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\begin{document}

\color{black}
\newpage
% You only need to change the NAME OF SCENARIO in the next line and delete the footnote. See above for instructions on toggling from Teacher Version to Student Version.
\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\MALARIA CONTROL}


\title{TEACHER VERSION\\MALARIA CONTROL}



\markboth{Malaria Control}{Malaria Control}






\author{
David Culver\\
United States Military Academy\\
West Point NY 10996 USA\\

}

\makeStitlePDFLaTex


\begin{abstract}

This project offers students a chance to make policy recommendations based on the analysis of models using both linear (exponential decay) and non-linear (logistic growth) differential equations. The scenario is based on the deployment of the United States Army's 62nd Engineer Battalion to West Africa in the fall of 2014. During this deployment the primary medical threat to Soldiers was malaria. It is up to the student to model and evaluate two methods of malaria prevention, malaria chemoprophylaxis and mosquito population control, and make recommendations to their commander.

\end{abstract}



%\section*{MOTIVATION}

%The intent of this project was to help students become more comfortable using mathematical modeling to assist in decision making. Our hope was for students to place the results of their mathematical analysis in the context of the situation, meaning that students consider other factors not captured in their mathematical model when formulating their recommendations. The project also introduces students to other aspects of decision making analysis such as formulating clear objectives, developing a system to compare policy options, and clearly communicating their results.


\teacher{\centerline{[All Teacher Comments are indented and within brackets.]}

\begin{quotation}
\section*{[ PROJECT OVERVIEW}

\textit{Timing}: 2-3 class periods and 9 total (in and out of class) hours per student

\textit{Prerequisites}: The ability to solve an ordinary differential equation analytically and numerically as well as the ability to use a spreadsheet, e.g., Excel.

\textit{Technology Needed}: Computer Algebra System or  numerical solving algorithm and spreadsheet

\textit{Background}: This project was developed in the fall of 2015 for a 4.5 credit hour calculus course with over 700 students. The project was an individual assignment and served as the capstone event for the differential equations block of the course. We devoted three class periods to the project and expected students to spend around 9 hours working on it.]
\end{quotation}}

\section*{SCENARIO DESCRIPTION}
\teacher{
\begin{quotation}
\begin{center}
\parbox{9 true cm}{[This portion includes the student version  with associated comments. The latter are indented and within brackets.]}
\end{center}
\end{quotation}
}

\subsection*{Background}
\teacher{
\begin{quotation}[This section motivates the problem and introduces the scenario to the student. The facts on malaria illustrate the significance of the malaria problem in Africa and also its impact on US military deployments in the region. The scenario is based on the actual deployment of the US Army 62nd Engineer Battalion from Ft. Hood, TX to Liberia in 2014. The battalion deployed to Liberia in response to the massive Ebola outbreak in the region and worked with the Liberian Army to construct medical treatment facilities throughout the country.]
\end{quotation}
}



Each year it is estimated that nearly 200 million people worldwide contract malaria, resulting in the deaths of approximately 584,000 people. Since 2005 more than 800 US service members have been diagnosed with malaria during overseas deployments. In 2014 a total of 44 service members, including 30 soldiers, contracted malaria \cite{Kime 2015}. Clearly, malaria is a continuing threat to soldiers operating overseas.

Malaria is a disease caused by a parasite that infects red blood cells. The parasite is transmitted from person to person through the bite of mosquitoes. Avoiding mosquito bites is the only sure method to prevent malaria infection. However, there are a number of medicines available to minimize the chance of contracting the illness in the event a mosquito bite occurs \cite{USAFRICOM 2015}.

You are the Battalion Medical Service Officer for the 62nd Engineer Battalion. Your unit has just been notified that you will deploy to Liberia within the next 45 days to assist the Liberian military in its construction of medical treatment facilities. Your commander is aware of the malaria threat in the region and has asked you to analyze malaria preventive measures for the soldiers in your unit.


