correct section 3.3.2

chapters/chap03/chap03.tex

@@ -240,48 +240,49 @@ reproduction number $\Rt$ on the German data of the RKI.
 \section{Estimating the Reproduction Number using PINNs   2}
 \label{sec:pinn:rsir}
 
-The previous section, shows the methodology we utilize to ascertain the
-non-time-dependent transmission and recovery rates from a data set obtained from
-the COVID-19 pandemic in Germany. In this section we employ PINNs to identify
-the time-dependent reproduction number $\Rt$, while reducing the number of state
-variables and the reliance on assumptions, by reducing the system of ODEs
+The previous section illustrates the methodology we employ to detemine the
+constant transmission and recovery rates from a data set obtained from
+the COVID-19 pandemic in Germany. In this section, we utilize PINNs to identify
+the time-dependent reproduction number, $\Rt$, while reducing the number of
+state variables and the reliance on assumptions, by reducing the system of ODEs
 comprising the SIR model. The methodology presented in this section is based on
 the approach developed by Millevoi \etal~\cite{Millevoi2023}.\\
 
-In real-world pandemics the rate of infection is affected by a multitude of
-factors. Events like the rising awareness for the disease in the population, the
-implementation of non-pharmaceutical mitigations such as social distancing
-policies, and the emergence of a new variants have an impact on the transmission
-rate $\beta$. Accordingly, a transmission rate that is not time-dependent and
-constant across the whole duration of the pandemic may not accurately reflect
-the dynamics of the spread of a real-world disease correctly. Although we set
-the transmission rate to be time-dependent, the recovery time is assumed to be
-relatively constant in time. The Robert Koch
+In real-world pandemics, the rate of infection is influenced by a multitude of
+factors. Events such as the growing awareness for the disease among the general
+population, the introduction of non-pharmaceutical mitigations such as social
+distancing policies, and the emergence of a new variants have an impact on the
+transmission rate $\beta$. Accordingly, a transmission rate that is not
+time-dependent and constant across the entire duration of the pandemic may not
+accurately reflect the dynamics of the spread of a real-world disease correctly.
+Although we set the transmission rate to be time-dependent, the recovery time
+is assumed to be relatively constant over time. The Robert Koch
 Institute\footnote{\url{https://github.com/robert-koch-institut/SARS-CoV-2-Infektionen_in_Deutschland.git}}
 posits that the typical recovery period for the illness under normal conditions
-is 14 days, while those individuals with severe cases take about 28 days to
-recover. Given the negligible number of severe cases compared to the number of
-normal cases, we can set the recovery time to $D=14$ resulting in $\alpha = \nicefrac{1}{14}$.
-The reproduction number, $\Rt$ (see~\Cref{sec:pandemicModel:rsir}), represents
-the number of infections that occur as a result of one infectious individual. It
-indicates if a pandemic is emerging or if it is spreading rapidly through the
-susceptible population. By inserting the definition~\Cref{eq:repr_num}, into the
-system 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.\\
+is 14 days, while those individuals with severe cases require approximately 28
+days to recover. In the light of the negligible number of severe cases in
+comparison to the number of normal cases, we can set the recovery time to
+$D=14$, which yields $\alpha = \nicefrac{1}{14}$. The reproduction number,
+$\Rt$ (see~\Cref{sec:pandemicModel:rsir}), represents the number of new
+infections that occur as a result of one infectious individual. It indicates
+whether a pandemic is emerging or if it is spreading rapidly through the susceptible
+population. By inserting the definition of~\Cref{eq:repr_num}, into the system
+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
 identifying the time-dependent reproduction number $\Rt$. The PINN approximates
-the size ,$\boldsymbol{I}$, with its model prediction $\hat{\boldsymbol{I}}$ by
+the size $\boldsymbol{I}$ with its model prediction $\hat{\boldsymbol{I}}$ by
 minimizing the term,
 \begin{equation}\label{eq:rSir_squared_err}
     \Big\|\hat{\boldsymbol{I}}^{(i)}-\boldsymbol{I}^{(i)}\Big\|^2,
 \end{equation}
 for each $i\in\{1,...,N_t\}$. In order to identify the reproduction number, the
 PINN minimizes the residuals of the ODE during the training process. The
-training process is analogous to the one of the PINN, which identifies $\beta$
-and $alpha$ (see~\Cref{sec:pinn:sir}). The distinction lies in the absence of
-trainable parameters and the inclusion of an additional state variable that
+training process is analogous to that of the PINN, which identifies $\beta$
+and $\alpha$ (see~\Cref{sec:pinn:sir}). However, there are two key differences. Firstly, the absence of
+trainable parameters. Secondly, the inclusion of an additional state variable that
 fluctuates in response to the input. While the state variable $\boldsymbol{I}$
 is approximated using the error between the training data and the predicted
 values, the state variable $\Rt$ is approximated exclusively based on the