add conclusions

chapters/conclusions/conclusions.tex

@@ -7,19 +7,62 @@
 %         summary of the content in this chapter
 % Version:  01.09.2024
 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\chapter{Conclusions  5}
+\chapter{Conclusions}
 \label{chap:conclusions}
 
-The states with the highest transmission rate
-values are Thuringia, Saxony Anhalt and Mecklenburg West-Pomerania. It is also,
-visible that all six of the eastern states have a higher transmission rate than
-Germany. These results may be explainable with the ratio of vaccinated individuals\footnote{\url{https://impfdashboard.de/}}.
-The eastern state have a comparably low complete vaccination ratio, accept for
-Berlin. While Berlin has a moderate vaccination ratio, it is also a hub of
-mobility, which means that contact between individuals happens much more often.
-This is also a reason for Hamburg being a state with an above national standard
-rate of transmission. Bremen has the highest ratio of vaccinated individuals,
-this might be a reason for the it having the lowest transmission of all states.\\
+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.
 
 % -------------------------------------------------------------------
 
@@ -62,7 +105,7 @@ 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 an pandemic.
+approximate and consequently quantify a pandemic.
 
 % -------------------------------------------------------------------
 

thesis.bbl

@@ -1,253 +1,230 @@
-\newcommand{\etalchar}[1]{$^{#1}$}
-\begin{thebibliography}{KSM{\etalchar{+}}21}
+\begin{thebibliography}{10}
 
-% this bibliography is generated by alphadin.bst [8.2] from 2005-12-21
+\bibitem{Anderson1991}
+R.~M. Anderson, Roy Malcolm;~May.
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+\newblock Oxford University Press, 1991.
 
-\providecommand{\url}[1]{\texttt{#1}}
-\expandafter\ifx\csname urlstyle\endcsname\relax
-  \providecommand{\doi}[1]{doi: #1}\else
-  \providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi
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+  hospitalization scenarios.
+\newblock {\em Advances in Continuous and Discrete Models}, 2022(1), Oct. 2022.
 
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-\newblock {In: }\emph{Scientific Reports} 13 (2023), M{\^^b a}rz, Nr. 1.
-\newblock \url{http://dx.doi.org/10.1038/s41598-023-31714-5}. --
-\newblock DOI 10.1038/s41598--023--31714--5. --
-\newblock ISSN 2045--2322
-
-\bibitem[SRS21]{Shaier2021}
-\textsc{Shaier}, Sagi ; \textsc{Raissi}, Maziar  ; \textsc{Seshaiyer},
-  Padmanabhan:
-\newblock \emph{Data-driven approaches for predicting spread of infectious
-  diseases through DINNs: Disease Informed Neural Networks}
-
-\bibitem[TP85]{Tenenbaum1985}
-\textsc{Tenenbaum}, Morris ; \textsc{Pollard}, Harry:
-\newblock \emph{Ordinary Differential Equations}.
-\newblock Harper and Row, Publishers, Inc., 1985
+  for analyzing the covid-19 outbreaks in the world.
+\newblock {\em Scientific Reports}, 13(1), Mar. 2023.
+
+\bibitem{Shaier2021}
+S.~Shaier, M.~Raissi, and P.~Seshaiyer.
+\newblock Data-driven approaches for predicting spread of infectious diseases
+  through dinns: Disease informed neural networks, 2021.
+
+\bibitem{Smirnova2017}
+A.~Smirnova, L.~deCamp, and G.~Chowell.
+\newblock Forecasting epidemics through nonparametric estimation of
+  time-dependent transmission rates using the seir model.
+\newblock {\em Bulletin of Mathematical Biology}, 81(11):4343--4365, May 2017.
+
+\bibitem{SRD}
+{Statista Research Department}.
+\newblock Anzahl infektionen und todesfälle in zusammenhang mit dem
+  coronavirus (covid-19) in deutschland seit februar 2020.
+\newblock
+  https://de.statista.com/statistik/daten/studie/1102667/umfrage/erkrankungs-und-todesfaelle-aufgrund-des-coronavirus-in-deutschland/.
+\newblock {Accessed: 2024-09-06}.
+
+\bibitem{Tenenbaum1985}
+M.~Tenenbaum and H.~Pollard.
+\newblock {\em Ordinary Differential Equations}.
+\newblock Harper and Row, Publishers, Inc., 1985.
+
+\bibitem{WHO}
+WHO.
+\newblock Coronavirus disease (covid-19).
+\newblock \url{https://www.who.int/health-topics/coronavirus#tab=tab_1}.
+\newblock {Accessed: 2024-09-06}.
 
