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\newpage

\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi
\\EBOLA OUTBREAK IN WEST AFRICA}




\markboth{Ebola Outbreak in West Africa}{Ebola Outbreak in West Africa}






\author{Lisa Driskell\\
Colorado Mesa University\\
Grand Junction  CO 81501 USA\\
ldriskel@coloradomesa.edu
}
\makeStitlePDFLaTex





\begin{abstract}
Students will use data published by the World Health Organization to model the 2014 outbreak of the Ebola virus in West Africa.  We begin with a simple exponential growth model and move through the modeling process to the logistic growth model.  Both are separable differential equations.  Students will investigate properties of the logistic growth model and will compare predicted values with observed values.  Both quantitative and qualitative questions are asked throughout.  This project can be used in a differential equations or modeling course as well as in the calculus sequence. The logistic growth model fits the data well and thus provides a simple introduction to modeling a real world situation. 


Note that in the supplemental materials, an Excel file containing a complete data set with more dates and records is available which can be used for additional modeling projects.  This Excel file includes the total number of cases of Ebola, the total number of deaths due to Ebola, the number of confirmed, probable, and suspected cases, and the number of cases per country for Guinea, Liberia, and Sierra Leone. 
\end{abstract}



\section*{SCENARIO DESCRIPTION}

Ebola virus disease or EVD is a potentially fatal disease affecting humans.  The virus is thought to be carried by animals such as fruit bats and was discovered in 1976 when the first human outbreaks occurred \cite{WHO}.  The World Health Organization (WHO) recognizes March 2014 to be the start of the worst Ebola outbreak in history.  The outbreak was primarily contained to West Africa, beginning in Guinea and spreading to Liberia and Sierra Leone \cite{WHO}.  

The outbreak and swift spread of Ebola in West Africa sparked fears worldwide for its potential to reach other regions.  In September 2014, the Center for Disease Control (CDC) cited model predictions for the number of Ebola cases  that reached as high as 550,000 cases by January 20, 2015 in Liberia and Sierra Leone alone \cite{CDC}.  When the model was corrected for under reporting, the number of cases jumped to $1.4$ million.  At the time of the prediction, Liberia and Sierra Leone combined for 2,407 cases while the model fairly accurately estimated 2,618 cases \cite{CDC}. 

In a January 2016 news release, the World Health Organization officially declared the Ebola outbreak over for all three countries \cite{WHONR} (although  few flare-ups resulted in 14 additional cases since that declaration). Because we have the data for the actual epidemic, we now know that the predictions by the CDC vastly overestimated the impact of the outbreak.  We will investigate our own models of the epidemic  based on the data that we now have.

We will use data published by the WHO (\cite{WHODON}, \cite{WHOSR}) to determine parameters for simple models that describe the outbreak.  The data for the total number of Ebola cases includes  cases confirmed by test results as well as suspected cases (those showing symptoms resistant to treatment) and probable cases (suspected cases that have been seen by a clinician).  Also note that the data provided for the total number of cases is a running total of the number of cases which includes those patients who may have since recovered or died from the disease. The data for the individual countries and the number of deaths due to Ebola is also included to provide the opportunity for additional modeling and comparisons. 

\subsection*{Part I: Exponential Growth Model}
Although some cases of Ebola were confirmed in countries such as Nigeria, Senegal, and Mali, the outbreak primarily occurred in Guinea, Liberia, and Sierra Leone.  Thus, as we study the Ebola outbreak in West Africa, we will consider only the data from these three countries.
  
\begin{enumerate}
\item Construct a model.
\begin{itemize}
\item Suppose the number of cases of persons  
in West Africa infected with the Ebola virus grows at a rate proportional to the total number of cases.  Write a differential equation to represent the situation described above.  Use $N$ for number of cases of Ebola and $t$ for the time in months.
\item In March 2014, there were $49$ cases and after three months, the number of cases had grown to $528$.  Solve the differential equation from above and use the given data to find an expression for the total number of Ebola cases, $N(t)$.
\end{itemize}

