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% Author:   Phillip Rothenbeck
% Title:    Your Thesis
% File:     conclusions/conclusions.tex
% Part:     conclusions
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%         summary of the content in this chapter
% Version:  01.09.2024
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\chapter{Conclusions}
\label{chap:conclusions}
The severe COVID-19 pandemic~\cite{WHO} infected millions of people, while hundreds of thousands
succumbed to it in Germany alone~\cite{SRD}. Over three years the pandemic
changed through the influence of various mitigation policies and numerous
emerging variants. In order to get a hold of the complex situation the necessity
for analysis arises. Therefore, the objective of this thesis is to measure the
COVID-19 pandemic in Germany and its 16 federal states by identifying several
epidemiological parameters that describe the spread of the disease. \\

We use the SIR model~\cite{1927} to describe the dynamics of the COVID-19
infection over time, offering an approximation of reality. In this model, the
transmission rate $\beta$ and recovery rate $\alpha$
describe the infectiousness and development of the disease that the respective
population experience. These rates serve as global evaluation measures
throughout the entire duration of the pandemic. Meanwhile, the time-dependent
reproduction number indicates the number of individuals infected by a single infectious
individual. The relations between parameters are defined in the system
of differential equations which governs the SIR model.\\

In order to obtain these epidemiological parameters and the reproduction number
for Germany, it is necessary to solve the system of ordinary differential equations (ODEs)
for real-world pandemic data recorded in each state and in Germany as a whole.
One method that has gained significant attention in recent years for solving
systems of differential equations is the data-driven approach known as
\emph{Physics-Informed Neural Networks} (PINN)~\cite{Raissi2019}. PINNs
integrate knowledge in form of physical models, while learning an approximation
the solution by fitting data points. We adapt previous epidemiological PINN
approaches~\cite{Shaier2021,Millevoi2023} to solve the set of ODEs of the SIR
model. The data for training is collected by the Robert Koch Institute and made
publicly available on GitHub~\cite{GHDead,GHInf}. After preprocessing, we
solve the inverse problem posed by the SIR model utilizing PINNs in order to find the
epidemiological parameters and the reproduction number for the given data. Using
this we conduct experiments on synthetic data and on the data for the federal
states and Germany itself. The results for the synthetic data yield a small
error, which demonstrates the efficacy of our approach on small datasets.\\

We divide our analysis of the real-world data into two groups. First, we have
the time-constant epidemiological parameters $\alpha$ and $\beta$, which
provide insight into the overall trajectory of the pandemic in a given region.
Given the assumed constant recovery period (see~\Cref{sec:preprocessing:rq}),
there is a dependency between the two parameters. Therefore, we focus our analysis on the
transmission rate $\beta$. The states with the highest estimated transmission rate values
are Thuringia, Saxony-Anhalt, and Mecklenburg-Western Pomerania, which means that
these states had a higher average number of infections during the pandemic.
Furthermore, it is evident that the six eastern states exhibit a higher transmission
rate than the overall German rate (see~\Cref{fig:alpha_beta_mean_std}).
Our results align with similarly observed differences in vaccination rates~\cite{FMH}
and highlight perceived discrepancies between the eastern and western federal
states~\cite{FMH,Desson2022}. We further substantiate this observation by
calculating the correlation coefficient between the vaccination
ratios $\nu$ of each state and our findings of $\beta$, which yields a strong
negative correlation. In other words, a lower vaccination rate is an indicator
for higher infection rates. The results from our second experiments,
underscore these findings. Here, we approximate the time-independent reproduction number $\Rt$
from the data. When $\Rt>1$, the disease spreads rapidly through the population.
Our results indicate a tendency for states with a high $\beta$ to experience
longer periods with $\Rt>1$. Furthermore, we can identify the time point on
which the most impactful events happened during the pandemic in Germany such as
the peak of the omicron variant~\cite{COVIDChronik} at around 700 days after
the start of data collection on 2020-03-09.\\

In conclusion, our approach has proven effective in yielding meaningful results
for the epidemiological parameters of $\alpha$ and $\beta$, as well as the
reproduction number $\Rt$ for Germany and its federal states. Despite the SIR
model being an approximation of reality, there is a clear connection between the
results and real-world data and events. We hope that this work will prove useful
in the analysis of the events of the COVID-19 pandemic in Germany.