\teacher{
\begin{quotation}
\begin{center}
\parbox{9 true cm}{[The project consists of two parts. Both parts are based on the same scenario but are not dependent on each other.]}
\end{center}
\end{quotation}
}

\subsection*{Part I: Numerical Methods and Pharmacokinetics of Malaria Chemoprophylaxis}

\teacher{
\begin{quotation}
[The intent of Part I is to have students answer pertinent questions that a battalion commander might ask by generating a simple differential equation model given only a description of the scenario and data. A secondary objective is to have students compare multiple numerical solution methods. These objectives are addressed by having students create and solve an exponential decay model for the pharmacokinetics of the anti-malarial drug Malarone using analytic and multiple numerical solution methods.]
\end{quotation}}


Your first task is to analyze the anti-malarial drug dosing regimen for the battalion. Your commander's primary concerns are how soon to start treatment before the battalion arrives in Liberia and the potential risks if soldiers miss one or two of their scheduled doses. These questions require an understanding of pharmacokinetics.  

A complete understanding of the pharmacokinetics of a drug requires information on the processes of absorption, distribution, metabolization, and excretion of the drug from the body. In order to simplify this problem we will assume that the anti-malarial medication is immediately absorbed into the blood stream and that all soldiers begin treatment with zero anti-malarial medication in their body. After the drug is absorbed into the blood, the rate of elimination of the drug from the body is proportional to the amount of drug currently in the blood. In other words, the more of the drug in the blood stream, the faster it is removed from the body. We will also assume that the rate of excretion of the drug is the same for all soldiers in the battalion.

In 2011 United States Africa Command (AFRICOM) issued a policy change recommending Malarone as the malaria chemoprophylaxis option for all US military operating in high malaria transmission areas of Africa \cite{Magill 2015}. Malarone is a pill taken at the same time each day that consists of two separate drugs, atovaquone and proguanil. A standard adult pill of Malarone contains 250mg of atovaquone and 100mg of proguanil. An individual must maintain at least 300mg of atovaquone and 30mg of proguanil in their blood at all times in order to prevent contracting malaria. A typical soldier's body eliminates atovaquone at a rate that corresponds to a half-life (time to eliminate one half of the original dose) of 48 hours. The half-life of proguanil is 12 hours \cite{FDA 2008}.


Pharmacokinetics is the study of what the body does to a drug. Specifically, it refers to the movement of the drug into, through, and out of the body. Understanding the pharmacokinetics of a drug allows medical professionals to develop appropriate drug dosing regimens for their patients.

\vspace*{.2 true cm}

\teacher{
\begin{quotation}
[All information on Malarone is accurate and was taken from the USDA website. The only fabricated information in this section is the minimum drug thresholds for malaria prevention. These values were selected so that the results obtained from the model would be similar to actual drug behavior. We also intentionally selected values so that the levels of proguanil and atovaquone would reach their thresholds at significantly different times when decaying from ``steady state''.

Most students required significant coaching from instructors before they were able to create a workable model. This was time intensive but resulted in very good class discussion on how and why differential equations are used. Once students had a workable model they generally did well finding an analytic solution and implementing the numerical solution methods. For those students that struggled implementing the numerical solution methods, instructors encouraged them to first replicate the example from the Numerical Methods Supplement to ensure their Excel syntax was correct before attempting to evaluate their pharmacokinetics model.]
\end{quotation}
}

\subsection*{Requirements:}

\teacher{
\begin{quotation}
[Below are the requirements asked of our students. However, the assignment can be tailored by only choosing a subset of these requirements.]
\end{quotation}
}

