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\label{sec:relatedWork}
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In \emph{Forecasting Epidemics Through Nonparametric Estimation of
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Time-Dependent Transmission Rates Using the SEIR Model}~\cite{Smirnova2017},
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-Smirnova \etal seek to find a stochastic method to estimate the time-dependend
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-transmission rate $\beta(t)$, that on the contrary to earlier studies does not
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-rely on parameters. They achieve this by projecting on a finite subspace, that
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-is defined by Legendre polynomials.
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+Smirnova \etal endeavor to identify a stochastic methodology for estimating the
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+time-dependent transmission rate $\beta(t)$. This is in response to the
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+limitations of earlier parametric estimation methods, which are prone
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+instability due to the difficulty in identifying parameter finding and a low
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+amount of available data. They achieve this by projecting the time-dependent
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+transmission rate onto a finite subspace, that is defined by Legendre
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+polynomials. Subsequently, they compare the three regularization techniques of
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+variational (Tikhonov’s) regularization, truncated singular value decomposition
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+(TSVD), and modified TSVD to ascertain the most reliable method for forecasting
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+with limited data. Their findings indicate that modified TSVD provides the most
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+stable forecasts on limited data, as demonstrated on both simulated data and
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+real-world data from the 1918 influenza pandemic and the 2014-2015 Ebola
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+epidemic.\\
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+
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+In their publication, entitled \emph{Data-driven approaches for predicting
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+ spread of infectious diseases through DINNs: Disease Informed Neural Networks},
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+Shaier \etal~\cite{Shaier2021} put forth a data-driven approach for identifying
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+the parameters of epidemiological models. The authors apply physics-informed
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+neural networks to the compartmental SIR models, and refer to their method as
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+disease informed neural networks (DINN). In their work, they demonstrate the
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+capacity of DINNs to forecast the trajectory of epidemics and pandemics. They
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+underpin the efficacy of their approach by applying it to 11 diseases, that have
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+previously been modeled, including examples such as COVID, HIV, Tuberculosis and
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+Ebola. In their experiments they employ the SIDR (susceptible, infectious, dead,
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+recovered) model. Finally, they present that this method is a robust and
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+effective means of identifying the parameters of a SIR model.\\
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+
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+In their article \emph{A physics-informed neural network to model COVID-19
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+ infection and hospitalization scenarios}, Berkhahn and Ehrhard~\cite{Berkhahn2022}
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+employ the susceptible, vaccinated, infectious, hospitalized and removed (SVIHR)
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+model. They solve the system of differential equations inherent to the SVIHR
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+model by the means of PINNs. The authors utilize a dataset of German COVID-19
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+data, covering the time span from the inceptions of the outbreak to the end of
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+2021. The proposed PINN methodology initially estimates the SVIHR model
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+parameters and subsequently forecasts the data. For comparative purposes,
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+Berkhahn and Ehrhard employ the method of non-standard finite differences (NSFD)
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+as well. In the validation process, the two forecasting methods project the
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+trajectory of COVID-19 from mid-April onwards. Berkhahn and Ehrhard find that
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+the PINN is able to adapt to varying vaccination rates and emerging variants.\\
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+
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+In their work, \emph{Data-Driven Deep-Learning Algorithm for Asymptomatic
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+ COVID-19 Model with Varying Mitigation Measures and Transmission Rate},
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+Olumoyin \etal~\cite{Olumoyin2021} employ an alternative methodology for
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+identifying the time-dependent transmission rate of an asymptomatic-SIR model.
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+On the premise that not all the infectious individuals are reported and included
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+in the data available. The algorithm they introduce, utilizes the cumulative and
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+daily reported infection cases and symptomatic recovered cases, to demonstrate
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+the effect of different mitigation measures and to ascertain the size of the
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+part of non-symptomatic individuals in the total number of infective individuals
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+and the proportion of asymptomatic recovered individuals. With this they can
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+illustrate the influence of vaccination and a set non-pharmaceutical mitigation
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+methods on the transmission of COVID-19 on data from Italy, South Korea, the
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+United Kingdom, and the United States.\\
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+
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+In \emph{A Physics-Informed Neural Network approach for compartmental
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+ epidemiological models} Millevoi \etal~\cite{Millevoi2023} address the issue
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+of describing the dynamically changing transmission rate, which is influenced by
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+the emergence of new variants or the implementation of non-pharmaceutical
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+measures. They employ a PINN to maintain an account of the changes of the
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+transmission rate included in the reproduction number and to approximate the
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+model state variables. To this end, Millevoi \etal employ the reproduction
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+number to reduce the system of differential equations to a single equation and
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+introduce a reduced-split version of the PINN, which initially trains on the
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+data and then trains to minimize the residual of the ODE. They test their
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+approach on five synthetic and two real-world scenarios from the early stages of
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+the COVID-19 pandemic in Italy. This method yields an increase in both accuracy
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+and training speed.
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+
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% -------------------------------------------------------------------
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Phillip Rothenbeck