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\section{Further Work}
\label{sec:furtherWork}

Our findings demonstrate that our methods enable the quantification of the
course of the COVID-19 pandemic in Germany using the data provided by the
Robert Koch Institute~\cite{GHDead,GHInf}. Here we present some limitations of
our work and propose future directions to address these points. First, we find
that our model does not accurately reconstruct the input data to the desired
level of precision. To address this, we propose a comprehensive hyperparameter
search to find the best configuration. Moreover, the SIR model does not
account for individuals, who may be immune due to the vaccination status or
those who are not infectious due to quarantine. In this section, we explore
epidemiological models that incorporate such dynamics observed in real-world
pandemics and recommend further investigation for Germany.

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\subsection{Further Compartmental Models}
As our results demonstrate, the SIR model is capable of approximating the
dynamics of real-world pandemics. However, the model is not without
limitations. The SIR model assumes that recovered
individuals remain immune and does not account for the reduction of exposure to
susceptible individuals through the introduction of non-pharmaceutical
mitigation policies, such as social distancing policies. These shortcomings can
be addressed by incorporating additional compartments and transmission rates
into the model. For example, the SEIRD model~\cite{Korolev2021} incorporates an
\emph{Exposed} group and subdivides the \emph{Removed} group into \emph{Dead}
and \emph{Recovered} compartments. Furthermore, the model is extended with four
additional parameters: the contact rate, the manifestation index, the incubation
rate, and the infection fatality rate. Doerre and Doblhammer~\cite{Doerre2022}
introduce an approach utilizing a SIERD model that they specialize to be age-
and gender-specific. For Germany, they show the impact of non-pharmaceutical
mitigation policies, solving the model using a numerical approximation method.\\

Additionally, Cooke and van den Driessche~\cite{Cooke1996}
propose the SEIRS model with two delays. This model is capable of
approximating diseases, that have an immune period, after which the recovered
individual becomes susceptible again. These are just a few examples of
the numerous modifications of the basic SIR model that can display the dynamics
of the real world in a higher degree of detail and may be used to approximate
and consequently quantify a pandemic.

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\subsection{Agent based models}

Compartmental models, such as the SIR model, look at the population as a
divided group. Each group represents a specific characterization that
all inhabitants of that group share. An \emph{Agent-Based Model} (ABM) sets its
focus on the individual. Each individual, or agent, has specific attributes
that determine its behavior and interactions with other agents during the
simulation. As Gilbert~\cite{Gilbert2010} states, ABMs simulate the behavior of
large groups. Each individual follows simple rules which leads to the emergence
of complex and stochastic behaviour on the mascroscopic level of the
system~\cite{Bodine2020}. With regard to COVID-19, Kerr \etal~\cite{Kerr2021}
put forth a simulation tool, \emph{Covasim}, which they base on an ABM. The ABM
employs local data, including demographic data, disease incidence data from the
region, and contact data for household, schools and workplaces, to define its
simulation for a specific region. Maziarz and Zach~\cite{Maziarz2020} address
the criticism levied against ABMs for simplifying the dynamics and lacking the
empirical support for the assumptions they make. The authors utilize an ABM and
the data specific to Australia to demonstrate the efficacy of ABMs in portraying
the dynamics of the COVID-19 pandemic. They state that ABMs can serve as a tool
for assessing the impact of non-pharmaceutical mitigation policies. This
illustrates that ABMs play a distinct role in analyzing the COVID-19 pandemic.
As the quantity of data has evolved, it is imperative to investigate the
potential of utilizing ABMs as a tool to assess the pandemic's course for
Germany in greater detail.

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