\begin{enumerate}
\renewcommand{\labelenumi}{\alph{enumi})}
\item At $t=0$ a typical soldier ingests one pill of Malarone. Write an initial value problem (IVP) that models the mass (mg) of the drug proguanil in the soldier's blood for $t \geq 0$. Solve this IVP analytically. After a soldier takes his first pill of Malarone, if he takes no further medication, for how many hours can he expect to have enough \emph{proguanil} in his blood to prevent contracting malaria?
\item 	Approximate the solution to the IVP you developed in part (a) over the next seven days (168 hours) by implementing Euler's method in Excel with two step sizes: $h = 6$ hours and $h = 1$ hour. Compute the Sum of Squared Error (SSE) for both step sizes by comparing your estimate to the analytic solution (see the Numerical Methods Supplement for a discussion on SSE).
\item 	Approximate the solution to the IVP over the next seven days by implementing Improved Euler's method in Excel with step size: $h = 1$ hour. Compute the SSE using your analytic solution.
\item 	Create a table summarizing the results of all numerical methods (parts c and d). This table should include columns for time, the actual mass of proguanil in the soldier's blood (from part b), and the approximate values of the solution for all methods and step sizes at $t = 0,24,48,...,168$ only. You should have a total of 5 columns. Include an additional entry at the bottom of each of the last 3 columns providing the SSE for that method and step size. Discuss your results. Your analysis should include a comparison of error for different step sizes and for different methods.
\item Write an IVP that models the mass (mg) of the drug \emph{atovaquone} in a typical soldier's blood for $t \geq 0$ if a soldier ingests only one pill of Malarone at $t = 0$. Approximate the solution to this IVP using Improved Euler's method in Excel with step size: $h = 1$ hour.
\item Using your solutions to parts (c) and (e), if a soldier ingests one pill of Malarone at the same time each day, how long before your unit's departure to Liberia should soldiers begin taking their pills? In other words, how long until the soldiers will consistently have enough proguanil AND atovaquone in their blood to prevent contracting malaria? Illustrate your result using a chart or table.
\item Assume a typical soldier reaches a "steady state" level of proguanil and atovaquone in their body after their 8th dose of Malarone. If after reaching "steady state" a soldier misses one dose, are they at risk of contracting malaria? How about after missing two consecutive doses? Explain.
\end{enumerate}

\subsection*{Part II: Mosquito Population Control}

\teacher{
\begin{quotation}
[The intent of Part II of the project is to help students become more comfortable using mathematical modeling to assist in decision making. Our hope was for students to place the results of their mathematical analysis in the context of the situation, meaning that students consider other factors not captured in their mathematical model when formulating their recommendations. This objective is addressed by having students use Improved Euler's method to implement a logistic population with harvesting model. The students are presented with three different courses of action that impact the model parameters in different ways. Each course of action has different environmental, soldier health, and monetary impacts that must be considered but are not included in the mathematical model. Students then must interpret the commander's objectives and recommend one of the courses of action based on their analysis.]
\end{quotation}
}


Unfortunately anti-malarial drugs are not 100\% effective at preventing the spread of malaria. The only sure method to prevent contracting malaria is to avoid mosquito bites. In light of this, your commander has asked you to look at methods to control the mosquito population in the battalion area near the living quarters. His objective is to reduce the mosquito population in both the short (less than 6 months) and long (greater than 6 months) term while also minimizing the environmental impact and potential soldier health problems associated with the mosquito control measures that are implemented.

There are two typical approaches to mosquito control. The first approach is to directly kill mosquitoes using an insecticide spray. The effectiveness of this method depends on the frequency of spraying and the concentration of the insecticide. However, frequent spraying of insecticide has potential negative health effects on both humans and nearby wildlife. This method is also temporary as the mosquito population will recover once routine spraying has stopped. The second approach to mosquito population control is to eliminate the resources that support the mosquito population. This is done through the destruction of mosquito breeding grounds and involves everything from filling in small puddles to the draining of swamps and marshes. Keeping in consideration manpower and budget constraints, you must find the combination of these two approaches that best meets your commander's objectives over the next six months.

Studies on mosquito control indicate that the rate of change of the mosquito population, $P$, (measured in millions of mosquitoes) in Liberia can be accurately modeled using the following Initial Value Problem:

\[\dfrac{dP}{dt}=kP \left(1-\dfrac{P}{M} \right)-EP,\qquad P(0) = P_0 \]

\noindent where $E$ is a constant that represents the effectiveness of insecticide spraying efforts, $M$ is the local carrying capacity and is affected by efforts to reduce mosquito breeding grounds, $k$ is the population growth constant, $P_0$ is the initial population, and time $t$ is measured in months. Current estimates suggest that there are approximately 10 million mosquitoes living near your unit's base camp in Liberia. Data indicates that $M = 11$ million and $k = 1.2$ for this mosquito population.

