update conclusions + small other problems

chapters/chap01-introduction/chap01-introduction.tex

@@ -54,7 +54,7 @@ $\Rt$ 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 epidemiological parameters $\beta$ and
-$alpha$, as well as the reproduction number $\Rt$ of COVID-19 over the first
+$\alpha$, 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

chapters/chap02/chap02.tex

@@ -556,7 +556,7 @@ leveraging the available knowledge about the problem in the form of a system of
 differential equations.\\
 
 In contrast to standard MLP models, PINNs are not solely data-driven. The differential
-equation,
+equation,\todo{Sai Paper cite? https://www.mdpi.com/1424-8220/23/21/8665}
 \begin{equation}
   \boldsymbol{y}=\mathcal{D}(\boldsymbol{x}),
 \end{equation}
@@ -629,7 +629,7 @@ observations.\\
   \label{fig:spring}
 \end{figure}
 In order to illustrate the working of a PINN, we use the example of a
-\emph{damped harmonic oscillator} taken from~\cite{Moseley}. In this problem, we
+\emph{damped harmonic oscillator} taken from~\cite{Moseley}. In this problem, we\todo{cite Raissi?}
 displace a body, which is attached to a spring, from its resting position. The
 body is subject to three forces: firstly, the inertia exerted by the
 displacement $u$, which points in the direction of the displacement; secondly,

chapters/chap04/chap04.tex

@@ -159,7 +159,7 @@ five iterations of our experiment. We can observe that the error $e_{SIR}$ is
 the highest for \emph{Saxony} and the lowest for \emph{Lower Saxony}.
 Furthermore, we include the distance $\Delta\beta_{\text{Germany}} = \beta_{\text{state}} - \beta_{\text{Germany}}$
 and the percentage of people that have a basic immunity through vaccination
-$\nu$ for each state provided by the Robert Koch Institute\footnote{{\tiny\url{https://impfdashboard.de/}}}.\\
+$\nu$ for each state provided by the Robert Koch Institute~\cite{FMH}.\\
 
 \begin{table}[h]
     \begin{center}
@@ -167,7 +167,7 @@ $\nu$ for each state provided by the Robert Koch Institute\footnote{{\tiny\url{h
             $\Delta\beta_{\text{Germany}} = \beta_{\text{state}} - \beta_{\text{Germany}}$
             across the 5 iterations, that we conducted for each German state and Germany
             as the whole country. Furthermore we include the vaccination percentage
-            $\nu$ provided from the RKI.}
+            $\nu$ provided from the RKIe~\cite{FMH}.}
         \label{table:state_mean_std}
         \begin{tabular}{lccccc}
             \toprule

chapters/conclusions/conclusions.tex

@@ -9,103 +9,114 @@
 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \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. \\
 
-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
+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 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 epidemiological parameters 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.\\
+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.\\
 
-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 $\Rt$
+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.\\
+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.\\
 
-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.
+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.
 
 % -------------------------------------------------------------------
 
 \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).
+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.
 
 % -------------------------------------------------------------------
 
 \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
+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 incorporates an \emph{Exposed}
+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, 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}
+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 be used to
-approximate and consequently quantify a pandemic.
+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.
 
 % -------------------------------------------------------------------
 
@@ -130,6 +141,7 @@ 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.
+potential of utilizing ABMs as a tool to assess the pandemic's course for
+Germany.
 
