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| 1 | +\documentclass{article} |
| 2 | +\usepackage{fancyhdr} |
| 3 | + |
| 4 | +% Set up the custom footer |
| 5 | +\pagestyle{fancy} |
| 6 | +\fancyfoot[R]{\thepage} % Centered page number in footer |
| 7 | +\fancyfoot[C]{\textbf{License:} CC-BY.\textbf{Copyright:} Andree \& Degoot, 2024 } |
| 8 | +\usepackage{tikz} |
| 9 | +\usetikzlibrary{arrows.meta, positioning} |
| 10 | +\usepackage{amsmath} |
| 11 | +\author{Andree Valle Campos and Abdoelnaser M Degoot \\ Epiverse-TRACE Team @ LSHTM } |
| 12 | +\title{Simple Introduction to Mathematical Modelling of Infectious Diseases} |
| 13 | +\begin{document} |
| 14 | +\maketitle |
| 15 | + |
| 16 | +\section{Introduction} |
| 17 | +This practical aims to assess your understanding of the fundamental |
| 18 | +principles of mathematical modeling while guiding you in constructing models using |
| 19 | +a simple SEIR framework for infectious disease outbreaks.\\ |
| 20 | + |
| 21 | + \textbf{Note: Please fill in the blanks.} |
| 22 | + |
| 23 | +\section{SEIR Model} |
| 24 | + |
| 25 | + In the SEIR model, we have four compartments (\( S \), \( E \), \( I \), \( R \)): |
| 26 | + |
| 27 | +\begin{itemize} |
| 28 | + \item \( S \) stands for \underline{\hspace{2cm}}, meaning \underline{\hspace{6cm}}. |
| 29 | +The parameter that explains the transition from (\( S \)) compartment |
| 30 | +to (\( E \)) compartment is \underline{\hspace{6cm}}. |
| 31 | +\item \(E\) stands for \underline{\hspace{2cm}}, meaning that it can |
| 32 | + \underline{\hspace{4cm}}. |
| 33 | + |
| 34 | + The rate that explains the transition from (\( E \)) to (\( I \)) is the rate of \underline{\hspace{1cm}}. |
| 35 | + |
| 36 | + \item \( I \) stands for \underline{\hspace{2cm}}, meaning that it can |
| 37 | + \underline{\hspace{3cm}}. |
| 38 | + The rate that explains the transition from (\( I \)) to (\( R \)) is the rate of \underline{\hspace{6cm}}. |
| 39 | + |
| 40 | + \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. |
| 41 | +\end{itemize} |
| 42 | + |
| 43 | +\section{\( R_0 \)} |
| 44 | +\( R_0 \) helps project the potential |
| 45 | +size of an epidemic and calculate the herd immunity threshold. |
| 46 | +It is defined as the average number of \underline{\hspace{2cm}} secondary cases |
| 47 | +generated from a primary case in a completely |
| 48 | +\underline{\hspace{3cm}} population. |
| 49 | + |
| 50 | +\section{\( R_t \)} |
| 51 | +\( R_t \) |
| 52 | +helps monitor the progress of the epidemic |
| 53 | +When the population is no longer \underline{\hspace{2cm}}, the instantaneous |
| 54 | +reproduction number \( R_t \) is used. This is defined as the average number |
| 55 | +of s\underline{\hspace{2cm}} in a population composed of |
| 56 | +\underline{\hspace{2cm}} and non-\underline{\hspace{2cm}} individuals at time \( t \). |
| 57 | + |
| 58 | +\section{A Diagram for Measles outbreak} |
| 59 | + |
| 60 | +Below is a typical SEIR model with demography (births and deaths). This is a simple |
| 61 | +model applicable to person-to-person infections in a homogeneously mixing population. |
| 62 | +Please carefully observe the model and examine the interactions with the equations |
| 63 | +in section \ref{eqs}. Use color codes or arrows to relate the diagram to the equations. |
| 64 | + |
| 65 | + |
| 66 | +\begin{center} |
| 67 | + \begin{tikzpicture}[ |
| 68 | + node distance=2cm, |
| 69 | + every node/.style={fill=blue!10, draw, minimum size=1cm, text centered}, |
| 70 | + arrow/.style={-Stealth, thick} |
| 71 | + ] |
| 72 | + |
| 73 | + % Nodes |
| 74 | + \node [circle, fill=green!75](S) {$S$}; |
| 75 | + \node [circle, fill=orange!75](E) [right=of S] { $E$}; |
| 76 | + \node [circle, fill=red!75](I) [right=of E] {$I$}; |
| 77 | + \node [circle, fill=blue!75](R) [right=of I] {$R$}; |
| 78 | + |
| 79 | + % Arrows for transitions |
