\documentclass{article}
\usepackage{fullpage}
\usepackage{parskip}
\usepackage{titlesec}
\usepackage{xcolor}
\usepackage[colorlinks = true,
linkcolor = blue,
urlcolor = blue,
citecolor = blue,
anchorcolor = blue]{hyperref}
\usepackage[natbibapa]{apacite}
\usepackage{eso-pic}
\AddToShipoutPictureBG{\AtPageLowerLeft{\includegraphics[scale=0.7]{powered-by-Authorea-watermark.png}}}
\renewenvironment{abstract}
{{\bfseries\noindent{\abstractname}\par\nobreak}\footnotesize}
{\bigskip}
\titlespacing{\section}{0pt}{*3}{*1}
\titlespacing{\subsection}{0pt}{*2}{*0.5}
\titlespacing{\subsubsection}{0pt}{*1.5}{0pt}
\usepackage{authblk}
\usepackage{graphicx}
\usepackage[space]{grffile}
\usepackage{latexsym}
\usepackage{textcomp}
\usepackage{longtable}
\usepackage{tabulary}
\usepackage{booktabs,array,multirow}
\usepackage{amsfonts,amsmath,amssymb}
\providecommand\citet{\cite}
\providecommand\citep{\cite}
\providecommand\citealt{\cite}
% You can conditionalize code for latexml or normal latex using this.
\newif\iflatexml\latexmlfalse
\AtBeginDocument{\DeclareGraphicsExtensions{.pdf,.PDF,.eps,.EPS,.png,.PNG,.tif,.TIF,.jpg,.JPG,.jpeg,.JPEG}}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\begin{document}
\title{Should Mathematical Modelling Be Used to Assist Efforts of Retarding the Spread of Yellow Fever in West Africa}
\author[ ]{James Moore-Stanley}
\affil[ ]{}
\vspace{-1em}
\date{}
\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup
\tableofcontents
\section{Introduction}
News of the most recent outbreak of yellow fever passing 300 deaths in Angola has prompted fears of the World Health Organisation (WHO) regarding its epidemic progression. Vaccination in some regions of the outbreak are very hard to accomplish, as the process of vaccinating is both very expensive, and hard to enforce on a population to satisfy the necessary threshold (to be explained later). Tracking and predicting progression of disease has always been a difficult aim to accomplish. However, there are several mathematical models that may assist with this problem. These models will be explained and adapted to accommodate a range of underlying assumptions about the population and the method of transforming the disease, which will enable the research question to be answered. This is certainly area that is worthy of exploration, both personally and on a greater scale. I have been an avid follower of how diseases have captivated the attention of global media and prevention efforts, namely the Ebola virus and the Zika virus. I have specifically chosen the Yellow Fever virus as it is a currently developing situation, and has manageable statistics for an exploration of this magnitude.
\section{Theory of the SIR model, basic reproductive number, and herd immunity}
\subsection{The 'basis' or basic SIR model}
The SIR model is a frequently used model to attempt to predict the rate at which a disease will spread through a population. SIR is an acronym for the three classes an individual may lie within: Susceptible, Infected, and Removed. An individual in the population may \textbf{only} exist within these three classes. There are also other assumptions in which the \textbf{basic} SIR model:
There are a number of assumptions that are important to note in the basic SIR model. These are the following:
\begin{enumerate}
\item \textbf{All} individuals in the susceptible class have never been in contact with the disease and are all therefore susceptible to it
\item The population is constant; there are no natural births and natural deaths
\item In order to move from one section, an individual has to enter another. This would mean that the population is assumed to be constant. (This is acceptable as the removed section includes deceased individuals).
\item When an individual is removed, he/she is assumed to be immune for the remainder of his/her life, and is therefore unable to rejoin susceptible.
\item Every individual that is infected has equal probability of encountering an individual that is susceptible per unit time.
\end{enumerate}
These assumptions will obviously limit the practical usages of the model, as the model's outcomes would not represent results in a real world environment. However, the model may be adapted in order to incorporate dimensions from a real world scenario, such as that of Yellow Fever in Angola. In order to adapt the model to fit the situation, understand of the underlying, or basic model is required. This will be explained below. As stated above, individuals may be in one of three categories. The equations below show the number of people in each of the three categories as a function of time.
\begin{align*}
S &= S(t) && \text{absolute number of people who are susceptible} \\
I &= I(t) && \text{absolute number of people who are infected} \\
R &= R(t) && \text{absolute number of people who are removed}
\end{align*}
The notation $N$ will be used for the total population value. It should be noted that as these are the only categories the population may be divided into, $S+I+R=N$ must be case for any value of $t$.