 % -------------------------------------------------------------------

thesis.bbl

@@ -1,166 +1,171 @@
 \begin{thebibliography}{10}
 
-\bibitem{Anderson1991}
-R.~M. Anderson, Roy Malcolm;~May.
-\newblock {\em Infectious diseases of humans : dynamics and control}.
-\newblock Oxford University Press, 1991.
-
-\bibitem{Berkhahn2022}
-S.~Berkhahn and M.~Ehrhardt.
-\newblock A physics-informed neural network to model covid-19 infection and
-  hospitalization scenarios.
-\newblock {\em Advances in Continuous and Discrete Models}, 2022(1), Oct. 2022.
-
-\bibitem{Cooke1996}
-K.~L. Cooke and P.~van~den Driessche.
-\newblock Analysis of an seirs epidemic model with two delays.
-\newblock {\em Journal of Mathematical Biology}, 35(2):240--260, Dec. 1996.
-
-\bibitem{Demtroeder2021}
-W.~Demtröder.
-\newblock {\em Experimentalphysik 1}, volume~1 of {\em Lehrbuch}.
-\newblock Springer Spektrum, Berlin, 9. auflage edition, 2021.
-\newblock Auf dem Umschlag: Mit über 2,5 h Lösungsvideos zu ausgewählten
-  Aufgaben.
-
-\bibitem{Doerre2022}
-A.~Doerre and G.~Doblhammer.
-\newblock The influence of gender on covid-19 infections and mortality in
-  germany: Insights from age- and gender-specific modeling of contact rates,
-  infections, and deaths in the early phase of the pandemic.
-\newblock {\em PLOS ONE}, 17(5):e0268119, May 2022.
-
-\bibitem{EdelsteinKeshet2005}
-L.~Edelstein-Keshet.
-\newblock {\em Mathematical Models in Biology}.
-\newblock Society for Industrial and Applied Mathematics, 2005.
+\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}.
 
-\bibitem{COVInfo}
-{Federal Centre for Health Education}.
-\newblock Ansteckung, Übertragung und krankheitsverlauf.
+\bibitem{RKI}
+RKI.
+\newblock Covid-19-strategiepapiere und nationaler pandemieplan.
 \newblock
-  \url{https://www.infektionsschutz.de/coronavirus/fragen-und-antworten/ansteckung-uebertragung-und-krankheitsverlauf/}.
-\newblock {Accessed: 2024-09-05}.
+  \url{https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/ZS/Pandemieplan_Strategien.html}.
+\newblock {Accessed: 2024-09-06}.
 
-\bibitem{COVIDChronik}
-{Federal Ministry of Health}.
-\newblock Coronavirus-pandemie: Was geschah wann?
+\bibitem{RKIa}
+RKI.
+\newblock Sars-cov-2: Virologische basisdaten sowie virusvarianten im zeitraum
+  von 2020 - 2022.
 \newblock
-  \url{https://www.bundesgesundheitsministerium.de/coronavirus/chronik-coronavirus.html}.
+  \url{https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Virologische_Basisdaten.html?nn=13490888#doc14716546bodyText10}.
 \newblock {Accessed: 2024-09-05}.
 
-\bibitem{Gilbert2010}
-G.~N. Gilbert.
-\newblock {\em Agent-based models}.
-\newblock Number 153 in Quantitative applications in the social sciences. Sage
-  Publ., Los Angeles [u.a.], 3. pr. edition, 2010.
-
-\bibitem{Goodfellow-et-al-2016}
-I.~Goodfellow, Y.~Bengio, and A.~Courville.
-\newblock {\em Deep Learning}.
-\newblock MIT Press, 2016.
-\newblock \url{http://www.deeplearningbook.org}.
-
-\bibitem{Hornik1989}
-K.~Hornik, M.~Stinchcombe, and H.~White.
-\newblock Multilayer feedforward networks are universal approximators.
-\newblock {\em Neural Networks}, 2(5):359--366, Jan. 1989.
+\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{1927}
-W.~O. Kermack and A.~G. McKendrick.
+William~Ogilvy Kermack and A.~G. McKendrick.
 \newblock A contribution to the mathematical theory of epidemics.
 \newblock {\em Proceedings of the Royal Society of London. Series A, Containing
-  Papers of a Mathematical and Physical Character}, 115(772):700--721, Aug.
+  Papers of a Mathematical and Physical Character}, 115(772):700--721, August
   1927.
 