\item Interpret and verify the model.
\begin{itemize}
\item Use your exponential growth model to estimate the time it will take for the outbreak to reach a total number of 5,000 cases.
\item Comment on how well your answer to the previous part matches the data in Table \ref{table:cases}. 
\item Use your model  to estimate the number of cases after 5 months, 7 months, and 10 months.
\item Compare the predicted values from above to the observed values in Table \ref{table:cases}.
\item Remark on whether you expect this model be accurate for the duration of the epidemic.
\end{itemize}

\item  Further examine the scenario. 
\begin{itemize}
\item Recall the predictions noted by the CDC for Liberia and Sierra Leone for January 2015.  How do your prediction and the actual data for all three countries compare to the prediction of the CDC?
\item Consider factors that may affect the behavior of the epidemic and its growth.  Describe at least four such factors and identify whether the influences are environmental, economic, social, cultural, political, or other.
\item Identify the inconsistency in the data for Liberia in Table \ref{table:cases}. Speculate reason(s) for the early discrepancy.
\item Describe the behavior you may expect for the epidemic as time progresses.  Sketch a rough graph to illustrate your expectations.  While the horizontal axis of your graph is time, what does the vertical axis represent?  
\end{itemize}
\end{enumerate}


\begin{enumerate}[resume]
\item Reflect on initial results.
\begin{itemize}
\item Give a brief summary of your findings from Part I including strengths and weaknesses of using exponential growth to model the Ebola virus outbreak in West African countries.  Consider the initial assumption that the virus spreads at a rate proportional to the number of infected individuals and address why this assumption is reasonable and include its faults, if any.
\end{itemize}
\end{enumerate}


\begin{table}[t]\label{table:cases} 
\begin{center}
\begin{tabular}{|r|c|r|>{\columncolor[gray]{0.90}}r|>{\columncolor[gray]{0.90}}r|>{\columncolor[gray]{0.90}}r|>{\columncolor[gray]{0.90}}r|}\hline
\multicolumn{1}{|c|}{}&\multicolumn{1}{c|}{$t$} & \multicolumn{1}{c|}{Total} & \multicolumn{1}{>{\columncolor[gray]{0.90}}c|}{Total} 
&\multicolumn{1}{>{\columncolor[gray]{0.90}}c|}{Cases in} & \multicolumn{1}{>{\columncolor[gray]{0.90}}c|}{Cases in} & \multicolumn{1}{>{\columncolor[gray]{0.90}}c|}{Cases in}\\ 
\multicolumn{1}{|c|}{Date}&\multicolumn{1}{c|}{(months)} & \multicolumn{1}{c|}{Cases} & \multicolumn{1}{>{\columncolor[gray]{0.90}}c|}{Deaths}
&\multicolumn{1}{>{\columncolor[gray]{0.90}}c|}{Guinea} & \multicolumn{1}{>{\columncolor[gray]{0.90}}c|}{Liberia} & \multicolumn{1}{>{\columncolor[gray]{0.90}}c|}{Sierra Leone}\\ \hline
3/22/2014 & 0 & 49 & 29 & 49 & - & - \\ \hline
4/14/2014 & 1 & 194 & 121 & 197 & 27 & - \\ \hline
5/12/2014 & 2 & 260 & 182 & 248	& 12 & - \\ \hline
6/16/2014 & 3 & 528 & 337 & 398	& 33 & 97 \\ \hline
7/14/2014 & 4 & 982 & 613 & 411	& 174	& 397 \\ \hline
8/13/2014 & 5 & 2,115 & 1,144 & 519	& 786	& 810 \\ \hline
9/14/2014 & 6 & 5,335 & 2,662 & 942	& 2,710	& 1,673 \\ \hline
10/14/2014 & 7 & 9,191 & 4,546 & 1,472	& 4,249	& 3,252 \\ \hline
11/11/2014 & 8 & 14,383 & 5,165 & 1,919	& 6,878	& 5,586 \\ \hline
12/14/2014 & 9 & 18,569 & 6,900 & 2,416 & 7,797 & 8,356 \\ \hline
1/11/2015 & 10 & 21,261 & 8,414 & 2,806	& 8,331	& 10,124 \\ \hline
\end{tabular}
\parbox{13 true cm}{\caption{Total number of Ebola cases in West Africa by date during the first ten months of the outbreak.  Also included are the number of deaths as well as cases reported by individual countries. All case totals are cumulative and do not reflect patient recovery or death.  Data retrieved from the World Health Organization Disease Outbreak News \cite{WHODON} (March 2014-August 2014) and Situation Reports \cite{WHOSR} (September 2014-January 2015). }}