\\In order to determine the \textbf{rate} at which individuals transition between \textit{susceptible and infected} and \texit{infected and removed}, a rate constant must be involved to state the percentage of the population that makes each of these transmissions. If values for $t$ are taken at discrete integer intervals, we can deduce the equation for the rate of transition, i.e. difference in the number of people in the susceptible category for time $t$ and $t+1$, or the number of people that have become infected in day $t$. This can be expressed in the equation below:
\begin{equation*}
S_{t+1} - S_t = \frac{\mathrm{d} S}{\mathrm{d} t} = -\beta S_t I_t
\end{equation*}
where $\beta$ is the aforementioned rate constant. $Beta$ is given a negative value, as the number of people who are classed as susceptible is decreasing, as they are moving into the infected classification.
A similar method may be applied to calculated the \textbf{rate} at which individuals move from the infected to the removed category. The rate of people becoming infected would logically be the same as the rate of decrease in susceptible individuals, as individuals can only from susceptible to infected.
\\However, people who are already infected may have either recovered or died at this time, meaning they are moved into removed classification. This is explained mathematically below:
\begin{align*}
I_{t+1} - I_t &= \frac{\mathrm{d} I}{\mathrm{d} t} = \beta S_t I_t && \text{This equation is incorrect as it does not take into account recovered individuals} \\
I_{t+1} - I_t &= \frac{\mathrm{d} I}{\mathrm{d} t} = \beta S_t I_t -\gamma S_t I_t && \text{ This equation includes individuals changing from infected to removed} \\
\end{align*}
$\beta$, in the second line equation, represents the rate constant for the percentage of the population moving from infected to removed per unit time.
\\Finally, the equation of the rate of individuals moving becoming removed (i.e. the rate of people who recover per unit time) is simply individuals who have recovered or died. This may be expressed as below:
\begin{equation*}
R_{t+1} - R_t = \frac{\mathrm{d} R}{\mathrm{d} t} = \gamma S_t I_t
\end{equation*}
\subsection{Basic Reproductive Number}
The basic reproductive number may be thought of as the number representing the number of secondary cases spawning from one case of an infectious disease in an otherwise unaffected population. It is represented by notation $R_0$. It would therefore be logical to say that if $R_0 \geq 1$, the disease is very likely to spread in a population. Conversely, if $R_0 \leq 1$, the disease is very likely to die out, and not affect the population.
Methods for calculating $R_0$ will be shown in the section below, however having knowledge of $R_0$ so that its use may be understood in the section \textit{Linking the basis SIR model, Basic Reproductive Number, and Herd Immunity.} In order to have a sufficient understanding of where $R_0$ comes from, the following factors are taken into account and processed into calculating $R_0$:
\begin{itemize}
\item The rate of contact in the host population
\item The probability of contracting the disease through contact
\item The latent period of the infection i.e. the length of time a person is still contagious.
\end{itemize}
\subsection{Herd Immunity}
Herd immunity may be defined as a method of indirect protection from a disease when a large percentage of the population is immunised from a disease, thereby protecting the minority of that population that cannot be immunised for a number of reasons, most commonly medical exclusion or religious pressures. Individuals who are immunised are said to be a barrier to a spread of a disease, as they either reduce the rate at which the disease spreads, or prevent the spread entirely. When a critical proportion of the population becomes immunised against the disease, the disease will cease to spread and exist. This is call the Herd Immunity Threshold (HIT). This threshold is able to be quantified, using the Basic Reproductive Rate.
\subsection{Linking the basis SIR model, Basic Reprodutive Rate, Herd Immunity}
The SIR model is certainly academically impressive, however when using any mathematical model, it is of vital importance in this case to recognise two things; will preventative action be able to be taken using these models, and what is the most significant part of the models, what aspect of the model can humans use to their advantage?