 \end{thebibliography}

thesis.bib

@@ -393,4 +393,94 @@
   publisher = {Elsevier BV},
 }
 
+@Misc{Kingma2014,
+  author    = {Kingma, Diederik P. and Ba, Jimmy},
+  title     = {Adam: A Method for Stochastic Optimization},
+  year      = {2014},
+  copyright = {arXiv.org perpetual, non-exclusive license},
+  doi       = {10.48550/ARXIV.1412.6980},
+  keywords  = {Machine Learning (cs.LG), FOS: Computer and information sciences, FOS: Computer and information sciences},
+  publisher = {arXiv},
+}
+
+@Misc{Paszke2019,
+  author    = {Paszke, Adam and Gross, Sam and Massa, Francisco and Lerer, Adam and Bradbury, James and Chanan, Gregory and Killeen, Trevor and Lin, Zeming and Gimelshein, Natalia and Antiga, Luca and Desmaison, Alban and Köpf, Andreas and Yang, Edward and DeVito, Zach and Raison, Martin and Tejani, Alykhan and Chilamkurthy, Sasank and Steiner, Benoit and Fang, Lu and Bai, Junjie and Chintala, Soumith},
+  title     = {PyTorch: An Imperative Style, High-Performance Deep Learning Library},
+  year      = {2019},
+  copyright = {arXiv.org perpetual, non-exclusive license},
+  doi       = {10.48550/ARXIV.1912.01703},
+  keywords  = {Machine Learning (cs.LG), Mathematical Software (cs.MS), Machine Learning (stat.ML), FOS: Computer and information sciences, FOS: Computer and information sciences},
+  publisher = {arXiv},
+}
+
+@Misc{GHInf,
+  author       = {RKI},
+  howpublished = {\url{https://github.com/robert-koch-institut/SARS-CoV-2-Infektionen_in_Deutschland}},
+  note         = {{Accessed: 2024-09-05}},
+  title        = {GitHub SARS-CoV-2 Infektionen in Deutschland},
+}
+
+@Misc{GHDead,
+  author       = {RKI},
+  howpublished = {\url{https://github.com/robert-koch-institut/COVID-19-Todesfaelle_in_Deutschland}},
+  note         = {{Accessed: 2024-09-05}},
+  title        = {GitHub COVID-19-Todesfälle in Deutschland},
+}
+
+@Misc{COVIDChronik,
+  author       = {{Federal Ministry of Health}},
+  howpublished = {\url{https://www.bundesgesundheitsministerium.de/coronavirus/chronik-coronavirus.html}},
+  note         = {{Accessed: 2024-09-05}},
+  title        = {Coronavirus-Pandemie: Was geschah wann?},
+}
+
+@Misc{COVInfo,
+  author       = {{Federal Centre for Health Education}},
+  howpublished = {\url{https://www.infektionsschutz.de/coronavirus/fragen-und-antworten/ansteckung-uebertragung-und-krankheitsverlauf/}},
+  note         = {{Accessed: 2024-09-05}},
+  title        = {Ansteckung, Übertragung und Krankheitsverlauf},
+}
+
+@Misc{WHO,
+  author       = {WHO},
+  howpublished = {\url{https://www.who.int/health-topics/coronavirus#tab=tab_1}},
+  note         = {{Accessed: 2024-09-06}},
+  title        = {Coronavirus disease (COVID-19)},
+}
+
+@Misc{RKI,
+  author       = {RKI},
+  howpublished = {\url{https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/ZS/Pandemieplan_Strategien.html}},
+  note         = {{Accessed: 2024-09-06}},
+  title        = {COVID-19-Strategiepapiere und Nationaler Pandemieplan},
+}
+
+@Misc{RKIa,
+  author       = {RKI},
+  howpublished = {\url{https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Virologische_Basisdaten.html?nn=13490888#doc14716546bodyText10}},
+  note         = {{Accessed: 2024-09-05}},
+  title        = {SARS-CoV-2: Virologische Basisdaten sowie Virusvarianten im Zeitraum von 2020 - 2022},
+}
+
+@Misc{SRD,
+  author       = {{Statista Research Department}},
+  howpublished = {https://de.statista.com/statistik/daten/studie/1102667/umfrage/erkrankungs-und-todesfaelle-aufgrund-des-coronavirus-in-deutschland/},
+  note         = {{Accessed: 2024-09-06}},
+  title        = {Anzahl Infektionen und Todesfälle in Zusammenhang mit dem Coronavirus (COVID-19) in Deutschland seit Februar 2020},
+}
+
+@Article{Kirchhoff1845,
+  author    = {Kirchhoff, Studiosus},
+  journal   = {Annalen der Physik},
+  title     = {Ueber den Durchgang eines elektrischen Stromes durch eine Ebene, insbesondere durch eine kreisförmige},
+  year      = {1845},
+  issn      = {1521-3889},
+  month     = jan,
+  number    = {4},
+  pages     = {497--514},
+  volume    = {140},
+  doi       = {10.1002/andp.18451400402},
+  publisher = {Wiley},
+}
+
 @Comment{jabref-meta: databaseType:bibtex;}