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% Author:   Phillip Rothenbeck
% Title:    Investigating the Evolution of the COVID-19 Pandemic in Germany Using Physics-Informed Neural Networks
% File:     chap01-introduction/chap01-introduction.tex
% Part:     introduction
% Description:
%         summary of the content in this chapter
% Version:  01.01.2012
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\chapter{Introduction   5}
\label{chap:introduction}

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\section{Related work   2}
\label{sec:relatedWork}
In \emph{Forecasting Epidemics Through Nonparametric Estimation of
    Time-Dependent Transmission Rates Using the SEIR Model}~\cite{Smirnova2017},
Smirnova \etal endeavor to identify a stochastic methodology for estimating the
time-dependent transmission rate $\beta(t)$. This is in response to the
limitations of earlier parametric estimation methods, which are prone
instability due to the difficulty in identifying parameter finding and a low
amount of available data. They achieve this by projecting the time-dependent
transmission rate onto a finite subspace, that is defined by Legendre
polynomials. Subsequently, they compare the three regularization techniques of
variational (Tikhonov’s) regularization, truncated singular value decomposition
(TSVD), and modified TSVD 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 2014-2015 Ebola
epidemic.\\

In their publication, entitled \emph{Data-driven approaches for predicting
    spread of infectious diseases through DINNs: Disease Informed Neural Networks},
Shaier \etal~\cite{Shaier2021} put forth a data-driven approach for identifying
the parameters of epidemiological models. The authors apply physics-informed
neural networks to the compartmental SIR models, and refer to their method as
disease informed neural networks (DINN). In their work, they demonstrate the
capacity of DINNs to forecast the trajectory of epidemics and pandemics. They
underpin the efficacy of their approach by applying it to 11 diseases, that have
previously been modeled, including examples such as COVID, HIV, Tuberculosis and
Ebola. In their experiments they employ the SIDR (susceptible, infectious, dead,
recovered) model. Finally, they present that this method is a robust and
effective means of identifying the parameters of a SIR model.\\

In their article \emph{A physics-informed neural network to model COVID-19
    infection and hospitalization scenarios}, Berkhahn and Ehrhard~\cite{Berkhahn2022}
employ the susceptible, vaccinated, infectious, hospitalized and removed (SVIHR)
model. They solve the system of differential equations inherent to the SVIHR
model by the means of PINNs. The authors utilize a dataset of German COVID-19
data, covering the time span from the inceptions of the outbreak to the end of
2021. The proposed PINN methodology initially estimates the SVIHR model
parameters and subsequently forecasts the data. For comparative purposes,
Berkhahn and Ehrhard employ the method of non-standard finite differences (NSFD)
as well. In the validation process, the two forecasting methods project the
trajectory of COVID-19 from mid-April onwards. Berkhahn and Ehrhard find that
the PINN is able to adapt to varying vaccination rates and emerging variants.\\

In their work, \emph{Data-Driven Deep-Learning Algorithm for Asymptomatic
    COVID-19 Model with Varying Mitigation Measures and Transmission Rate},
Olumoyin \etal~\cite{Olumoyin2021} employ an alternative methodology for
identifying the time-dependent transmission rate of an asymptomatic-SIR model.
On the premise that not all the infectious individuals are reported and included
in the data available. The algorithm they introduce, utilizes the cumulative and
daily reported infection cases and symptomatic recovered cases, to demonstrate
the effect of different mitigation measures and to ascertain the size of the
part of non-symptomatic individuals in the total number of infective individuals
and the proportion of asymptomatic recovered individuals. With this they can
illustrate the influence of vaccination and a set non-pharmaceutical mitigation
methods on the transmission of COVID-19 on data from Italy, South Korea, the
United Kingdom, and the United States.\\

In \emph{A Physics-Informed Neural Network approach for compartmental
    epidemiological models} Millevoi \etal~\cite{Millevoi2023} address the issue
of describing the dynamically changing transmission rate, which is influenced by
the emergence of new variants or the implementation of non-pharmaceutical
measures. They employ a PINN to maintain an account of the changes of the
transmission rate included in the reproduction number and to approximate the
model state variables. To this end, Millevoi \etal employ the reproduction
number to reduce the system of differential equations to a single equation and
introduce a reduced-split version of the PINN, which initially trains on the
data and then trains to minimize the residual of the ODE. They test their
approach on five synthetic and two real-world scenarios from the early stages of
the COVID-19 pandemic in Italy. This method yields an increase in both accuracy
and training speed.





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