adding .Tex version of hands-out file (#56)

* adding .Tex version of hands-out file * add more questions * Update learners/files/hands-on.tex Co-authored-by: Andree Valle Campos <[email protected]> * Update learners/files/hands-on.tex Co-authored-by: Andree Valle Campos <[email protected]> * Update learners/files/hands-on.tex Co-authored-by: Andree Valle Campos <[email protected]> * Update learners/files/hands-on.tex Co-authored-by: Andree Valle Campos <[email protected]> * Update learners/files/hands-on.tex Co-authored-by: Andree Valle Campos <[email protected]> * Update learners/files/hands-on.tex Co-authored-by: Andree Valle Campos <[email protected]> * Update learners/files/hands-on.tex Co-authored-by: Andree Valle Campos <[email protected]> * Update learners/files/hands-on.tex Co-authored-by: Andree Valle Campos <[email protected]> * Update learners/files/hands-on.tex Co-authored-by: Andree Valle Campos <[email protected]> * Update learners/files/hands-on.tex Co-authored-by: Andree Valle Campos <[email protected]> --------- Co-authored-by: Andree Valle Campos <[email protected]>
epiverse-trace · Nov 20, 2024 · bdbc25a · bdbc25a
1 parent eb29b8f
commit bdbc25a
Show file tree

