Thesis/chapters/conclusions/conclusions.tex

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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}. Across three years the pandemic
changed through the influence of various mitigation policies and numerous
emerging variant. 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 disease 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 relation between parameters and values is defined in the system
of differential equations which governs the the SIR model. In order to obtain these values
for Germany, it is necessary to solve the system of ordinary differential equations (ODEs)
for real-world pandemic data which was recorded in each state and in Germany as a whole.
The data-driven approach of \emph{Physics-Informed Neural Networks} (PINN)~\cite{Raissi2019}
to solve systems of differential equations has gained  attention in the last
years. These integrate the knowledge in form of physical
models, while they learn the solution by fitting data points. We adapt previous
epidemiological PINN approaches~\cite{Shaier2021,Millevoi2023} to solve the
ODE's. The data on which we train is collected by the Robert Koch Institute and
made publicly available on GitHub~\cite{GHInf,GHDead}. After a
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 demonstrate 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-Vorpommern which means that on
average these states had a high number of infections during the pandemic.
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}).
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 March 9. 2020.\\

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 some
limitations during training, there is a clear connection between the results
and real-world data and events are evident. 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 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~\cite{GHDead,GHInf}. Here we present some limitations of
our work and propose future directions to remedy these point. First we find
that our model does not reconstruct the input data as precisely as possible.
To address this, we propose a comprehensive hyperparameter search to find the
best configuration. Furthermore, 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.

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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 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~\cite{Korolev2021} 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, the manifestation index, the incubation rate, and
the infection fatality rate. As Doerre and Doblhammer~\cite{Doerre2022} show
for Germany using a numerical approximation method, for a 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 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}

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 for
Germany.

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