include cooments and change transition rates -> epidemiological parameters

chapters/chap01-introduction/chap01-introduction.tex

@@ -29,7 +29,7 @@ tools that allows for the comparison of disease courses across different disease
 and time points. The common approach in epidemiology to address this is the
 utilization of epidemiological models that approximate the dynamics by focusing
 on specific factors and modeling these using mathematical tools. These models
-provide transition rates and parameters that determine the behavior of a disease
+provide epidemiological parameters that determine the behavior of a disease
 within the boundaries of the model. A seminal epidemiological model, is the
 \emph{SIR model}, which was first proposed by Kermack and McKendrick~\cite{1927}
 in 1927. The SIR model is a compartmentalized model that divides the entire
@@ -39,21 +39,21 @@ In the context of the SIR model, the constant parameters of the transmission
 rate $\beta$ and the recovery rate $\alpha$ serve to quantify and determine the
 course of a pandemic. However, pandemic is not a static entity, therefor, Liu
 and Stechlinski~\cite{Liu2012}, and Setianto and Hidayat~\cite{Setianto2023},
-propose an SIR model with time-dependent transition rates and reproduction number $\Rt$. The SIR model
+propose an SIR model with time-dependent epidemiological parameters and reproduction number $\Rt$. The SIR model
 is defined by a system of differential equations, that incorporate
-the transition rates, thereby depicting the fluctuation between the three
-compartments. For a given set of data, the transition rate can be identified by
+the parameters $\alpha$ and $\beta$, thereby depicting the fluctuation between the three
+compartments. For a given set of data, the epidemiological parameters can be identified by
 solving the set of differential systems. Recently, the data-driven approach of
 \emph{physics-informed neural networks} (PINN) has gained attention due to its
 capability of finding solutions to differential equations by fitting its
 predictions to both given data and the governing system of differential
 equations. By employing this methodology, Shaier \etal~\cite{Shaier2021} were
-able to find the transition rate on data for different diseases. Additionally,
+able to find the epidemiological parameters on data for different diseases. Additionally,
 Millevoi \etal~\cite{Millevoi2023} were able to identify the reproduction number
 $\Rt$ 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 transition rates $\beta$ and
+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
 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
@@ -61,10 +61,10 @@ moralities from the beginning of the outbreak in Germany to the present. We
 utilize and preprocess this data according to the required format of our
 approaches. As the raw data lacks information on recovery incidence, we
 introduce the recovery queue that simulates a recovery period. To estimate the
-transition rates we adopt the approach of Shaier \etal~\cite{Shaier2021}, which
+epidemiological parameters we adopt the approach of Shaier \etal~\cite{Shaier2021}, which
 utilizes a physics-informed neural network learning the data, which consists of
 time point with their respective sizes of  the $S, I$ and $R$ compartments, to
-predict the transition rates based on the data and the governing system of
+predict the epidemiological parameters based on the data and the governing system of
 differential equations. Moreover, we utilize the methodology proposed by
 Millevoi \etal~\cite{Millevoi2023} that estimates the reproduction number for
 each day across the 1200-day span for each German state and Germany as a whole,
@@ -95,7 +95,7 @@ to ascertain the most reliable method for forecasting with limited data. Their
 findings indicate that modified TSVD provides the most stable forecasts on
 limited data, as demonstrated on both simulated data and real-world data from
 the 1918 influenza pandemic and the Ebola epidemic. In contrast, we
-utilize physics-informed neural networks (PINN) to find the constant transition rates
+utilize physics-informed neural networks (PINN) to find the constant epidemiological parameters
 and the reproduction number for Germany and its states\\
 
 Some related works similarly to us apply PINN approaches to COVID-19 and other
@@ -156,7 +156,7 @@ we present the PINN approaches, which are inspired by the work of Shaier \etal~\
 and Millevoi \etal~\cite{Millevoi2023}.~\Cref{chap:evaluation}
 presents the setups and results of the experiments that we conduct. This chapter
 is divided into two sections. The first section presents and discusses the
-results concerning the transition rates of $\beta$ and $\alpha$. The subsequent
+results concerning the epidemiological parameters of $\beta$ and $\alpha$. The subsequent
 section presents the results concerning the reproduction value $\Rt$. Finally,
 in \Cref{chap:conclusions}, we connect our results with the events of the
 real-world and give an overview of potential further work.

chapters/chap02/chap02.tex

@@ -201,7 +201,7 @@ or carry the disease.
   \label{fig:sir_model}
 \end{figure}
 As visualized in the~\Cref{fig:sir_model} the
-individuals may transition between groups based on transition rates. The
+individuals may transition between groups based on epidemiological parameters. The
 transmission rate $\beta$ is responsible for individuals becoming infected,
 while the rate of removal or recovery rate $\alpha$ (also referred to as
 $\delta$ or $\nu$, \eg,~\cite{EdelsteinKeshet2005,Millevoi2023}) moves
@@ -334,7 +334,7 @@ by the occurrence of deaths. One assumption, stated in the SIR model is that
 the size of the population, $N$, remains constant, as the daily change is
 negligible to the total population. Other examples include the impossibility
 for individuals to be susceptible again, after having recovered, or the
-possibility for the transition rates to change due to new variants or the
+possibility for the epidemiological parameters to change due to new variants or the
 implementation of new countermeasures. We address this latter option in the
 next~\Cref{sec:pandemicModel:rsir}.
 
