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\begin{document}

\newpage


\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\MODELING A NONLETHAL INFLUENZA EPIDEMIC}



\markboth{Epidemic Modeling}{Epidemic Modeling}






\author{Sheila Miller, Formerly of\\
New York City College of Technology\\
City University of New York\\
Brooklyn  NY 11201 USA\\
c/o Director@simiode.org
}

 \makeStitlePDFLaTex


\begin{abstract}
This paper presents real-world data, a problem statement, and discussion of a common approach to modeling that data, including student responses.  In particular, we provide time-series data on the number of boys bedridden due to an outbreak of influenza at an English boarding school and ask students to build a mathematical model, either discrete or continuous, of this epidemic, and to estimate the parameters in their model and validate it against the data.  Students will need access to a computer or computer lab with spreadsheet software, a computer algebra system, or a sufficient statistical analysis system such as R.

\end{abstract}

\section*{SCENARIO DESCRIPTION}


A boarding school is a relatively closed community in which all students live on campus, teachers tend to live on or near campus, and students do not regularly interact with people not in the boarding school community. Table~\ref{table:Bedriddens} gives data for an influenza outbreak at a boarding school in England during which there were no fatalities. These data were compiled from the Communicable Disease Surveillance Center \cite[p.~587]{CDSC1978} and are given as an example by Murray in his book on mathematical biology \cite{Murray1993}. The data values were extracted from the graph found in \cite{Switkes2003}.

\begin{table}[h]
\small
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline
\hspace*{-.1 true cm} Time (in days) \hspace*{-.1 true cm} & \hspace*{-.1 true cm}  0\hspace*{-.1 true cm} & \hspace*{-.1 true cm}  1\hspace*{-.1 true cm} & \hspace*{-.1 true cm}  2 \hspace*{-.1 true cm} & \hspace*{-.1 true cm} 3\hspace*{-.1 true cm} & \hspace*{-.1 true cm}  4 \hspace*{-.1 true cm} & \hspace*{-.1 true cm} 5\hspace*{-.1 true cm} & \hspace*{-.1 true cm}  6 \hspace*{-.1 true cm} & \hspace*{-.1 true cm} 7 \hspace*{-.1 true cm} & \hspace*{-.1 true cm} 8\hspace*{-.1 true cm} & \hspace*{-.1 true cm}  9\hspace*{-.1 true cm} & \hspace*{-.1 true cm}  10 \hspace*{-.1 true cm} & \hspace*{-.1 true cm} 11\hspace*{-.1 true cm} & \hspace*{-.1 true cm}  12 \hspace*{-.1 true cm} & \hspace*{-.1 true cm} 13\\ \hline
\hspace*{-.1 true cm} Number of Bedridden Boys \hspace*{-.1 true cm} & \hspace*{-.1 true cm}  1\hspace*{-.1 true cm} & \hspace*{-.1 true cm}  3 \hspace*{-.1 true cm} & \hspace*{-.1 true cm} 25 \hspace*{-.1 true cm} & \hspace*{-.1 true cm} 72\hspace*{-.1 true cm} & \hspace*{-.1 true cm}  222\hspace*{-.1 true cm} & \hspace*{-.1 true cm}  282 \hspace*{-.1 true cm} & \hspace*{-.1 true cm} 256 \hspace*{-.1 true cm} & \hspace*{-.1 true cm} 233\hspace*{-.1 true cm} & \hspace*{-.1 true cm}  189 \hspace*{-.1 true cm} & \hspace*{-.1 true cm} 123 \hspace*{-.1 true cm} & \hspace*{-.1 true cm} 70 \hspace*{-.1 true cm} & \hspace*{-.1 true cm} 25\hspace*{-.1 true cm} & \hspace*{-.1 true cm}  11\hspace*{-.1 true cm} & \hspace*{-.1 true cm}  4\\ \hline
\end{tabular}

