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% Author:   Phillip Rothenbeck
% Title:    Your Thesis
% File:     conclusions/conclusions.tex
% Part:     conclusions
% Description:
%         summary of the content in this chapter
% Version:  01.09.2024
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\chapter{Conclusions}
\label{chap:conclusions}

The objective of this thesis is to identify quantifying measures for the
COVID-19 pandemic in Germany and its 16 federal states. We use the SIR model to
describe the dynamics of the disease over time, offering an approximation of the
reality. In this model, the transmission rate $\beta$ and recovery rate $\alpha$
describe the infectiousness and resolution of the disease that the respective
population experience. These rates serve as constant evaluation measures
throughout the entire duration of the pandemic. The time-dependent reproduction
number indicates the number of individuals infected by a single infectious
individual. The SIR model is defined on a system of differential equations that
elucidates the relations between these rates. In order to obtain these values
for Germany, it is necessary to solve the ordinary differential equations (ODEs)
for the data pertaining to the pandemic in each state and in Germany as a whole.
We employ a physics-informed neural network in our approach to solve the ODE's.
The data on which we train is collected by the Robert Koch Institute and made
publicly available on GitHub, where they can be accessed for download. We
preprocess the data to fit have the required format for the PINNs to reconstruct
it, and at the same time predicts the transition rates and the reproduction
number for the given data. Using this we conduct experiments on synthetic data
and on the data for the German states and Germany itself. The results for the
synthetic data demonstrate the efficacy of our data on small datasets.\\

The results of our work regarding the real-world data are divided into two
groups. First we have the constant transmission rates, which provide insight
into the overall trajectory of the pandemic in a given region. A high
transmission rate indicates that, on average, the significant number of
individuals were infected during the pandemic. Conversely, a high recovery rate
indicates that individuals either recovered or died from the disease at a faster
rate. Due to this contradiction in positive or negative meaning in $\alpha$
paired with the uncertainty of a possible dependency on $\beta$ during training,
we want to shift the focus on our results of $\beta$. The states with the
highest transmission rate values are Thuringia, Saxony-Anhalt and
Mecklenburg-Vorpommern. Furthermore, it is evident the six  eastern
states exhibit a higher transmission rate than the overall German rate
(see~\Cref{fig:alpha_beta_mean_std}). These results align with the ongoing
narrative of the COVID-19 pandemic in Germany, which has highlighted a perceived
discrepancy in vaccination rates between the eastern and western federal states.
This assertion which can be substantiated by a comparison of the vaccination
ratios $\nu$ of each state and our findings. We find a strong negative
correlation between $\nu$ and $\beta$. The results from our second experiments,
underscore these findings. Here, we approximate the 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.\\

Although larger events are visible, smaller, less impactful events that are
still visible on the raw data, do not appear in our results. This discrepancy
can be attributed to the less precise reconstruction of the input data. The
predicted version is smooth and does not contain any smaller peaks. To address
these implementational limitations of our method, we intend to conduct
comprehensive hyperparameter search to find the best configuration of our models
to fit the data. Further optimizations can be applied to the epidemiological
model that we employ, for which we present options in the subsequent section.

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\section{Further Work}
\label{sec:furtherWork}
Our findings demonstrate that with our methods enable the quantification of the
course of the COVID-19 pandemic in Germany using the data provided by the
Robert Koch Institute. Additionally, we present the limitations of our work.
The SIR model is subject to numerous limitations. For instance, it 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 illustrate these dynamics observed in real-world
pandemics and recommend further investigation for Germany. First, we examine
extensions of the SIR models, then we focus on agent-based models (ABMs).

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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. As previously stated, the SIR model assumes that recovered
individuals remain immune and does not account for the reduction of exposure of
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 incorporates an \emph{Exposed}
group and subdivides the \emph{Removed} group into \emph{Dead} and
\emph{Recovered} compartments. Furthermore, this adds four additional rates to
the model: the contact rate, representing the average number of contacts
between infectious and susceptible people with a high probability of infection;
the manifestation index, indicating the proportion of individuals exposed to
the disease who will become infectious; the incubation rate, measuring the time
required for exposed individuals to become infectious; and the infection
fatality rate, quantifying the fraction of individuals who succumb to the
disease. As Doerre and Doblhammer~\cite{Doerre2022} show for Germany using a
numerical approximation method, for an SIERD model that they specialize to be
age- and gender-specific, that it shows the impact of non-pharmaceutical
mitigation policies. In their work, Cooke and van den Driessche~\cite{Cooke1996}
propose the SEIRS model with two delays. This is 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 be used to
approximate and consequently quantify a pandemic.

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\subsection{Agent based models}

While compartmental models, such as the SIR model, look at the population as a
divided group, with each group representing 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, with each individual following simple rules. 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. In their work,
Maziarz and Zach~\cite{Maziarz2020} address the criticism levied against ABMs
for simplifying the dynamics and lacking the empirical support for the
assumptions it 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 further state that ABMs can serve as 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 data situation has evolved, it is imperative to investigate the
potential of utilizing ABMs as a tool to assess the pandemic's course.

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