| 80 | + \draw[arrow] (S) -- node[above, draw=none] {$\beta S \frac{I}{N}$} (E); |
| 81 | + \draw[arrow] (E) -- node[above, draw=none] {$\sigma E$} (I); |
| 82 | + \draw[arrow] (I) -- node[above, draw=none] {$\gamma I$} (R); |
| 83 | + |
| 84 | + % Natural birth and death rates |
| 85 | + \draw[arrow] (-2,0.0) -- node[above, draw=none] {$\Lambda N$} (S); |
| 86 | + \draw[arrow] (S) -- +(0,-1.2) node[below, draw=none] {$\mu$}; |
| 87 | + \draw[arrow] (E) -- +(0,-1.2) node[below, draw=none] {$\mu$ }; |
| 88 | + \draw[arrow] (I) -- +(0,-1.2) node[below, draw=none] {$\mu$ }; |
| 89 | + \draw[arrow] (R) -- +(0,-1.2) node[below, draw=none] {$\mu$}; |
| 90 | + |
| 91 | + \end{tikzpicture} |
| 92 | +\end{center} |
| 93 | + |
| 94 | +Where: |
| 95 | +\begin{itemize} |
| 96 | + \item \( \beta \): Transmission rate |
| 97 | + \item \( \sigma \): Rate of progression from exposed to infectious |
| 98 | + \item \( \gamma \): Recovery rate |
| 99 | + \item \( \mu \): Death rate (natural death rate) |
| 100 | + \item \( N \): Total population size, \( N = S + E + I + R \). |
| 101 | +\end{itemize} |
| 102 | + |
| 103 | + The parameter $\beta$ is derived from the multiplication of $p$ |
| 104 | + and $c$, where $p$ is the probability of transmission during contact, and $c$ |
| 105 | + is the contact rate, defined as the average number of contacts per unit of time.\\ |
| 106 | + |
| 107 | +Model parameters are often (but not always) specified as rates. |
| 108 | +The rate at which an event occurs is the inverse of the average time until that event. |
| 109 | +For example, in the SEIR model, the recovery rate $\gamma$ is the inverse of the average infectious period.\\ |
| 110 | + |
| 111 | +Values of these rates can be determined from the natural history of the disease. |
| 112 | +For example, if people are on average infectious for 8 days, then in the model, |
| 113 | +1/8 of currently infectious people would recover each day |
| 114 | +(i.e. the rate of recovery, $\gamma=1/8=0.125$). |
| 115 | + |
| 116 | +\section{Equations}\label{eqs} |
| 117 | +Note that in the diagram, arrows entering compartments are expressed as positive |
| 118 | +terms in the equations, while arrows exiting compartments are represented with negative terms. |
| 119 | +Based on the above diagram,deduce the following equations that describe this system: |
| 120 | + |
| 121 | +\section*{Compartment Equations} |
| 122 | + |
| 123 | +\begin{itemize} |
| 124 | + \item \textbf{S compartment:} |
| 125 | + \[ |
| 126 | + \frac{dS}{dt} = |
| 127 | + \] |
| 128 | + |
| 129 | + \item \textbf{E compartment:} |
| 130 | + \[ |
| 131 | + \frac{dE}{dt} = |
| 132 | + \] |
| 133 | + \item \textbf{I compartment:} |
| 134 | + \[ |
| 135 | + \frac{dI}{dt} = \ |
| 136 | + \] |
| 137 | + |
| 138 | + \item \textbf{R compartment:} |
| 139 | + \[ |
| 140 | + \frac{dR}{dt} = |
| 141 | + \] |
| 142 | +\end{itemize} |
| 143 | + |
| 144 | +\section{Computing $R_0$} |
| 145 | +The expression for the basic reproduction number ($R_0$) in the above system is given by: |
| 146 | + |
| 147 | +\begin{equation*} R_0 = \frac{\mu}{(\mu + \alpha)} \frac{\beta}{(\mu + \gamma)}. \end{equation*} |
| 148 | + |
| 149 | +To calculate the $R_0$ value for given parameter values, write an R function |
| 150 | +called Measles$R_0$ that implements this formula. The function will use the following parameter values: |
| 151 | + |
| 152 | + \begin{itemize} |
| 153 | + \item $\mu = \frac{1}{75}$ (natural mortality rate) |
| 154 | + \item $\alpha = \frac{1}{10}$ (rate of progression from the exposed to the infectious stage) |
| 155 | + \item $\gamma = 1/8$ (recovery rate) |
| 156 | + \item $\beta = 1.8$ (transmission rate) |
| 157 | + \end{itemize} |
| 158 | +Then compute the final size of such epidemic. |
| 159 | +\end{document} |
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