% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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, as of the end of June in 2023~\cite{SRD}. In light of these
figures the need for an analysis arises.\\

The dynamics of the spread 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 allows 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 transition rates and parameters that determine the behavior of a disease
within the boundaries of the model. A fundamental epidemiological model, is the
\emph{SIR model}, which was first proposed by Kermack and McKendrick~\cite{1927}
in 1927. The SIR model is a compartmentalized 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, pandemic is not a static entity, therefor, Liu
and Stechlinski~\cite{Liu2012}, and Setianto and Hidayat~\cite{Setianto2023},
propose an SIR model with time-dependent transition rates and reproduction number $\Rt$. The SIR model
is defined by a system of differential equations, that incorporate
the transition rates, thereby depicting the fluctuation between the three
compartments. For a given set of data, the transition rate 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 transition rate 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 transition rates $\beta$ and
$alpha$, 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) 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
transition rates we adopt the approach of Shaier \etal~\cite{Shaier2021}, which
utilizes a physics-informed neural network learning the data, which consists of
time point with their respective sizes of  the $S, I$ and $R$ compartments, to
predict the transition rates based on the data and the governing system of
differential equations. Moreover, we utilize the methodology proposed by
Millevoi \etal~\cite{Millevoi2023} that estimates the reproduction number for
each day across the 1200-day span for each German state and Germany as a whole,
in 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 to demonstrate the relevance of these numbers.
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.\\


% -------------------------------------------------------------------

\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
limited data, as demonstrated on both simulated data and real-world data from
the 1918 influenza pandemic and the Ebola epidemic. In contrast, we
utilize physics-informed neural networks (PINN) to find the constant transition rates
and the reproduction number for Germany and its states\\

Some related works similarly to us apply PINN approaches to COVID-19 and other
real-world disease data such as~\cite{Shaier2021,Berkhahn2022,Olumoyin2021,Millevoi2023}.
Specifically in~\cite{Shaier2021}, Shaier \etal put forth a data-driven
approach which they refer to as disease informed neural networks (DINN). In their work,
they demonstrate the capacity of DINNs 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 in~\cite{Berkhahn2022}, Berkhahn and Ehrhard 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 employ both methods
the two forecasting methods project the trajectory of COVID-19 from mid-April
2023 onwards. Berkhahn and Ehrhard find that the PINN is 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 vaccination 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 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 ODE. 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 rates and the reproduction number for Germany for the entirety of the span
from early March in 2020 to late June in 2023.

% -------------------------------------------------------------------

\section{Overview}

This thesis is comprised of four chapters. \Cref{chap:background}
presents 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 physics-informed neural
networks (PINN) 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 Robert Koch Institute (RKI). 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}
presents the setups and results of the experiments that we conduct. This chapter
is divided into two sections. The first section presents and discusses the
results concerning the transition rates of $\beta$ and $\alpha$. The subsequent
section presents the results concerning the reproduction value $\Rt$. Finally,
in \Cref{chap:conclusions}, we connect our results with the events of the
real-world and give an overview of potential further work.

% -------------------------------------------------------------------