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@@ -17,13 +17,13 @@ the \emph{Robert Koch Institute} (RKI). The second section outlines the
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techniques we use to process this data to fit our project's requirements.
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Subsequently, we give a theoretical overview of the PINN's that we employ.
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These latter sections, establish the foundation for the implementations
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-described in~\Cref{sec:sir:setup} and~\Cref{sec:rsir:setup}.
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+used in~\Cref{sec:sir:setup} and~\Cref{sec:rsir:setup}.
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% -------------------------------------------------------------------
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\section{Epidemiological Data}
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\label{sec:preprocessing}
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-In this thesis we want to analyze the COVID-19 pandemic in Germany utilizing
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+In this thesis, we want to analyze the COVID-19 pandemic in Germany utilizing
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the SIR model and PINNs. For a PINN to learn the parameters of the SIR model,
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we need pandemic data in the correct format for the approach. Let $N_t$ be the
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number of training points, then let $i\in\{1, ..., N_t\}$
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@@ -44,7 +44,7 @@ employ to transform it into the correct structure.
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\subsection{RKI Data}
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\label{sec:preprocessing:rki}
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The RKI is a biomedical institute in Germany responsible for
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-the on monitoring and prevention of diseases. As the central institution of the
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+the monitoring and prevention of diseases. As the central institution of the
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German government in the field of biomedicine, one of its tasks during the
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COVID-19 pandemic was to track the number of infections and death cases in
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Germany. The data was collected by university hospitals, research facilities
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@@ -89,7 +89,7 @@ date is equivalent to the report date.\\
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The RKI assumes that the duration of the illness under normal conditions is 14 days,
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while the duration of severe cases is assumed to be 28 days. The recovery cases
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in the dataset are calculated using these assumptions, by adding the duration on
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-the reference date if it is given. As stated, the recovery data should be used
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+the reference date if it is given. As the RKI states, the recovery data should be used
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with caution. Since we require the recovery data for further calculations, the
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following section presents the solutions we employed to address this issue.
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@@ -99,7 +99,7 @@ following section presents the solutions we employed to address this issue.
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\label{sec:preprocessing:rq}
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At the outset of this section, we establish the format of the data, that is
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-necessary for training the PINNs. In this subsection, we present the method, that we
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+necessary for training the PINNs. In this section, we present the method, that we
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employ to preprocess and transform the RKI data (see~\Cref{sec:preprocessing:rki})
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into the training data. \\
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@@ -132,7 +132,7 @@ and releases them into the removed group $D$ days later.\\
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In order to solve the reduced SIR model, we employ a similar algorithm to that
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used for the SIR model. However, in contrast to the recovery queue, we utilize
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-a set recovery rate $\alpha$ to transfer a portion $\alpha\boldsymbol{I}^{(i)}$
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+a constant recovery rate $\alpha$ to transfer a portion $\alpha\boldsymbol{I}^{(i)}$
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of infections, which have recovered or died on the $i$'th day and put them into
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the $\boldsymbol{R}^{(i+1)}$ compartment of the next day, which is irrelevant to
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our purposes. The transformed data for both the SIR model and the reduced SIR
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@@ -172,7 +172,7 @@ for a set of time points, with the aim of minimizing the term,
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\begin{equation}\label{eq:SIR_obs_term}
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\Big\|\hat{\boldsymbol{S}}^{(i)}-\boldsymbol{S}^{(i)}\Big\|^2 + \Big\|\hat{\boldsymbol{I}}^{(i)}-\boldsymbol{I}^{(i)}\Big\|^2 + \Big\|\hat{\boldsymbol{R}}^{(i)}-\boldsymbol{R}^{(i)}\Big\|^2,
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\end{equation}
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-for each data point in the set of training dataset of a cardinality $N_t$ and with
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+for each data point in the set of training data points of a cardinality $N_t$ and with
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$i\in\{1, ..., N_t\}$. Moreover, the aforementioned parameters must satisfy the system
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of differential equations that govern the SIR model. For this reason, Shaier
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\etal~\cite{Shaier2021} utilize a PINN framework to satisfy both requirements.