Hide file tree

Showing 2 changed files with 230 additions and 67 deletions.
diff --git a/learners/files/hands-on.tex b/learners/files/hands-on.tex
@@ -0,0 +1,159 @@
+\documentclass{article}
+\usepackage{fancyhdr}
+
+% Set up the custom footer
+\pagestyle{fancy}
+\fancyfoot[R]{\thepage} % Centered page number in footer
+\fancyfoot[C]{\textbf{License:} CC-BY.\textbf{Copyright:} Andree \& Degoot, 2024 }
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta, positioning}
+\usepackage{amsmath}
+\author{Andree Valle Campos and Abdoelnaser M Degoot \\ Epiverse-TRACE Team @ LSHTM }
+\title{Simple Introduction to Mathematical Modelling of Infectious Diseases}
+\begin{document}
+\maketitle
+
+\section{Introduction}
+This practical aims to assess your understanding of the fundamental 
+principles of mathematical modeling while guiding you in constructing models using 
+a simple SEIR framework for infectious disease outbreaks.\\
+
+ \textbf{Note: Please fill in the blanks.}
+
+\section{SEIR Model}
+
+ In the  SEIR model, we have four compartments (\( S \), \( E \), \( I \), \( R \)):
+
+\begin{itemize}
+    \item \( S \) stands for \underline{\hspace{2cm}}, meaning \underline{\hspace{6cm}}.
+The parameter that explains the transition from  (\( S \)) compartment 
+to  (\( E \)) compartment is \underline{\hspace{6cm}}.
+\item \(E\) stands for \underline{\hspace{2cm}}, meaning that it can 
+    \underline{\hspace{4cm}}. 
+
+    The rate that explains the transition from  (\( E \)) to  (\( I \)) is the rate of \underline{\hspace{1cm}}.
+
+    \item \( I \) stands for \underline{\hspace{2cm}}, meaning that it can 
+    \underline{\hspace{3cm}}.
+    The rate that explains the transition from  (\( I \)) to  (\( R \)) is the rate of \underline{\hspace{6cm}}.
+
+    \item \( R \) stands for \underline{\hspace{3cm}}. This compartment includes those who have ceased to be infectious and acquire immunity against infection, regardless of the clinical course.
+\end{itemize}
+
+\section{\( R_0 \)}
+\( R_0 \) helps project the potential 
+size of an epidemic and calculate the herd immunity threshold.
+It is defined as the average number of \underline{\hspace{2cm}} secondary cases 
+generated from a primary case in a completely 
+\underline{\hspace{3cm}} population. 
+
+\section{\( R_t \)}
+\( R_t \) 
+helps monitor the progress of the epidemic 
+When the population is no longer \underline{\hspace{2cm}}, the instantaneous 
+reproduction number \( R_t \) is used. This is defined as the average number 
+of s\underline{\hspace{2cm}} in a population composed of 
+\underline{\hspace{2cm}} and non-\underline{\hspace{2cm}} individuals at time \( t \).
+
+\section{A Diagram for Measles outbreak}
+
+Below is a typical SEIR model with demography (births and deaths). This is a simple 
+model applicable to person-to-person infections in a homogeneously mixing population.
+Please carefully observe the model and examine the interactions with the equations 
+in section \ref{eqs}.  Use color codes or arrows to relate the diagram to the equations.
+
+
+\begin{center}
+    \begin{tikzpicture}[
+        node distance=2cm, 
+        every node/.style={fill=blue!10, draw, minimum size=1cm, text centered},
+        arrow/.style={-Stealth, thick}
+    ]
+
+    % Nodes
+    \node [circle, fill=green!75](S) {$S$};
+    \node [circle, fill=orange!75](E) [right=of S] { $E$};
+    \node [circle, fill=red!75](I) [right=of E] {$I$};
+    \node [circle, fill=blue!75](R) [right=of I] {$R$};
+
+    % Arrows for transitions
+    \draw[arrow] (S) -- node[above, draw=none] {$\beta S  \frac{I}{N}$} (E);
+    \draw[arrow] (E) -- node[above, draw=none] {$\sigma  E$} (I);
+    \draw[arrow] (I) -- node[above, draw=none] {$\gamma  I$} (R);
+
+    % Natural birth and death rates
+    \draw[arrow] (-2,0.0) -- node[above, draw=none] {$\Lambda N$} (S);
+    \draw[arrow] (S) -- +(0,-1.2) node[below, draw=none] {$\mu$};
+    \draw[arrow] (E) -- +(0,-1.2) node[below, draw=none] {$\mu$ };
+    \draw[arrow] (I) -- +(0,-1.2) node[below, draw=none] {$\mu$ };
+    \draw[arrow] (R) -- +(0,-1.2) node[below, draw=none] {$\mu$};
+
+    \end{tikzpicture}
+\end{center}
+
+Where:
+\begin{itemize}
+    \item \( \beta \): Transmission rate
+    \item \( \sigma \): Rate of progression from exposed to infectious
+    \item \( \gamma \): Recovery rate
+    \item \( \mu \): Death rate (natural death rate)
+    \item \( N \): Total population size, \( N = S + E + I + R \).
+\end{itemize}
+
+ The parameter $\beta$ is derived from the multiplication of $p$
+  and $c$, where $p$ is the probability of transmission during contact, and $c$ 
+  is the contact rate, defined as the average number of contacts per unit of time.\\
+
+Model parameters are often (but not always) specified as rates.
+The rate at which an event occurs is the inverse of the average time until that event.
+For example, in the SEIR model, the recovery rate $\gamma$ is the inverse of the average infectious period.\\
+
+Values of these rates can be determined from the natural history of the disease.
+For example,  if people are on average infectious for 8 days, then in the model, 
+1/8 of currently infectious people would recover each day
+(i.e. the rate of recovery, $\gamma=1/8=0.125$).
+
+\section{Equations}\label{eqs}
+Note that in the diagram, arrows entering compartments are expressed as positive 
+terms in the equations, while arrows exiting compartments are represented with negative terms.
+Based on the above diagram,deduce the following equations that describe this system:
+
+\section*{Compartment Equations}
+
+\begin{itemize}
+    \item \textbf{S compartment:}
+    \[
+    \frac{dS}{dt} = 
+    \]
+
+    \item \textbf{E compartment:}
+    \[
+    \frac{dE}{dt} = 
+    \]
+    \item \textbf{I compartment:}
+    \[
+    \frac{dI}{dt} = \
+    \]
+
+    \item \textbf{R compartment:}
+    \[
+    \frac{dR}{dt} =
+    \]
+\end{itemize}
+
+\section{Computing $R_0$}
+The expression for the basic reproduction number ($R_0$) in the above system  is given by:
+
+\begin{equation*} R_0 = \frac{\mu}{(\mu + \alpha)} \frac{\beta}{(\mu + \gamma)}. \end{equation*}
+
+To calculate the $R_0$ value for given parameter values,  write an R function 
+called Measles$R_0$ that implements this formula. The function will use the following parameter values:
+
+    \begin{itemize}
+        \item $\mu = \frac{1}{75}$ (natural mortality rate)
+        \item $\alpha = \frac{1}{10}$ (rate of progression from the exposed to the infectious stage)
+        \item $\gamma = 1/8$ (recovery rate)
+        \item $\beta = 1.8$ (transmission rate)
+    \end{itemize}
+Then compute the final size of such epidemic.
+\end{document}