\\The answer to the first of these two questions is yes, it can be used in real-world action. The answer to the second, is that $R_0$ is undoubtedly the most vital component of the model. \textit{The Bill & Melinda Gates Foundation} does use these models, and is spending vast resources on trying to reduce the $R_0$ value for malaria. This section will build on previously mentioned equations by suggesting methods for calculating $R_0$, and manipulating $R_0$ to display new, useful equations. It is not important, and frankly far beyond the level of this exploration, to understand \textit{how} these derived, but that all variables in the equation are understood, and that useful data is collected from these new equations.
\\If $R_0$ can be thought of as the number of secondary cases resulting from one primary case, it can be said to be related to the rate of contact between persons and the rate at which they recover. $R_0$ can therefore be related to the SIR model, by saying that a good approximation of $R_0$ is equal to:
\begin{equation*}
R_0 = \frac{\beta}{\gamma}
\end{equation*}
with as in the SIR, $\beta$ representing the contact rate of the infected and susceptible, and $\gamma$, where an individual recovers with parameter $\gamma$.
With $R_0$ being calculated using the method above or otherwise, the aforementioned herd immunity threshold can be calculated. This is through the formula:
\begin{equation*}
P_{critical} = 1 - \frac{1}{R_0}
\end{equation*}
Plotting the graph of $y=1-\frac{1}{x}$ leads to the graph shaped as below:
*insert graph here*
The graph shows that as $R_0$ rises, the proportional of the population that must be vaccinated rises very quickly. Measles can be used as an example for showing the equation in use, as the $R_0$ value for measles is widely accepted as accurate.
The accepted value for $R_0$ of measles is 15. The calculation using this value follows:
\begin{equation*}
P_{critical} = 1-\frac{1}{R_0} = 93\%
\end{equation*}
This shows that for diseases with a high $R_0$ value, the critical proportion of the population is very high, so high that it would be considered impossible in countries that are less economically developed.
\\This is however, only an approximation of $R_0$. One of the main problems with this approximation is that it does not take the period of time when one is moved into the removed category, however will act as an appropriate approximation.
\\Before the next equation is explained, it should be noted that the use of S, I and R as a fraction of the population or as the real values are interchangeable. In the equation below, we are dealing with \textbf{fractions} of the population, not the whole population, hence instead of $N$ being used, $n$ is used, and since we are using fractions of the population, $n$ can be said to equal $1$ (100\% of the population).
\\A new notation $\rho$ will be introduced. $\rho$ is related to $R_0$, as $\rho$ is the reciprocal of $R_0$. This manipulation is shown below:
\begin{align*}
R_0 &= \frac{\beta}{\gamma} && \text{This is the equation for $R_0$} \\
\rho &= \frac{n \gamma}{\beta} && \text{Note that in this equation, $\gamma$ is on top, and $\beta$ is on the bottom. Remember that $n$ is equal to 1, so we can discard it.} \\
\end{align*}
We can therefore state the equation:
\begin{equation*}
\rho = \frac{1}{R_0}
\end{equation*}
The purpose of deriving $\rho$ is so that when it is used in the formula below, it is clear that it is in fact $R_0$ that is an important factor.
This equation shows the maximum proportion of a population that can be infected throughout the course of the epidemic. Notation varies among sources, so $x$ and $y$ have been used here.
\begin{equation*}
y_{max} = y_0 + x_0 - \rho -\rho log\frac{x_0}{\rho}
\end{equation*}
where $y$ represents people as infected, and $x$ represents people classed as susceptible.
\\This equation proves that if $R_0$ is small, the maximum proportion of infected people, $y_{max}$ is also small. This is because in order to minimise $y_{max}$ according to the equation above, $\rho$ should be as large as possible.
\\Due to the face that $\rho = 1/R_0$, if $R_0$ is smaller, $\rho$ will be larger, and therefore $y_{max}$ will decrease.
\section{Methodology of estimating $R_0$ for Yellow Fever}
It is very difficult to determine an accurate value for $R_0$ for any disease. This is especially true in the case of yellow fever, where there is not a plentiful source of data on transmission rate statistics, merely the occasional update from the World Health Organisation.
\\However, taking into account local geographical considerations, combined with other estimated values for diseases of similar transmission, progression, and transmission, one can make an estimation for the $R_0$ values for yellow fever by using $R_0$ values for similar diseases, where there is a larger amount of better quality data than yellow fever. Although this is an imperfect estimation, it is still a reasonably accurate assumption to make so that its results may be applied. The $R_0$ value for malaria, a disease with similar geographical considerations and transmission method, is approximately 20, so an $R_0$ value of 20 shall be used here.