\caption{\label{table:Bedriddens} Total Number of Bedridden Boys on Day $t$.}
\end{center}
\vspace*{-.6 true cm}
\end{table}
There were 763 boys at the English boarding school from which our data was obtained. We can see from Table~\ref{table:Bedriddens} that the initial number of bedridden students is one. The data given is number of bedridden boys rather than number of boys with influenza; let us assume that the number of boys who are bedridden on day $t$ consists exactly of all those who are newly infected and all continuing infections (students who are still on bedrest after being infected on a prior day), namely that no boys are bedridden for any other reason during this period of time and also that all boys with influenza will be not only symptomatic but bedridden\footnote{One may wish to test the effects of this last assumption on the model in the analysis phase.  Another perhaps more realistic assumption is that the bedridden boys represent a fixed percentage of the infected students.}.  These assumptions, together with excluding teachers and staff from the population, imply that the number of students infected on day zero is one, or, $I(0) =1$, where $I(t)$ is the number of students symptomatic on day $t$ (either infectious or infected).

\begin{enumerate}
\item Plot the data points of the number infected students on day $t$  against time in days, $t$. Comment on what you see and relate it to how you expect influenza to spread in a small, semi-closed, community of healthy, school-age children.  Recall that influenza is spread primarily through social contact, but moisture droplets containing the virus can also contaminate shared surfaces, so influenza can be transmitted from one person to another without direct contact.
\item  To track an epidemic we need to know how many people are susceptible, how many are infected, and how many have recovered (and are therefore, in the case of a virus such as influenza, immune). Build a differential or difference equation model of $S(t)$, the number of susceptible students on day $t$, $I(t)$, the number of infectious or infected students on day $t$, and $R(t)$, the number of recovered students on day $t$, for $t = 0, 1, \ldots , 13$.  Defend each term in each equations, and define the parameters and their units.
\item Play with the parameters in the model you created in (2) to see which parameter values might produce the behavior demonstrated in the plot of the given data you constructed in (1).  Are all of those parameter values reasonable?
\item Adjust the parameters to find a best fit to model the data. Remember to define your criterion for ``best fit.''
\item Confirm or validate that your model is reasonable. Conduct a sensitivity analysis for at least one of your assumptions.
\item  Compare the values of your model from (4) with the data. Use all available and appropriate techniques when assessing your model, including quantities that can be computed, such as sum of squared error and coefficient of determination and also other comparisons, such as the predictive or visual power of the model. How well does the general shape of the model fit the data? Assess and comment upon the fit of your model and on discrepancies between your model and the data.  Consider whether your model predicts infections late or early (compared to the data) and what consequences this might have for an attempt to manage an epidemic. Conduct a sensitivity analysis for one of your parameters.
\end{enumerate}

\teacher{
\section*{NOTES FOR TEACHERS}

A published paper on this work appears in \cite{MillerHelms2010}.

The students will likely produce, either through their own initiative or through guidance from the teacher, an SIR (Susceptible, Infected, Recovered) model, a standard for modeling epidemics.  The  visualization in Figure~\ref{fig:6-1-ArrowSIRDiagram} of the model could be helpful. 

\begin{figure}[h]
\centering
\includegraphics[width=10cm]{6-1-ArrowSIRDiagram.eps}


\parbox{10.8 true cm}{\caption{\label{fig:6-1-ArrowSIRDiagram}  A visualization of the flow from susceptible to infected to recovered, illustrating that some interactions between susceptible and infected individuals lead to new infections, and some infected individuals recover.}}

\end{figure}

First we give the continuous form (\ref{eq:cont}) of the model using ordinary\footnote{As opposed to partial.} differential equations:

\vspace*{-.4 true cm}
\begin{eqnarray}\label{eq:cont}  \nonumber
S'(t) &=& - a S(t) I(t)\\
I'(t) &=& a S(t) I(t) - b I(t)\\  \nonumber
R'(t) &=& b I(t).
\end{eqnarray}
where $S(t)$ is the number of students susceptible to the disease at time $t$, $I(t)$ includes all students infected or infectious at time $t$, and $R(t)$ is the number of students who have recovered at time $t$.

The discrete time (\ref{eq:disc}) form of the model using difference equations is:
\begin{eqnarray}\label{eq:disc}  \nonumber
S_{n+1} &=& S_n - a S_n I_n\\
I_{n+1} &=& I_n + a S_{n} I_n - b I_n \\  \nonumber
R_{n+1} &=& R_n + b I_n. 
\end{eqnarray}
where $S_n$ is the number of susceptible students at time $n$ (which for a virus can be assumed to include all people who are neither infected nor recovered from this particular virus; this assumption imposes that the only way to obtain immunity to the virus is to contract it as vaccination has not been considered), $I_n$ is the number of students infected or infectious at time $n$, and $R_n$ is the number of students who have recovered at time $n$.