\end{center}
\vspace*{-.6 true cm}
\end{table}


\subsection*{Part II: Limited Growth Model}
From the explorations in the previous section, you likely realized that an exponential model is not appropriate for predicting the behavior of the epidemic beyond the first few months after the initial outbreak. A limited growth model can be used to more accurately describe the total number cases of Ebola in West Africa. 

\begin{enumerate}
\item Limiting the spread of the disease.
\begin{itemize} 
\item Suppose the countries of Guinea, Liberia, and Sierra Leone are quarantined so that no person is allowed to enter or leave the countries.  What would be an upper bound on the number of total cases of Ebola?  Are you making additional assumptions about the disease when determining this upper bound?
\item List a few other factors that could limit the total number of individuals who may become infected with the disease during the outbreak. 
\end{itemize}
\end{enumerate}

One model that incorporates limited growth is the logistic or Verhulst model: 
\begin{equation} 
\dfrac{dN}{dt} = kN\left(1-\frac{N}{M}\right).
\label{logdiffeq}
\end{equation} 

In the logistic growth model (\ref{logdiffeq}), $k$ is the \emph{intrinsic growth} rate while $M$ is referred to as the \emph{carrying capacity}. This model is often associated with population growth, however it is also sometimes used to reflect the spread of contagious diseases. 
\begin{enumerate}[resume]
\item Basic properties of the logistic growth model.
\begin{itemize}
\item Based on the differential equation (\ref{logdiffeq}) above, for what intervals of $N$ will the function $N(t)$ be increasing? Decreasing?
 
For the given situation and interpretation of $N$, is it realistic for $N$ to decrease?  
\item When the number of cases $N$ is close to the carrying capacity $M$,  
what can you say about the rate ($dN/dt$) at which the disease is spreading? Write your answer in words and mathematically as a limit. 
\item How does your upper bound for the total number of cases found in Part I relate to $M$ in the differential equation (\ref{logdiffeq})?  Which value will be greater?
\item When the number of cases is small, what can you say about the rate ($dN/dt$) at which the disease is spreading? Compare to the exponential growth model from Part I.
\end{itemize}
\end{enumerate} 

From the last observation and our work from the previous section, we can conclude that in the early months after the outbreak, the spread of the Ebola virus is exponential.  Thus, we may use the constant of proportionality determined in Part I for the parameter $k$ here.

However, we know that the outbreak of Ebola in West Africa did not reach the entire populations of the three countries discussed here.  Thus, to develop a model with better accuracy, we must determine a more realistic value for $M$.  To do so, we continue investigating the properties of the logistic growth model. 

In order to demonstrate limited growth, or a maximum number of total cases, the growth rate must transition from a rate that increases to a decreasing rate.  That is, the number of new cases per month must begin to decline.  This transition occurs at an inflection point on the graph of $N(t)$.

\begin{enumerate}[resume]
\item  Inflection point and carrying capacity in the logistic growth model.

\begin{itemize}
\item Use a plot of the data from Table \ref{table:cases} or compute the average growth rate from month to month to determine the coordinates $(t, N)$ for the inflection point in the data.
\item Use implicit differentiation and substitution to determine $\dfrac{d^2N}{dt^2}$.  Determine the value(s) of $N$ at which the inflection point(s) occur.  
\item Combine the previous two parts to give an estimate for the carrying capacity, $M$.
\end{itemize}
\end{enumerate}

\begin{enumerate}[resume]
\item Expression for $N(t)$.