@@ -354,7 +354,7 @@ infectivity or deadliness, or the administration of a vaccination that provides
 previously susceptible individuals with immunity without ever being infected.
 As these fine details of disease progression are missed in the constant rates,
 Liu and Stechlinski~\cite{Liu2012}, and Setianto and Hidayat~\cite{Setianto2023},
-introduce time-dependent transition rates and the time-dependent reproduction
+introduce time-dependent epidemiological parameters and the time-dependent reproduction
 number to address this issue. Millevoi \etal~\cite{Millevoi2023} present a
 reduced version of the SIR model.\\
 
@@ -384,7 +384,7 @@ defined as,
   \Rt=\frac{\beta(t)}{\alpha(t)}\cdot\frac{S(t)}{N},
 \end{equation}
 on the time interval $\mathcal{T}$ and the population site $N$. This definition
-includes the transition rates for information about the spread of the disease
+includes the epidemiological parameters for information about the spread of the disease
 and information of the decrease of the ratio of susceptible individuals in the
 population. In contrast to $\beta$ and $\alpha$, $\Rt$ is not a parameter but
 a state variable in the model, which gives information about the reproduction of the disease

chapters/chap03/chap03.tex

@@ -162,7 +162,7 @@ provides insight into the progression of the COVID-19 pandemic in Germany.
 The objective is to identify a function that solves the system of differential
 equations of the SIR model, by returning the size of each compartment at a
 specific point in time. This function is supposed to be able to reconstruct the
-training data and is defined by the values of the transition rates $\beta$ and
+training data and is defined by the values of the epidemiological parameters $\beta$ and
 $\alpha$. From a mathematical and semantic perspective, it is essential to
 determine these values of the parameter.\\
 
@@ -180,7 +180,7 @@ of differential equations that govern the SIR model. For this reason, Shaier
 \etal~\cite{Shaier2021} utilize a PINN framework to satisfy both requirements.
 Their approach, which they refer to as the \emph{disease-informed neural network}
 (see~\Cref{sec:pinn:dinn}), takes epidemiological data as the input and returns
-the two transition rates $\alpha$ and $\beta$. This method
+the two epidemiological parameters $\alpha$ and $\beta$. This method
 achieves this by finding an approximate solution of to the inverse problem of
 physics-informed neural networks (see~\Cref{sec:pinn}). In terms of the terms of
 the SIR model, a PINN addresses the inverse problem in two ways. First, it minimizes the mean of~\Cref{eq:SIR_obs_term}
@@ -192,7 +192,7 @@ inverse problem presets that a parameter is unknown. Thus, we designate the para
 $\beta$ and $\alpha$ as free, learnable parameters, $\widehat{\beta}$ and
 $\widehat{\alpha}$. These separate trainable parameters are values that are
 optimized during the training process and must fit the equations of the set of
-ODEs. Assuming that the values of the transition rates stay below
+ODEs. Assuming that the values of the epidemiological parameters stay below
 1~\cite{Shaier2021}, we force the value of both rates to be in a
 range of $[-1, 1]$. Therefor, we regularize the parameters using the
 \emph{tangens hyperbolicus}. This results in the terms,
@@ -271,7 +271,7 @@ of ODEs of the SIR model, we can derive one~\Cref{eq:reduced_sir_ODE}. In order
 to solve this, we must identify a function that maps a time point to the size
 of the infectious compartment and the specific reproduction number.\\
 
-As with the constant transition rates, we employ a data-driven approach for
+As with the constant epidemiological parameters, we employ a data-driven approach for
 identifying the time-dependent reproduction number $\Rt$. The PINN approximates
 the size $\boldsymbol{I}$ with its model prediction $\hat{\boldsymbol{I}}$ by
 minimizing the term,

chapters/chap04/chap04.tex

@@ -18,11 +18,11 @@ examine the reproduction number in synthetic and real-world data of Germany.
 