\section{Using the SIR Model to Model Yellow Fever}
\subsection{Estimating parameters}
The recovery time for yellow fever is between 4-6 days. The graphs below have time intervals of one day on the x-axis, hence the parameter $\gamma$ should be calculated as the rate of recovery per day. Using the 4-6 day recovery time data, taking 5 as an average, it can be estimated the the parameter $\gamma$ is $\frac{1}{5}$.
\subsection{Graphical Representation}
\subsubsection{The Basic SIR model}
Public ASU has developed multiple programs in which parameters can be entered to produce a graph that shows the transition of individuals in a population between the three classification of susceptible, infected and removed. Having calculated $R_0$ to be approximately 20, and taking gamma to be $\frac{1}{5}$, we are able to calculate the value for $\beta$. This can be done such:
\begin{equation*}
\beta = R_0\cdot \gamma
\end{equation*}
This calculation estimates $\beta$, the effective contact rate between individuals in the population, as 3. This parameter is a reasonable estimate for the average number of secondary cases emerging from one primary case, as the majority of the population is spread out in the country side, instead of largely concentrated in urban areas. This approximation for $\beta$ will be used throughout. The first recorded case of yellow fever in Angola was dated 5th December 2015 in the Luanda province. The population of this province is approximately 6.5 million people, hence the number of people in the susceptible category should be the population -1 person (who is infected).
\\Inserting values into the graphing software gives the following graph:\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Basis-SIR-model1/Basis-SIR-model1}
\caption{{A graph to show the distribution and change in classification of individuals according to the basis SIR model with approximated parameters used%
}}
\end{center}
\end{figure}
\\However, the World Health Organisation statistics for this particular outbreak would not concur with predictions using the basis SIR model; the actual spread has been significantly different to that offered by the above graph. This would indicate that the basis SIR model would not be useful for predicting future trends.
\\This leads to the question 'why has the model failed?'. The answer lies in the assumptions made when asserting the model upon a population. The first of these is that the value for $R_0$ simply cannot be said to be true for all members of a population. Their differing socio-economic backgrounds may lead to these factors:
\begin{enumerate}
\item Some backgrounds are more likely to be more densely populated
\item Some backgrounds may have restricted access to mosquito resistant materials
\item Some backgrounds may have a lower propensity to be vaccinated.
\end{enumerate}
It is very different to try to quantify these factors and then adapt the basis model accordingly. There is however one other major factor that can be quantified. The basis SIR model naturally assumes that the transmission of the disease occurs between humans. However, this not the case for yellow fever. The virus is transmitted by the (FILL ME!) mosquito only (CORRECT ME!). It therefore makes the makes the parameters used in the SIR model, such as $\beta$ irrelevant, and will of course cause the model to be inaccurate., as well as calculations involving $\beta$, such as $R_0$ wrong.
\subsubsection{The Ross-Macdonald Model}
However, there is a model that does take this 'vector' factor into account. This is called the Ross-Macdonand model. This model also relies on several assumptions that are listed below, some of which are also incorporated into the SIR model:
\begin{enumerate}
\item The number of mosquitoes is constant
\item Mosquitoes are only permitted to be classed as susceptible or infectious
\item There is a constant population size
\item Incubation periods are ignored
\item The rate at which mosquitoes bite humans is proportional to the number of mosquitoes.
\item The rate at which mosquitoes bite humans is independent of the human population
\item Populations are homogeneous
\item The equation is applied to deterministic situations
\end{enumerate}
\\Number (8) in the assumptions makes reference to deterministic methods. All this means is that instead of using a small data set and using probability to apply it to larger numbers is not permitted (this is known as stochastic method). The data that is inputted must have been taken from a large population. This would make sense, as one of the problems with the basic SIR model is that not all parameters can be assumed for all individuals. Using large data ensures that such discrepancies will have been inputted into the large data.