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@@ -199,7 +199,7 @@ range of $[-1, 1]$. Therefore, we regularize the parameters using the
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\begin{equation}
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\tilde{\beta} = \tanh(\hat{\beta}),\quad \tilde{\alpha} = \tanh(\hat{\alpha}),
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\end{equation}
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-where $\tilde{\alpha}$ are regularized model predictions.\\
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+where $\tilde{\alpha}$ and $\hat{\beta}$ are regularized model predictions.\\
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The input data must include the time point $\boldsymbol{t}^{(i)}$ and its
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corresponding measured true values of $\Psi^{(i)}$.
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@@ -218,7 +218,7 @@ residual of each ODE. The mean square error of the residuals constitutes the
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physics loss
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$\mathcal{L}_{\text{physics}}(\boldsymbol{t}, \Psi, \hat{\Psi})$.
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The residuals are calculated using the model predictions $\hat{\Psi}$
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-and the regularized model predictions of the parameters, $\tilde{\beta}$ and $\tilde{\alpha}$.
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+and the regularized model predictions of the parameters, $\tilde{\alpha}$ and $\tilde{\beta}$.
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The residuals are given by,
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\begin{equation}
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0=\frac{d\hat{\boldsymbol{S}}}{d\boldsymbol{t}}+ \tilde{\beta}\frac{\hat{\boldsymbol{S}}\hat{\boldsymbol{I}}}{N}, \quad 0=\frac{d\hat{\boldsymbol{I}}}{d\boldsymbol{t}} - \tilde{\beta}\frac{\hat{\boldsymbol{S}}\hat{\boldsymbol{I}}}{N} + \tilde{\alpha}\hat{\boldsymbol{I}}, \quad 0=\frac{d\hat{\boldsymbol{R}}}{d\boldsymbol{t}} + \hat{\alpha}\hat{\boldsymbol{I}}.
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@@ -243,7 +243,7 @@ reproduction number $\Rt$ on the German data of the RKI.
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\label{sec:pinn:rsir}
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The previous section illustrates the methodology we employ to determine the
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-constant transmission and recovery rates from a data set obtained from
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+constant transmission and recovery rates from a data set obtained during
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the COVID-19 pandemic in Germany. In this section, we utilize PINNs to identify
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the time-dependent reproduction number, $\Rt$, while reducing the number of
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state variables and the reliance on assumptions, by decreasing the number of ODEs
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@@ -277,8 +277,8 @@ minimizing the term,
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\end{equation}
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for each $i\in\{1,...,N_t\}$. In order to identify the reproduction number, the
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PINN minimizes the residuals of the ODE during the training process. The
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-training process is analogous to the PINN, which identifies $\beta$
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-and $\alpha$ (see~\Cref{sec:pinn:sir}). However, there are two key differences. Firstly, the absence of
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+training process is analogous to the PINN, which identifies $\alpha$
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+and $\beta$ (see~\Cref{sec:pinn:sir}). However, there are two key differences. Firstly, the absence of
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free, trainable parameters. Secondly, the inclusion of an additional state variable that
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fluctuates in response to the input. While the state variable $\boldsymbol{I}$
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is approximated using the error between the training data and the predicted
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@@ -307,10 +307,10 @@ Then we train on composite loss function given by,
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to achieve a better solution.\\
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Although we set the transmission rate to be time-dependent, we define the
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-recovery time constant over time to reduce the complexity of the problem. The
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+recovery time to be constant over time to reduce the complexity of the problem. The
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RKI~\cite{GHInf} posits that the typical recovery period for the illness under
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normal conditions is 14 days, while those individuals with severe cases require
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-approximately 28 days to recover. As we assume the case with normal condition,
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+approximately 28 days to recover. As we assume the case with a normal condition,
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we can set the recovery time to $D=14$, which yields $\alpha = \nicefrac{1}{14}$.\\
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We perform extensive empirical evaluations of the methodology employed to
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