-\bibitem{Kerr2021}
-C.~C. Kerr, R.~M. Stuart, D.~Mistry, R.~G. Abeysuriya, K.~Rosenfeld, G.~R.
-  Hart, R.~C. Núñez, J.~A. Cohen, P.~Selvaraj, B.~Hagedorn, L.~George,
-  M.~Jastrzębski, A.~S. Izzo, G.~Fowler, A.~Palmer, D.~Delport, N.~Scott,
-  S.~L. Kelly, C.~S. Bennette, B.~G. Wagner, S.~T. Chang, A.~P. Oron, E.~A.
-  Wenger, J.~Panovska-Griffiths, M.~Famulare, and D.~J. Klein.
-\newblock Covasim: An agent-based model of covid-19 dynamics and interventions.
-\newblock {\em PLOS Computational Biology}, 17(7):e1009149, July 2021.
-
-\bibitem{Kirchhoff1845}
-S.~Kirchhoff.
-\newblock Ueber den durchgang eines elektrischen stromes durch eine ebene,
-  insbesondere durch eine kreisförmige.
-\newblock {\em Annalen der Physik}, 140(4):497--514, Jan. 1845.
-
-\bibitem{Lagaris1997}
-I.~E. Lagaris, A.~Likas, and D.~I. Fotiadis.
-\newblock Artificial neural networks for solving ordinary and partial
-  differential equations.
-\newblock 1997.
-
 \bibitem{Liu2012}
-X.~Liu and P.~Stechlinski.
+Xinzhi Liu and Peter Stechlinski.
 \newblock Infectious disease models with time-varying parameters and general
   nonlinear incidence rate.
 \newblock {\em Applied Mathematical Modelling}, 36(5):1974--1994, May 2012.
 
-\bibitem{Maziarz2020}
-M.~Maziarz and M.~Zach.
-\newblock Agent‐based modelling for sars‐cov‐2 epidemic prediction and
-  intervention assessment: A methodological appraisal.
-\newblock {\em Journal of Evaluation in Clinical Practice}, 26(5):1352--1360,
-  Aug. 2020.
+\bibitem{Setianto2023}
+Setianto Setianto and Darmawan Hidayat.
+\newblock Modeling the time-dependent transmission rate using gaussian pulses
+  for analyzing the covid-19 outbreaks in the world.
+\newblock {\em Scientific Reports}, 13(1), March 2023.
+
+\bibitem{Shaier2021}
+Sagi Shaier, Maziar Raissi, and Padmanabhan Seshaiyer.
+\newblock Data-driven approaches for predicting spread of infectious diseases
+  through dinns: Disease informed neural networks, 2021.
 
 \bibitem{Millevoi2023}
-C.~Millevoi, D.~Pasetto, and M.~Ferronato.
+Caterina Millevoi, Damiano Pasetto, and Massimiliano Ferronato.
 \newblock A physics-informed neural network approach for compartmental
   epidemiological models.
 \newblock 2023.
 
-\bibitem{Minsky1972}
-M.~Minsky and S.~A. Papert.
-\newblock {\em Perceptrons}.
-\newblock The MIT Press, Cambridge/Mass. [u.a.], 2. print. with corr edition,
-  1972.
-\newblock Literaturangaben.
+\bibitem{Smirnova2017}
+Alexandra Smirnova, Linda deCamp, and Gerardo 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{Moseley}
-B.~Moseley.
-\newblock So, what is a physics-informed neural network?
-\newblock
-  \url{https://benmoseley.blog/my-research/so-what-is-a-physics-informed-neural-network/}.
-\newblock {Accessed: 2024-09-08}.
+\bibitem{Berkhahn2022}
+Sarah Berkhahn and Matthias Ehrhardt.
+\newblock A physics-informed neural network to model covid-19 infection and
+  hospitalization scenarios.
+\newblock {\em Advances in Continuous and Discrete Models}, 2022(1), October
+  2022.
+
+\bibitem{Olumoyin2021}
+K.~D. Olumoyin, A.~Q.~M. Khaliq, and K.~M. Furati.
+\newblock Data-driven deep-learning algorithm for asymptomatic covid-19 model
+  with varying mitigation measures and transmission rate.
+\newblock {\em Epidemiologia}, 2(4):471--489, September 2021.
+
+\bibitem{Rudin2007}
+Walter Rudin.
+\newblock {\em Analysis}.
+\newblock Oldenbourg Wissenschaftsverlag GmbH, 2007.
+
+\bibitem{Tenenbaum1985}
+Morris Tenenbaum and Harry Pollard.
+\newblock {\em Ordinary Differential Equations}.
+\newblock Harper and Row, Publishers, Inc., 1985.
+
+\bibitem{Demtroeder2021}
+Wolfgang Demtröder.
+\newblock {\em Experimentalphysik 1}, volume~1 of {\em Lehrbuch}.
+\newblock Springer Spektrum, Berlin, 9. auflage edition, 2021.
+\newblock Auf dem Umschlag: Mit über 2,5 h Lösungsvideos zu ausgewählten
+  Aufgaben.
+
+\bibitem{Kirchhoff1845}
+Studiosus Kirchhoff.
+\newblock Ueber den durchgang eines elektrischen stromes durch eine ebene,
+  insbesondere durch eine kreisförmige.
+\newblock {\em Annalen der Physik}, 140(4):497--514, January 1845.
 