Use the information provided and Improved Euler's method to evaluate the three mosquito population control courses of action (COA) that have been proposed by the battalion staff. Recommend a COA to your commander that you feel best supports his objectives and support your recommendation with mathematics. Be sure to include other considerations (not just what your mathematical model tells you) in your analysis. Tables, graphs, and other visual representations that summarize and support your analysis and recommendation are encouraged and should be included in an appendix.

\begin{enumerate}
\renewcommand{\labelenumi}{COA \Alph{enumi} -}
\item In this COA the battalion would devote most of its available resources for mosquito control to the use of insecticide. Soldiers would spray highly concentrated and expensive insecticide in the soldier living areas daily. Due to the emphasis on the use of insecticide the battalion would be limited to destroying small mosquito breeding grounds only on the base camp. You estimate a value for $E$ of $E_A$ (see table below) and a new carrying capacity of $M_A$ million mosquitoes that would be in effect immediately.
\newline
\item In this COA the battalion would devote most of its resources towards destroying mosquito breeding grounds. This COA would involve the permanent draining of a nearby marsh that is believed to be the primary breeding ground for mosquitoes at the base camp. Draining the marsh would take time to complete, delaying any impact on the mosquito population by two months, but would lower the carrying capacity to $M_B$ million mosquitoes. The battalion would still spray insecticide but less frequently and with less potency than in COA A. It is estimated that spraying would immediately result in a value for $E$ of $E_B$.
\newline
\item This COA seeks to balance the insecticide and breeding ground destruction approaches. In this COA the battalion would use the same, more potent, insecticide as COA A but spray less frequently. This would free up manpower to destroy small mosquito breeding grounds in both the base camp and in the areas surrounding the base camp. You estimate a new value for $E$ of $E_C$ and a new carrying capacity of $M_C$ million mosquitoes that would be in effect immediately.
\end{enumerate}

\begin{table}[h]
\centering
\begin{tabular}{|c|cc|cc|cc|}
\hline
               & \multicolumn{2}{c|}{\textbf{COA A}} & \multicolumn{2}{c|}{\textbf{COA B}} & \multicolumn{2}{c|}{\textbf{COA C}} \\ \cline{2-7} 
               & \textbf{$M_A$}   & \textbf{$E_A$}   & \textbf{$M_B$}   & \textbf{$E_B$}   & \textbf{$M_C$}   & \textbf{$E_C$}   \\ \hline
\textbf{Set 1} & 10.5             & 0.55             & 6.0              & 0.10             & 9.0              & 0.40             \\
\textbf{Set 2} & 10.0             & 0.60             & 5.5              & 0.10             & 8.5              & 0.45             \\
\textbf{Set 3} & 10.5             & 0.50             & 6.5              & 0.10             & 9.0              & 0.40             \\ \hline
\end{tabular}
\end{table}





\teacher{
\begin{quotation}
[We developed three sets of parameters for Part II of the project in order to minimize the opportunity of students copying from one another and to keep discussions among groups and individual students  about process, not just answers.

Although this part of the project is based on a standard population model and references real mosquito population control techniques, the values of all parameters and initial conditions were created by us. We selected plausible values that prevented one course of action from being decisively better than the others. For example, course of action B is more effective than course of action C at permanently eliminating the mosquito population. However, course of action C is less damaging to the environment than course of action B. Trade-offs such as this force students to use the minimal guidance they have received in the prompt and their judgment to develop and weigh evaluation criteria.

After completing Part I of the project most students had a solid understanding of how to implement Improved Euler's method in Excel and required very little assistance on their calculations. However, students struggled with understanding how to balance their mathematical results with the other considerations in the scenario to best meet the commander's objectives. Many students simply ignored all factors besides long term mosquito population and recommended course of action B. Other students understood that they needed to incorporate factors besides mosquito population into their analysis, but were unsure how to do so. After coaching, most students in this category simply listed the pros and cons of the course of action that they recommended. Some top students, however, were able to derive a very basic Pugh matrix to support their recommendation. We recommend that instructors provide their students a brief overview of decision analysis if they decide to use this part of the project.]
\end{quotation}
}

\subsection*{Hardcopy Submission Requirements and Guidance:}

\teacher{
\begin{quotation}
[This section of the prompt provides guidance on what we expected from our students. We intentionally limited the written submission to a one page executive summary of Part II in order to force students to remain concise in their explanations. Typically decision makers do not have time to thoroughly read an entire decision analysis paper, so this is a good opportunity to prepare students for the real world. Forcing students to be concise also has the additional benefit of reducing the grading burden on the instructor. We also asked students to submit all electronic files that they used for their calculations to assist with assigning partial credit during grading.]
\end{quotation}
}