\begin{itemize}
\item Solve the separable differential equation (\ref{logdiffeq}). 
\item Use your growth rate from the exponential model for $k$ and the carrying capacity from above for $M$ to write $N(t)$ representing the total number of cases of Ebola in West Africa at time $t$ months. Graph $N(t)$ for $t = 0, 1, 2, \ldots , 24$.
\item Compare the graph of $N(t)$ to your sketch (in the previous section on exponential growth) depicting the behavior of the epidemic that you expect.   
\end{itemize}
%
\item Model predictions and verification.
\begin{itemize}
\item Use your equation for $N(t)$ to estimate the total number of cases of Ebola for times $t = 5, 7,$ and $10$ months.  
\item Compare the predicted values from the previous part to the data in Table \ref{table:cases} and to the results from the exponential model in Part I.  
\item Remark on the performance of the logistic model when used to depict the Ebola outbreak during its first ten months.   
State your expectations for the performance and accuracy of the model for the duration of the epidemic.  
\item Compare the results of the logistic growth model to the data reported by the World Health Organization for $1 \leq t \leq 24$ in Tables \ref{table:cases} and \ref{table:cases2}. Show comparisons using both a table and a graph.

\end{itemize}
 
\begin{table}[h!]
\begin{center}
\begin{tabular}{|r|c|r|>{\columncolor[gray]{0.90}}r|>{\columncolor[gray]{0.90}}r|>{\columncolor[gray]{0.90}}r|>{\columncolor[gray]{0.90}}r|}\hline
\multicolumn{1}{|c|}{}&\multicolumn{1}{c|}{$t$} & \multicolumn{1}{c|}{Total} & \multicolumn{1}{>{\columncolor[gray]{0.90}}c|}{Total} 
&\multicolumn{1}{>{\columncolor[gray]{0.90}}c|}{Cases in} & \multicolumn{1}{>{\columncolor[gray]{0.90}}c|}{Cases in} & \multicolumn{1}{>{\columncolor[gray]{0.90}}c|}{Cases in}\\ 
\multicolumn{1}{|c|}{Date}&\multicolumn{1}{c|}{(months)} & \multicolumn{1}{c|}{Cases} & \multicolumn{1}{>{\columncolor[gray]{0.90}}c|}{Deaths}
&\multicolumn{1}{>{\columncolor[gray]{0.90}}c|}{Guinea} & \multicolumn{1}{>{\columncolor[gray]{0.90}}c|}{Liberia} & \multicolumn{1}{>{\columncolor[gray]{0.90}}c|}{Sierra Leone}\\ \hline
2/15/2015 & 11 & 23,218 & 9,365 & 3,108	& 9,007	& 11,103 \\ \hline
3/15/2015 & 12 & 24,666  & 10,179 & 3,389	& 9,526	& 11,751 \\ \hline
4/12/2015 & 13 & 25,791 & 10,689 & 3,548	& 10,042	& 12,201 \\ \hline
5/10/2015 & 14 & 26,724 & 11,065 & 3,597	& 10,604	& 12,523 \\ \hline
6/14/2015 & 15 & 27,305 & 11,169 & 3,674	& 10,666	& 12,965 \\ \hline
7/12/2015 & 16 & 27,642 & 11,261 & 3,760	& 10,673	& 13,209 \\ \hline
8/16/2015 & 17 & 27,952 & 11,284 & 3,786	& 10,672	& 13,494 \\ \hline
9/13/2015 & 18 & 28,220 & 11,291 & 3,792	& 10,672	& 13,756 \\ \hline
10/11/2015 & 19 & 28,454 & 11,297 & 3,800	& 10,672	& 13,982 \\ \hline
11/15/2015 & 20 & 28,598 & 11,299 & 3,804	& 10,672	& 14,122 \\ \hline
12/13/2015 & 21 & 28,604 & 11,300 & 3,807	& 10,675	& 14,122 \\ \hline
1/17/2016 & 22 & 28,602 & 11,301 & 3,804	& 10,675	& 14,123 \\ \hline
2/14/2016 & 23 & 28,603 & 11,301 & 3,804	& 10,675	& 14,124 \\ \hline
3/13/2016 & 24 & 28,603 & 11,301 & 3,804	& 10,675	& 14,124 \\ \hline
\end{tabular}
\parbox{13 true cm}{\caption{Total number of Ebola cases in West Africa by date during the second year of the epidemic.
Also included are the number of deaths as well as cases reported by individual countries. All case totals are cumulative and do not reflect patient recovery or death.  Data retrieved from the World Health Organization Situation Reports \cite{WHOSR} (February 2015-March 2016). }}
\label{table:cases2}
\end{center}
\vspace*{-.6 true cm}
\end{table}
\item Conclusion.
\begin{itemize}
\item Give a brief summary of your findings from Part II including strengths and weaknesses of using logistic growth to model the Ebola virus outbreak in West African countries. 
Consider the impact of the carrying capacity on the accuracy of the logistic model and address 
whether it is reasonable to develop and use this model during an outbreak, in particular early on before the inflection point appears in the data.