\newpage
As mentioned for other equations, it is not important to understand the derivation of these formulas, but to have understand the effect of changing a variable in the equation to ultimately affect the quantity of what you are trying to measure. The Ross-Macdonald model is stated below in the form of two equations, one showing the number of infected individuals, and the other showing the number of infected mosquitoes.
\begin{align*}
\dot{Y} &= abI \left ( \frac{H-Y}{H}\right )-\varepsilon Y && \text{Infective humans} \\
\dot{I} &= ac(V-I)\frac{Y}{H}-\delta I && \text{Infective mosquitoes} \\
\end{align*}
Where: \\
$\dot{Y} = \textup{the number of infected humans}$
$\dot{I} = \textup{the number of infected mosquitos}$
$a = \textup{mosquito biting rate}$ \\
$b = \textup{mosquito to human transition probability}$ \\
$c = \textup{human to mosquito mosquito transmission probability}$ \\
$\varepsilon = \textup{human recovery rate}$ \\
$\delta = \textup{mosquito death rate}$ \\
These are then said to lead to this 'behavior'in the population, with behavior being represented by $R_0$:
\begin{equation*}
R_0 = \frac{ma^2bc}{\varepsilon \delta}
\end{equation*}
With these three equations in mind, there is something that most obviously stands out; the importance of $a$, the mosquito biting rate. This is particularly important in the 'behavior' equation, as this factor is squared. This means that $R_0$ is very dependent upon the biting rate.
\\Unfortunately, there is no readily available graphing algorithm for this precise model. Even if such a model were available, it would be extremely difficult to estimate values for required parameters, and impossible for parameters such as the average number of bites each mosquito will perform.
\section{Conclusions}
The above paragraphs may have indicated that using mathematical modeling is not particularly useful in attempting to map the spread of a disease, and indeed this is partially true. It does not however, indicate that these models are not useful for understanding effects of changing variables that humans \textit{can} control. Within this conclusion, we can ultimately say what the models can and cannot be used for, and then answer the research question, \textit{overall} are they useful.
\subsection{Innapropriate use of the model}
The models should not be used for attempting to predict the spread of a disease nationwide, or an a large scale. The main reason for this is because it is (1) difficult to calculate parameters that are needed to use the model, such as contact rate or probabilities of infection and (2) As already mentioned, the $R_0$ value for calculated from one region may not be applied nationally. Even within one region there will be a variety of factors deciding if it is appropriate for all individuals in a region. The overall message is that attempting to calculate all parameters will be unsuccessful.
\subsection{Appropriate use of the model and actions to be taken from it}
There are estimates of $R_0$ used by the World Health Organisation, among others, to aid with their preparations. This means that $R_0$ can simply be put to one side and cast as useless. If $R_0$ is taken to be an approximation over a sustained period of time (i.e. for a disease that is a constant threat and has years and years of data) it tells the relevant assisting body information such as prioritising which epidemics need to be tended to first, and whether vaccination is likely to be an effective retardant.
\\Relevant organisations should also try to also take other easier to measure parameters into account, for example the number of mosquitoes (or whatever the vector may be).
\\The models tell us two solutions. In the short term, if the $R_0$ is high, there is ultimately not much that can be done. Pharmaceutial companies are generously willing to provide vaccinations at 8 per person for yellow fever. Luckily, yellow fever immunity for life only requires one vaccination. The WHO rather unsurprisingly is unable to issue vaccinations to all, for political and monetary (rather lack of) reasons.
\\However, for diseases that have a constantly high $R_0$ over a long period of time, it seems that the only long-term solutions will be affecting the parameters in which $R_0$ is calculated from. This has been the aim of the \textit{Bill and Melinda Gates Foundation}, with their aim is to try to slow the effect. This is comparable to yellow fever, as mosquitoes are the vector in both diseases. They aim to eventually reduce the number of mosquitoes, which, according to the models, specifically the Ross-Macdonald model, will have a large effect, particularly as the model places great emphasis on the bite rate $(a^2)$. This means that in the long term, these models should be used to understand that by changing the variables, the disease will eventually begin to die out. One can conclude from the model that \textit{any} attack on mosquitoes will reduce the rate of spread, meaning that mathematical models should lead to action reducing the bite rate consisting of:
\begin{itemize}
\item attempt to reduce the number of mosquitoes in the environment
\item provide insect repellent
\item providing mosquito nets
\end{itemize}
\selectlanguage{english}
\FloatBarrier
\end{document}