Your submission should follow this general format:
\begin{enumerate}
\item Cover Sheet
\item Executive Summary - The executive summary consists of a single spaced 1 page paper formatted with Times New Roman, 12 point font, and documented in accordance with the MLA style. The executive summary should summarize your work for Part II in such a way that the reader can rapidly become acquainted with the material. It should contain a brief description of the problem, important background information, a discussion of pertinent assumptions, a short description of your methodology, concise analysis, and your main conclusions. Assume the reader is familiar with the basics of calculus and differential equations, so there is no need to walk through every step of your solution process or include equations. However, you should still describe the processes and mathematical techniques you used to reach your conclusions and explain why you used them. To assist the reader, be sure to present your Bottom Line Up Front (BLUF). Refer to appendices as necessary.
\item Appendices - Appendices should be neatly formatted and present information in a logical manner. Do NOT simply print out Mathematica code or a long Excel spreadsheet. Consolidate your results and provide a short explanation of what it is the reader is seeing while also highlighting key pieces of information in the appendix.
\begin{enumerate}
\item Appendix A - Answers and analysis for Part I
\item Appendix B - Provide a table that summarizes the results of your analysis for all three COAs in Part II. Also include a copy of all work you used to analyze your recommended COA.
\item Additional Appendices - Include additional appendices as necessary.
\end{enumerate}
\item Works Cited
\end{enumerate}



\subsection*{Appendix A: Numerical Methods Supplement \cite{Mugg 2014}}

\teacher{
\begin{quotation}
[Prior to the project all students had experience in the classroom with Euler's method but had not been exposed to the concepts of Sum of Squared Error and Improved Euler's method. When the project was first given to the students each instructor spent approximately ten minutes providing an overview of these two new concepts. We then provided this supplement to assist students with implementing these numerical techniques in Excel. This proved to be a critical resource as many students struggled initially with implementing Improved Euler's method. Instead of tackling the more complicated problems in the project many students began by replicating the examples in this supplement. Once they confirmed that their Excel code correctly executed the numerical technique they were able to use it as a template to solve the more complicated problems.]
\end{quotation}
}




Many differential equations cannot be solved analytically. When presented with a differential equation that cannot be solved explicitly we can turn to numerical methods to provide a near approximation of the solution. This is similar to a situation in calculus: if we cannot find an antiderivative in terms of elementary functions, we turn to a numerical method such as area estimation using endpoints or midpoints to evaluate a definite integral.

With numerical methods, as with qualitative methods, we obtain information about the solution - quantitative and qualitative respectively - without the use of a solution formula.





\subsection*{Euler's Method}

The simplest numerical solution method was developed by Euler and is based on the tangent line approximation to a function. Given the initial value $y(t_0) = y_0$, we define $t_n$ and $y_n$ recursively by the following:

\vspace{-.6 true cm}

\[n=0,1,2,\ldots \newline \]
\[t_{n+1}=t_n + h \newline \]
\[y_{n+1} = y_n + hf(t_n,y_n) \newline \]

\noindent where $f(t_n,y_n) = y'_n$ and $h = t_{n+1} - t_n$.
The approximations $y_{n+1}$ become better as $h$ (step size) is reduced.\\


\vspace{-.2 true cm}\noindent \textit{Example 1} \quad Consider the initial value problem:
\[\frac{dy}{dt} = \frac{t}{y} \qquad y(0) = 1. \]

We wish to approximate $y(0.3)$ using Euler's method with step size $h=0.1$. After three steps we find $y(0.3)=1.0298$. See Figure~\ref{fig:Euler1b}. 




\begin{figure}[!h]
\begin{center}
\includegraphics[width=11.5cm]{1-24-exceleuler1b.eps} 

\parbox{14.5 true cm}{\caption{\label{fig:Euler1b} Using Euler's method with step size $h=0.1$ to approximate $y(0.3)$ with three steps. }}
\end{center}

\vspace*{-.5 true cm}
\end{figure}

Next use $h=0.05$ and six steps. Notice that $y(0.3)=1.036982$. See Figure~\ref{fig:Euler2b}. 

\begin{figure}[!h]
\begin{center}
\includegraphics[width=11.5cm]{1-24-exceleuler2b.eps} 

\parbox{14.5 true cm}{\caption{\label{fig:Euler2b} Using Euler's method with step size $h=0.05$ to approximate $y(0.3)$ with six steps. }}
\end{center}
\end{figure}


In this example, cutting the step size in half had the effect of cutting the error approximately in half. As expected the global error in the Euler's method is proportional to $h$.