 % -------------------------------------------------------------------
 
-\section{Identifying the Transition Rates}
+\section{Identifying the Transmission and Recovery Rates}
 \label{sec:sir}
 In this section, we aim to identify the transmission rate $\beta$ and the
 recovery rate $\alpha$ from either synthetic or preprocessed real-world data.
-The methodology that we employ to identify the transition rates is described
+The methodology that we employ to identify the epidemiological parameters is described
 in~\Cref{sec:pinn:sir}. Meanwhile, the methods we utilize to preprocess the
 real-world data are detailed in~\Cref{sec:preprocessing:rq}. In the first part
 we present the setup of our experiments, then we provide the results including a
@@ -106,15 +106,15 @@ hidden layers with twenty neurons each, and an activation function of ReLU. We
 follow the hyperparameter setting in~\cite{Shaier2021} but change the base
 learning rate to $\expnumber{1}{-3}$. And employ a polynomial scheduler
 implementation from the PyTorch library~\cite{Paszke2019} instead. We train the
-model for 10000 epochs to extract the parameters. For each set of parameters, we
-conduct five iterations to demonstrate stability of the values. For measuring the
-accuracy, we calculate the error $e$, using the 2-Norm. Let $G$ be the set of
+model for 10000 iterations to extract the parameters. For each set of parameters, we
+conduct five runs to demonstrate stability of the values. For measuring the
+accuracy, we calculate the \emph{Relative L2 Error} $e$. Let $G$ be the set of
 compartment training data the SIR model with $\boldsymbol{g}\in G$ and $\hat{\boldsymbol{g}}$ be the
 corresponding model prediction, then,
 \begin{equation}
     e_{G} = \frac{1}{|G|}\sum_{g\in G}^{}\frac{\Big\|\hat{\boldsymbol{g}} - \boldsymbol{g}\Big\|_2}{\Big\|\boldsymbol{g}\Big\|_2},
 \end{equation}
-is the average error across all three groups.
+is the average error across all three compartments.
 
 % -------------------------------------------------------------------
 
@@ -217,7 +217,7 @@ the same states that exhibit a transmission rate exceeding the national value,
 have a higher recovery rate than the national standard, with the exception of
 Saxony. It is noteworthy that the recovery rates of all states exhibit a
 tendency to align with the recovery rate of $\alpha=\nicefrac{1}{14}$, which is
-equivalent to a recovery period of $D=\nicefrac{1}{\alpha}=14$ days. When
+equivalent to a recovery period of 14 days. When
 calculating the correlation coefficient between the predicted transmission rate
 and the vaccination ratio, we get a value of $-0.5134$. The strong negative
 correlation indicates that the transmission rate is high when the vaccination
@@ -234,7 +234,12 @@ function of the transmission rate $\beta$. This phenomenon occurs because the
 transmission rate determines the number of individuals that get infected per
 day, and the recovery queue moves a proportional number of people to the
 removed compartment. Consequently, a number of people defined by $\beta$ move
-to the $R$ compartment 14 days after they were infected.\\
+to the $R$ compartment 14 days after they were infected. Furthermore,
+in~\Cref{sec:pandemicModel:rsir}, we discussed the reproduction number $\Rt$,
+which describes the number of individuals infected by one infectious individual.
+This can be another reason for the observed correlation, as $\Rt$ depends on
+both $\alpha$ and $\beta$ (see~\Cref{eq:repr_num}), which illustrates that both
+parameters are influenced by changes to the reproductivity of the disease.\\
 
 This issue can be addressed by reducing the SIR model, thereby eliminating the
 significance of the $R$ compartment size. In the following section, we present
@@ -312,7 +317,7 @@ using a base learning rate of $\expnumber{1}{-3}$, with the same scheduler and
 optimizer as we describe in~\Cref{sec:sir:setup}. We train the model for the
 states 20000 epochs and start the physics training after 10000 epochs, while we
 train for Germany for 25000 and start the physics training after 15000 epochs.
-To reduce the standard deviation, each experiment is conducted 15 times. For
+To ensure the reliability of the results, we conduct ten trials of each experiment. For
 evaluation, we use the error $e_G$ as we do in the subsequent section.\\
 
 % -------------------------------------------------------------------
@@ -349,8 +354,8 @@ exhibits an upward trend, while during the final 120 days, the predictions
 demonstrate remarkable precision.\\
 
 In~\Cref{fig:state_results}, we present the graphs of $\Rt$ for the state with
-the highest value of $\beta$, namely Thuringia, and for the state with the lowest
-transmission rate $\beta$, namely Bremen. Further visualizations of the results
+the highest value of $\beta$, namely Thuringia, and for the state with the
+$\beta$, namely Bremen. Further visualizations of the results
 can be found in~\Cref{chap:appendix}. In all datasets, the graphs with $\alpha =
     \nicefrac{1}{5}$ are of a smaller size than those with
 $\alpha = \nicefrac{1}{14}$. This is due to the fact that the individuals are

chapters/conclusions/conclusions.tex

@@ -26,7 +26,7 @@ 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
+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.\\