 \bibitem{Oksendal2000}
-B.~Oksendal.
+Bernt Oksendal.
 \newblock {\em Stochastic Differential Equations}.
 \newblock Universitext Ser. Springer Berlin / Heidelberg, Berlin, Heidelberg,
   5th ed. edition, 2000.
 \newblock Description based on publisher supplied metadata and other sources.
 
-\bibitem{Olumoyin2021}
-K.~D. Olumoyin, A.~Q.~M. Khaliq, and K.~M. Furati.
-\newblock Data-driven deep-learning algorithm for asymptomatic covid-19 model
-  with varying mitigation measures and transmission rate.
-\newblock {\em Epidemiologia}, 2(4):471--489, Sept. 2021.
+\bibitem{EdelsteinKeshet2005}
+Leah Edelstein-Keshet.
+\newblock {\em Mathematical Models in Biology}.
+\newblock Society for Industrial and Applied Mathematics, 2005.
 
-\bibitem{Paszke2019}
-A.~Paszke, S.~Gross, F.~Massa, A.~Lerer, J.~Bradbury, G.~Chanan, T.~Killeen,
-  Z.~Lin, N.~Gimelshein, L.~Antiga, A.~Desmaison, A.~Köpf, E.~Yang, Z.~DeVito,
-  M.~Raison, A.~Tejani, S.~Chilamkurthy, B.~Steiner, L.~Fang, J.~Bai, and
-  S.~Chintala.
-\newblock Pytorch: An imperative style, high-performance deep learning library,
-  2019.
+\bibitem{Anderson1991}
+Robert~M. Anderson, Roy Malcolm;~May.
+\newblock {\em Infectious diseases of humans : dynamics and control}.
+\newblock Oxford University Press, 1991.
+
+\bibitem{Rumelhart1986}
+David~E. Rumelhart, Geoffrey~E. Hinton, and Ronald~J. Williams.
+\newblock Learning representations by back-propagating errors.
+\newblock {\em Nature}, 323(6088):533--536, October 1986.
+
+\bibitem{Goodfellow-et-al-2016}
+Ian Goodfellow, Yoshua Bengio, and Aaron Courville.
+\newblock {\em Deep Learning}.
+\newblock MIT Press, 2016.
+\newblock \url{http://www.deeplearningbook.org}.
+
+\bibitem{Rosenblatt1958}
+F.~Rosenblatt.
+\newblock The perceptron: A probabilistic model for information storage and
+  organization in the brain.
+\newblock {\em Psychological Review}, 65(6):386--408, 1958.
+
+\bibitem{Minsky1972}
+Marvin Minsky and Seymour~A. Papert.
+\newblock {\em Perceptrons}.
+\newblock The MIT Press, Cambridge/Mass. [u.a.], 2. print. with corr edition,
+  1972.
+\newblock Literaturangaben.
+
+\bibitem{Hornik1989}
+Kurt Hornik, Maxwell Stinchcombe, and Halbert White.
+\newblock Multilayer feedforward networks are universal approximators.
+\newblock {\em Neural Networks}, 2(5):359--366, January 1989.
+
+\bibitem{Lagaris1997}
+I.~E. Lagaris, A.~Likas, and D.~I. Fotiadis.
+\newblock Artificial neural networks for solving ordinary and partial
+  differential equations.
+\newblock 1997.
 
 \bibitem{Raissi2019}
-M.~Raissi, P.~Perdikaris, and G.~Karniadakis.
+M.~Raissi, P.~Perdikaris, and G.E. Karniadakis.
 \newblock Physics-informed neural networks: A deep learning framework for
   solving forward and inverse problems involving nonlinear partial differential
   equations.
-\newblock {\em Journal of Computational Physics}, 378:686--707, Feb. 2019.
+\newblock {\em Journal of Computational Physics}, 378:686--707, February 2019.
 