The squared error in the tables above were determined by squaring the difference between the exact value $y_a$ and the experimentally determined value $y_n$ from Euler's method.
\vspace*{-.2 true cm}

\[ {\rm SquaredError} = (y_a(t) - y_n)^2 \]

\vspace*{.2 true cm}
The Sum of Squared Error (SSE) was determined by adding up the squared error values,

$$ SSE = \sum_{i=0}^{n-1} (y_a(t) - y_n)^2\, . $$

\subsection*{Improved Euler's Method or Huen's Method}

We again start with the tangent line approximation using Euler's method; however, we now replace the slope $y'(t_n)$ by the average of the two slopes $y'(t_n)$ and $y'(t_{n+1})$, such that:

\[n=0,1,2,\ldots \newline \]
\[t_{n+1}=t_n + h \newline \]
\[y_{n+1} = y_n + h \frac{(f(t_n,y_n)+f(t_{n+1},y_{n+1}^*))}{2},  \]

\vspace*{.2 true cm}

\noindent where $f(t_n,y_n)=y'_n$, and $f(t_{n+1},y_{n+1}^*)=y'_{n+1}$, given the initial value $y(t_0)=y_0$. Notice that $y'_{n+1}$ requires $y_{n+1}^*$. Therefore we first compute one Euler's method step to approximate $y_{n+1}^*$, such that $y_{n+1}^*=y_n+hf(t_n,y_n)$. We will call this solution a ``predictor'' for $y_{n+1}$. We then ``correct'' this solution using what is referred to as the improved Euler's method or Huen's Method\cite{HuenWiki}, which simplifies to:

\[y_{n+1} = y_n + h \frac{(f(t_n,y_n)+f(t_{n+1},y_n+hf(t_n,y_n)))}{2}, \]

\noindent \textit{Example 2} \quad We again consider the initial value problem from Example 1 and find an approximation to $y(0.3)$ with $h=0.1$. We obtain $y(0.3)=1.0441$. See Figure \ref{fig:Euler3b}.  



\begin{figure}[!h]
\begin{center}
\includegraphics[width=11.5cm]{1-24-exceleuler3b.eps} 

\parbox{14.5 true cm}{\caption{\label{fig:Euler3b} Using Improved Euler's method with step size $h=0.1$ to approximate $y(0.3)$. }}
\end{center}
\end{figure}


Notice that the SSE in the improved Euler method with $h=0.1$ is considerably less than the SSE in the Euler method with $h=0.05$. The global error in the improved Euler method is proportional to $h^2$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{thebibliography}{98}



\bibitem{Kime 2015} Kime, Patricia. 23 April 2015. Troops get malaria during Ebola deployment. {\it Military Times\/}. \url{http://www.militarytimes.com/story/military/benefits/health-care/2015/04/23/us-military-ebola-deployment-malaria/26236769/}. Accessed 26 August 2015.

\bibitem{Magill 2015} Magill, Alan J. 10 July 2015. Special Considerations for US Military Deployments. {\it Centers for Disease Control and Prevention\/}. \url{http://wwwnc.cdc.gov/travel/yellowbook/2016/advising-travelers-with-specific-needs/special-considerations-for-us-military-deployments}. Accessed 26 August 2015.

\bibitem{USAFRICOM 2015} Malaria. {\it United States Africa Command\/}. \url{http://www.africom.mil/malaria}. Accessed 26 August 2015.

\bibitem{FDA 2008} Malarone: Prescribing Information. {\it US Food and Drug Administration\/}. \url{http://www.fda.gov/ohrms/dockets/ac/05/briefing/2005-4089b1_05_05_atovaquone.pdf}. Accessed 26 August 2015.

\bibitem{Mugg 2014} Mugg, Patrick. 7 Oct 2014. Numerical Methods Supplement. {\it MA205 Project 1\/}.

\bibitem{HuenWiki} Wikipedia contributors.  Heun's method. Wikipedia, The Free Encyclopedia. \url{https://en.wikipedia.org/w/index.php?title=Heun\%27s_method\&oldid=666543258}. Accessed 6 May 2016.  


\end{thebibliography}



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