\end{itemize}
\end{enumerate}

\begin{thebibliography}{98}


\bibitem{WHO} World Health Organization. 2016. \emph{Ebola virus disease}. World Health Organization (Fact sheet No.103).  Available at  
\href{http://www.who.int/mediacentre/factsheets/fs103/en/}{http://www.who.int/mediacentre/factsheets/fs103/en/}. 
Accessed January 2016.
%
\bibitem{CDC} Meltzer, M. I., Atkins, C. Y., Santibanez, S., et al. 2014. \emph{Estimating the Future Number of Cases in the Ebola Epidemic - Liberia and Sierra Leone, 2014-2015}. Centers for Disease Control and Prevention 
(MMWR; 63,Suppl-3). Available at    
\href{http://www.cdc.gov/mmwr/preview/mmwrhtml/su6303a1.htm}{http://www.cdc.gov/mmwr/preview/mmwrhtml/su6303a1.htm}. Accessed January 2016.

\bibitem{WHONR} World Health Organization. 2016. \emph{Latest Ebola outbreak over in Liberia; West Africa is at zero, but new flare-ups are likely to occur}. World Health Organization (News Release; January 14, 2016). Available at 
\href{http://www.who.int/mediacentre/news/releases/2016/ebola-zero-liberia/en/}{http://www.who.int/mediacentre/news/releases/2016/ebola-zero-liberia/en/}. 
Accessed January 2016.
%
\bibitem{WHODON} World Health Organization. 2014. \emph{Ebola virus disease}. World Health Organization (Disease Outbreak News; March 22, 2014-August 31, 2014). Available at 
\href{http://www.who.int/csr/don/archive/disease/ebola/en/}{http://www.who.int/csr/don/archive/disease/ebola/en/}. 
%Accessed 
%
\bibitem{WHOSR} World Health Organization. 2014-2016. \emph{Ebola virus disease}. World Health Organization (Situation Reports; September 5, 2014-June 10, 2016).
Available at
\href{http://www.who.int/csr/disease/ebola/situation-reports/archive/en/}{http://www.who.int/csr/disease/ebola/situation-reports/archive/en/}.
%
\bibitem{WHOsummary} World Health Organization. 2016. \emph{Ebola virus disease}. World Health Organization (Situation Summary; April 6, 2016 - May 11, 2016). 
Available at
\href{http://apps.who.int/gho/data/view.ebola-sitrep.ebola-summary-latest?lang=en}{http://apps.who.int/gho/data/view.ebola-sitrep.ebola-summary-latest?lang=en}. Accessed June 2016.

\end{thebibliography}



\teacher{
\section*{NOTES FOR TEACHERS}



This modeling scenario can be used in class or as an out of class project.  You may wish to have students produce a report or simply give short answer responses to the questions and prompts. The logistic growth portion of the project is set up as an introduction to this model for those who have not seen or worked with it previously. Of course the prompts can be removed or adjusted for use with students who have prior experience with population models. Excel or another software may be desired for plotting predicted and observed data points.  Some plots are included in this section and can be distributed to students if desired or if project completed in class.  