-\bibitem{RKI}
-RKI.
-\newblock Covid-19-strategiepapiere und nationaler pandemieplan.
+\bibitem{Moseley}
+Ben Moseley.
+\newblock So, what is a physics-informed neural network?
 \newblock
-  \url{https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/ZS/Pandemieplan_Strategien.html}.
-\newblock {Accessed: 2024-09-06}.
+  \url{https://benmoseley.blog/my-research/so-what-is-a-physics-informed-neural-network/}.
+\newblock {Accessed: 2024-09-08}.
 
 \bibitem{GHDead}
 RKI.
@@ -176,64 +181,83 @@ RKI.
   \url{https://github.com/robert-koch-institut/SARS-CoV-2-Infektionen_in_Deutschland}.
 \newblock {Accessed: 2024-09-05}.
 
-\bibitem{RKIa}
-RKI.
-\newblock Sars-cov-2: Virologische basisdaten sowie virusvarianten im zeitraum
-  von 2020 - 2022.
+\bibitem{Paszke2019}
+Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory
+  Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban
+  Desmaison, Andreas Köpf, Edward Yang, Zach DeVito, Martin Raison, Alykhan
+  Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu~Fang, Junjie Bai, and Soumith
+  Chintala.
+\newblock Pytorch: An imperative style, high-performance deep learning library,
+  2019.
+
+\bibitem{FMH}
+{Federal Ministry of Health}.
+\newblock Übersicht zum impfstatus - covid-19-impfung in deutschland bis zum
+  8. april 2023.
+\newblock \url{https://impfdashboard.de/}.
+\newblock {Accessed: 2024-09-08}.
+
+\bibitem{COVInfo}
+{Federal Centre for Health Education}.
+\newblock Ansteckung, Übertragung und krankheitsverlauf.
 \newblock
-  \url{https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Virologische_Basisdaten.html?nn=13490888#doc14716546bodyText10}.
+  \url{https://www.infektionsschutz.de/coronavirus/fragen-und-antworten/ansteckung-uebertragung-und-krankheitsverlauf/}.
 \newblock {Accessed: 2024-09-05}.
 
-\bibitem{Rosenblatt1958}
-F.~Rosenblatt.
-\newblock The perceptron: A probabilistic model for information storage and
-  organization in the brain.
-\newblock {\em Psychological Review}, 65(6):386--408, 1958.
-
-\bibitem{Rudin2007}
-W.~Rudin.
-\newblock {\em Analysis}.
-\newblock Oldenbourg Wissenschaftsverlag GmbH, 2007.
+\bibitem{COVIDChronik}
+{Federal Ministry of Health}.
+\newblock Coronavirus-pandemie: Was geschah wann?
+\newblock
+  \url{https://www.bundesgesundheitsministerium.de/coronavirus/chronik-coronavirus.html}.
+\newblock {Accessed: 2024-09-05}.
 
-\bibitem{Rumelhart1986}
-D.~E. Rumelhart, G.~E. Hinton, and R.~J. Williams.
-\newblock Learning representations by back-propagating errors.
-\newblock {\em Nature}, 323(6088):533--536, Oct. 1986.
+\bibitem{Desson2022}
+Zachary Desson, Lukas Kauer, Thomas Otten, Jan~Willem Peters, and Francesco
+  Paolucci.
+\newblock Finding the way forward: Covid-19 vaccination progress in germany,
+  austria and switzerland.
+\newblock {\em Health Policy and Technology}, 11(2):100584, June 2022.
 
-\bibitem{Setianto2023}
-S.~Setianto and D.~Hidayat.
-\newblock Modeling the time-dependent transmission rate using gaussian pulses
-  for analyzing the covid-19 outbreaks in the world.
-\newblock {\em Scientific Reports}, 13(1), Mar. 2023.
+\bibitem{Korolev2021}
+Ivan Korolev.
+\newblock Identification and estimation of the seird epidemic model for
+  covid-19.
+\newblock {\em Journal of Econometrics}, 220(1):63--85, January 2021.
 
-\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{Doerre2022}
+Achim Doerre and Gabriele Doblhammer.
+\newblock The influence of gender on covid-19 infections and mortality in
+  germany: Insights from age- and gender-specific modeling of contact rates,
+  infections, and deaths in the early phase of the pandemic.
+\newblock {\em PLOS ONE}, 17(5):e0268119, May 2022.
 
