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@@ -177,7 +177,7 @@ individual. This is possible due to the distinction between infected individuals
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who are carriers of the disease and the part of the population, which is
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susceptible to infection. In the model, the mentioned groups are capable to
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change, \eg, healthy individuals becoming infected. The model assumes the
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-size $N$ of the population remains constant throughout the duration of the
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+size $N$ of the population to remain constant throughout the duration of the
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pandemic. The population $N$ comprises three distinct compartments: the
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\emph{susceptible} group $S$, the \emph{infectious} group $I$ and the
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\emph{removed} group $R$ (hence SIR model). Let $\mathcal{T} = [t_0, t_f]\subseteq
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@@ -319,7 +319,7 @@ in its demise before it can exert a significant influence on the population. In
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contrast, an early and high peak means that the disease is rapidly transmitted
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through the population, with a significant proportion of individuals having been
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infected.~\Cref{fig:sir_model} illustrates this effect by varying the values of
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-$\beta$ or $\alpha$ while simulating a pandemic using a model such
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+$\alpha$ and $\beta$ while simulating a pandemic using a model such
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as~\Cref{eq:modSIR}. It is evident that both the transmission rate $\beta$
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and the recovery rate $\alpha$ influence the height and time of the peak of $I$.
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When the number of infections exceeds the number of recoveries, the peak of $I$
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@@ -335,7 +335,7 @@ in the real-world is subject to a number of factors that can contribute to
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change. The population is increased by the occurrence of births and decreased
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by the occurrence of deaths. One assumption, stated in the SIR model is that
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the size of the population, $N$, remains constant, as the daily change is
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-negligible to the total population. Other examples include the impossibility
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+negligible compared to the total population. Other examples include the impossibility
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for individuals to be susceptible again, after having recovered, or the
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possibility for the epidemiological parameters to change due to new variants or the
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implementation of new countermeasures. We address this latter option in the
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@@ -345,7 +345,7 @@ next~\Cref{sec:pandemicModel:rsir}.
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\subsection{Reduced SIR Model and the Reproduction Number}
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\label{sec:pandemicModel:rsir}
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-The~\Cref{sec:pandemicModel:sir} presents the classical SIR model. This model
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+The~\Cref{sec:pandemicModel:sir} presents the classical SIR model by Kermack and McKendrick~\cite{1927}. This model
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contains two scalar parameters $\alpha$ and $\beta$, which describe the course
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of a pandemic over its duration. This is beneficial when examining the overall
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pandemic; however, in the real world, disease behavior is dynamic, and the
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@@ -366,18 +366,18 @@ they alter the definition of $\alpha$ and $\beta$ to be time-dependent,
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\begin{equation}
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\beta: \mathcal{T}\rightarrow\mathbb{R}_{\geq0}, \quad\alpha: \mathcal{T}\rightarrow\mathbb{R}_{\geq0}.
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\end{equation}
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-Another crucial element is $D(t) = \frac{1}{\alpha(t)}$, which represents the initial time
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+Another crucial element is $D(t) = \frac{1}{\alpha(t)}$, which represents the time
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span an infected individual requires to recuperate. Subsequently, at the initial time point
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$t_0$, the \emph{reproduction number},
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\begin{equation}
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\RO = \beta(t_0)D(t_0) = \frac{\beta(t_0)}{\alpha(t_0)},
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\end{equation}
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represents the number of susceptible individuals, that one infectious individual
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-infects at the onset of the pandemic. In light of the effects of $\beta$ and
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-$\alpha$ (see~\Cref{sec:pandemicModel:sir}), $\RO < 1$ indicates that the
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+infects at the onset of the pandemic. In light of the effects of $\alpha$ and
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+$\beta$ (see~\Cref{sec:pandemicModel:sir}), $\RO < 1$ indicates that the
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pandemic is emerging. In this scenario $\alpha$ is relatively low due to the
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limited number of infections resulting from $I(t_0) << S(t_0)$. Further,
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-$\RO > 1$ leads to the disease spreading rapidly across the population, with an
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+$\RO > 1$ leads to the disease spreading rapidly among the population, with an
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increase in $I$ occurring at a high rate. Nevertheless, $\RO$ does not cover
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the entire time span. For this reason, Millevoi \etal~\cite{Millevoi2023}
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introduce $\Rt$ which has the same interpretation as $\RO$, with the exception
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@@ -386,7 +386,7 @@ defined as,
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\begin{equation}\label{eq:repr_num}
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\Rt=\frac{\beta(t)}{\alpha(t)}\cdot\frac{S(t)}{N},
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\end{equation}
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-on the time interval $\mathcal{T}$ and the population site $N$. This definition
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+on the time interval $\mathcal{T}$ and the population size $N$. This definition
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includes the epidemiological parameters for information about the spread of the disease
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and information of the decrease of the ratio of susceptible individuals in the
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population. In contrast to $\alpha$ and $\beta$, $\Rt$ is not a parameter but
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@@ -419,7 +419,7 @@ variable $I$, results in,
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\end{equation}
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which is a further reduced version of~\Cref{eq:sir}. This less complex
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differential equation results in a less complex solution, as it entails the
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-elimination of a parameter ($\beta$) and the two state variables ($S$ and $R$).
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+elimination of a parameter ($\beta$) and two state variables ($S$ and $R$).
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The reduced SIR model is more precise due to fewer input variables, making it
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advantageous in situations with limited data, such as when recovery data is
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missing.
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