The differential equations portion of Part I and Part II of the project require separation of variables.  For use in a calculus course, students will need to know the calculus of transcendental functions, integration by partial fractions, and  implicit differentiation.  Other topics covered include intervals of increasing and decreasing, limits, and inflection points. 



Students are provided with monthly data for the number of cases and deaths however a supplementary file with more frequent and comprehensive data is available.  These additional records of Ebola numbers can be used for further explorations. 

Also included in the file are plots of the data by day and by month as well as plots that correspond to the project questions which can be altered as needed.

 
Figure \ref{fig:cases}  is a plot of the data from Table \ref{table:cases} which can be used to determine the location of the inflection point (at $t=8$) asked for in Part II of the project. 

\begin{figure}[h!]
\begin{center}
\includegraphics[width=4in]{1-38-Table1plot.png}
\parbox{7.8 true cm}{\caption{Total number of Ebola cases in West Africa by month during the first $10$ months of the outbreak.}
\label{fig:cases}}
\end{center}
\end{figure}


The plot in Figure \ref{fig:EXPmodel} shows the values predicted by the exponential growth model in Part I alongside the data from Table \ref{table:cases}.  


\begin{figure}[h!]
\centering
\includegraphics[width=4in]{1-38-EXPmodel.png}
\parbox{7.8 true cm}{\caption{The exponential growth model from Part I plotted along with data from WHO.}
\label{fig:EXPmodel}}
\end{figure}

Figure \ref{fig:LogisticModel} shows the observed data with the predicted values from the logistic growth model in Part II using $k = \ln\left(528/49\right)/3 \approx .7924$ computed from the exponential model in Part I and $M=2(14{,}383) = 28{,}766$. 



\begin{figure}[h!]
\centering
\includegraphics[width=4in]{1-38-LogisticModel.png} 
\parbox{7.8 true cm}{\caption{The logistic growth model from Part II plotted along with data from WHO.}
\label{fig:LogisticModel}}
\end{figure} 

Using a logistic growth model to represent the spread of disease disregards certain aspects of the disease.  A few features of infectious diseases that are not taken into account include whether a recovered subject can become infected again, the duration of the infection and its contagious stage, and the potential for the rate of disease transmission to change due to factors such as immunization. Nevertheless, the logistic growth model fits the data fairly well and thus provides a simple introduction to modeling a real world situation.

The data used in this project was gathered from publications by the World Health Organization.  The data from March 2014 through August 2014 was retrieved from the Disease Outbreak News (DON) provided by the WHO \cite{WHODON}. This data varies in time between records, sometimes with several records for one week.  The numbers beginning in September 2014 were retrieved from the Situation Reports published weekly by WHO \cite{WHOSR}.  Data after March 2016 is from the Situation Summary on the WHO website \cite{WHOsummary}.  The specific date of publication for each Situation Report and Disease Outbreak News that corresponds to each line of data can be found in the supplemental Excel file.  


For reference, the World Health Organization declared Guinea free of Ebola virus transmission on June 1, 2016 (\cite{WHOSR}, published June 2, 2016) after initially making the declaration on December 29, 2015.  
After several previous declarations from May 2015 to January 2016, WHO's most recent (at time of publication) declaration for Liberia being free of Ebola virus transmission was on June 9, 2016 (\cite{WHOSR}, published June 10, 2016).  
Sierra Leone was initially declared free of Ebola virus transmission on November 7, 2015 and after two additional cases the country was declared  free of Ebola virus transmission again on March 17, 2016 (\cite{WHOSR}, published March 16, 2016).  According to WHO, for a country or region to be declared free of Ebola virus transmission, 42 days (two 21-day incubation cycles of the virus) must pass since the last confirmed Ebola patient from that region tested negative for the disease for the second time. 
}


\end{document}