-\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{Cooke1996}
+K.~L. Cooke and P.~van~den Driessche.
+\newblock Analysis of an seirs epidemic model with two delays.
+\newblock {\em Journal of Mathematical Biology}, 35(2):240--260, December 1996.
 
-\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{Gilbert2010}
+G.~Nigel Gilbert.
+\newblock {\em Agent-based models}.
+\newblock Number 153 in Quantitative applications in the social sciences. Sage
+  Publ., Los Angeles [u.a.], 3. pr. edition, 2010.
 
-\bibitem{Tenenbaum1985}
-M.~Tenenbaum and H.~Pollard.
-\newblock {\em Ordinary Differential Equations}.
-\newblock Harper and Row, Publishers, Inc., 1985.
+\bibitem{Kerr2021}
+Cliff~C. Kerr, Robyn~M. Stuart, Dina Mistry, Romesh~G. Abeysuriya, Katherine
+  Rosenfeld, Gregory~R. Hart, Rafael~C. Núñez, Jamie~A. Cohen, Prashanth
+  Selvaraj, Brittany Hagedorn, Lauren George, Michał Jastrzębski, Amanda~S.
+  Izzo, Greer Fowler, Anna Palmer, Dominic Delport, Nick Scott, Sherrie~L.
+  Kelly, Caroline~S. Bennette, Bradley~G. Wagner, Stewart~T. Chang, Assaf~P.
+  Oron, Edward~A. Wenger, Jasmina Panovska-Griffiths, Michael Famulare, and
+  Daniel~J. Klein.
+\newblock Covasim: An agent-based model of covid-19 dynamics and interventions.
+\newblock {\em PLOS Computational Biology}, 17(7):e1009149, July 2021.
 
-\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}.
+\bibitem{Maziarz2020}
+Mariusz Maziarz and Martin Zach.
+\newblock Agent‐based modelling for sars‐cov‐2 epidemic prediction and
+  intervention assessment: A methodological appraisal.
+\newblock {\em Journal of Evaluation in Clinical Practice}, 26(5):1352--1360,
+  August 2020.
 
 \end{thebibliography}

thesis.bib

@@ -493,4 +493,39 @@
   title        = {So, what is a physics-informed neural network?},
 }
 
+@Misc{FMH,
+  author       = {{Federal Ministry of Health}},
+  howpublished = {\url{https://impfdashboard.de/}},
+  note         = {{Accessed: 2024-09-08}},
+  title        = {Übersicht zum Impfstatus - COVID-19-Impfung in Deutschland bis zum 8. April 2023},
+}
+
+@Article{Desson2022,
+  author    = {Desson, Zachary and Kauer, Lukas and Otten, Thomas and Peters, Jan Willem and Paolucci, Francesco},
+  journal   = {Health Policy and Technology},
+  title     = {Finding the way forward: COVID-19 vaccination progress in Germany, Austria and Switzerland},
+  year      = {2022},
+  issn      = {2211-8837},
+  month     = jun,
+  number    = {2},
+  pages     = {100584},
+  volume    = {11},
+  doi       = {10.1016/j.hlpt.2021.100584},
+  publisher = {Elsevier BV},
+}
+
+@Article{Korolev2021,
+  author    = {Korolev, Ivan},
+  journal   = {Journal of Econometrics},
+  title     = {Identification and estimation of the SEIRD epidemic model for COVID-19},
+  year      = {2021},
+  issn      = {0304-4076},
+  month     = jan,
+  number    = {1},
+  pages     = {63--85},
+  volume    = {220},
+  doi       = {10.1016/j.jeconom.2020.07.038},
+  publisher = {Elsevier BV},
+}
+
 @Comment{jabref-meta: databaseType:bibtex;}

thesis.tex

@@ -81,7 +81,7 @@
 
 \interlinepenalty10000 % so no bib-entry will be separated by a pagebreak
 \bibliography{thesis.bib}
-\bibliographystyle{abbrv} % change the bib-style if you want to
+\bibliographystyle{unsrt}%abbrv} % change the bib-style if you want to
 
 \listoffigures
 \listoftables