 1918 influenza pandemic and the Ebola epidemic. In contrast, we
utilize PINNs to find the constant epidemiological parameters and the
reproduction number for Germany and its states.\\

Some related works similar to our method apply PINN approaches to COVID-19 and
other real-world disease examples~\cite{Shaier2021,Millevoi2023,Berkhahn2022,Olumoyin2021}.
Specifically Shaier \etal~\cite{Shaier2021} put forth a data-driven method
which they refer to as \emph{Disease-Informed Neural Networks} (DINN). In their
work, they demonstrate the capacity of PINNs to forecast the trajectory of
epidemics and pandemics. They underpin the efficacy of their approach by
applying it to 11 diseases, that have previously been modeled. In their
experiments they employ the SIDR (susceptible, infectious, dead, recovered)
model. Finally, they present that this method is a robust and effective means
of identifying the parameters of a SIR model.\\

Similarly  Berkhahn and Ehrhard~\cite{Berkhahn2022}, employ the susceptible,
vaccinated, infectious, hospitalized and removed (SVIHR) model. The proposed
PINN methodology initially estimates the SVIHR model parameters for German
COVID-19 data, covering the time span from the inceptions of the outbreak to
the end of 2021. For comparative purposes, Berkhahn and Ehrhard employ the
method of non-standard finite differences (NSFD) as well.  The authors utilize
both forecasting methods to project the trajectory of COVID-19 from mid-April
2023 onwards. Berkhahn and Ehrhard find that PINNs are able to adapt to varying
vaccination rates and emerging variants.\\

Furthermore, Olumoyin \etal~\cite{Olumoyin2021} employ an alternative
methodology for identifying the time-dependent transmission rate of an
asymptomatic-SIR model accounting for unreported infectious cases. The PINN
approach they introduce, utilizes the cumulative and daily reported infection
cases and symptomatic recovered cases, to demonstrate the effect of different
mitigation measures and to ascertain the proportion of non-symptomatic
individuals and asymptomatic recovered individuals. With this they can
illustrate the influence of vaccinations and a set non-pharmaceutical
mitigation methods on the transmission of COVID-19 on data from Italy, South
Korea, the United Kingdom, and the United States.\\

Finally, Millevoi \etal~\cite{Millevoi2023} address the issue of the changes in
the transmission rate due to the dynamics of a pandemic.  The authors employ the
reproduction number $\Rt$ to reduce the system of differential equations to a
single equation and introduce a reduced-split version of the PINN, which
initially trains on the data and then trains to minimize the residual of the
ordinary differential equation. They test their approach on five synthetic and
two real-world scenarios from the early stages of the COVID-19 pandemic in
Italy. This method yields an increase in both accuracy and training speed. In
contrast, to these works, we estimate the epidemiological of $\alpha$ and
$\beta$ and the reproduction number $\Rt$ for Germany for the entirety of the
span from early March in 2020 to late June in 2023.

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\section{Overview}

This thesis is comprised of four chapters. \Cref{chap:background}
starts with the theoretical overview of mathematical modeling in epidemiology,
with a particular focus on the SIR model. Subsequently, it shifts its focus to
neural networks, specifically on the background of PINNs and their use in
solving ordinary differential equations.~\Cref{chap:methods} outlines the
methodology employed in this thesis. First, we present the data, that was
collected by the RKI and our preprocessing. Then, we present the PINN
approaches, which are inspired by the work of Shaier \etal~\cite{Shaier2021}
and Millevoi \etal~\cite{Millevoi2023}.~\Cref{chap:evaluation} provides the
setups and results of the experiments that we conduct. This chapter is divided
into two sections. The first section shows and discusses the results concerning
the epidemiological parameters of $\alpha$ and $\beta$. The subsequent section
presents the results concerning the reproduction value $\Rt$. Finally, in
\Cref{chap:conclusions}, give a conclusion of our work and provide an overview
of potential further work.

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