In both models, the parameter $a$ measures the rate (or proportion) of infection.  Notice that the spread of the disease depends not only on the respective numbers of people susceptible and infectious, but on their interactions, so the parameter $a$ depends on both the virus and  the social dynamics of the population.  The maximum possible interactions between the susceptible and infectious groups are if every person interacts with every other: $S(t)I(t)$ or $S_n I_n$.  The proportional mixing $aS(t)I(t)$ or $aS_nI_n$ represents the encounters between susceptible and infectious individuals that results in new infections.  The parameter $b$ is the recovery rate of infected individuals. For the sake of disambiguating the continuous model from the discrete one, we reserve the notation $S(t)$ for the continuous case and use $S_n$ for difference equations.

The parameters for the continuous model found in \cite{Murray1993} are $a = 0.00218$ and $b = 0.441$.  In the discrete time approach one is assuming, in essence, that all changes in status (e.g., from infected to recovered) and all interactions between categories happen at the same time each day.  The continuous time model allows for people to interact, become well, and become ill at any time.

We supply two Mathematica notebooks (and pdf versions for those who do not read Mathematica), 6-001-T-Mma-Epidemic-TeacherVersion-Continuous and 6-001-T-Mma-Epidemic-TeacherVersion-Discrete as well as an Excel File, 6-001-T-EXCEL-Epidemic-TeacherVersion.xlsx.

\subsection*{Difference equation model using Excel spreadsheet}

We  offer a difference equation (discrete) model approach in 6-001-T-Mma-Epidemic-TeacherVersion-Discrete.nb and .pdf and in 6-001-T-Excel-Epidemic-TeacherVersion.xlsx. 

A common method for improving parameter estimates to fit a model to data is to calculate
and minimize the sum of squared errors (SSE), 
\begin{equation}\label{eq:SSEMin}
{\rm SSE}  = \sum_{i=1}^{n}(y_i-\hat{y}_i)^2\, , 
\end{equation} where $n$ is the number of observations, $y_i$ is the data value of the time-dependent observation, and $\hat{y}_i$ is the model prediction of $y_i$.  

We offer a result in Figure~\ref{fig:6-1-EXCELFirstModel} which uses Excel's Solver command  to determine the parameters $a$ and $b$ which minimize the sum of square errors between the results from the difference equation model 
(\ref{eq:disc}) and the data as computed in (\ref{eq:SSEMin}).  Here the iteration method is simply Euler's Method with $\Delta t = 0.1$ and the differences between the model and the data taken only at integer values of time. See the Excel  file 6-001-T-Excel-Epidemic-TeacherVersion.xls found at the modeling scenario site for this model in {\tt www.simiode.org} for the code, execution, and results. 





\begin{figure}[h]
\centering
\includegraphics[width=8cm]{6-1-EXCELFirstModel.eps}

\vspace*{- 6.0 true cm}
\parbox{13.2 true cm}{\caption{\label{fig:6-1-EXCELFirstModel}  Model of number of infected patients at the English boarding school plotted over the actual data (blue diamonds), using SIR difference equation model (\ref{eq:disc}) (red squares) with model parameters, $a = 0.003618$ and $b = 0.64867$, determined by Excel's Solver command which minimizes the sum of square errors between the model and the data.  The sum of square errors (SSE) is 99,445.52. }}
\vspace*{- .2 true cm}
\end{figure}

Solver  determines the two parameters $a$ and $b$ which minimize the SSE:

$$a = 0.003618, \qquad  b = 0.64867, \quad {\rm and} \quad   SSE = 99,44552\, .$$
We see in Figure~\ref{fig:6-1-EXCELFirstModel} that the model  does not look good when plotted over the data.



Can we reason our way to good parameters $a$ and $b$?  From the information available in Table \ref{table:Bedriddens}, we know: the size of the population, 763; that one person---typically called patient zero---is infected on day zero; that three people are infected on day one; and the total number of bedridden boys each day.  We do
not know how quickly people recover and are removed from the infected category.  Therefore, some assumptions will be necessary in order to make parameter estimation possible. We assume that no one in the population starts off with immunity to the disease and that an infected person requires exactly two days to recover (which we identify with leaving bedrest).

With the given information and our assumptions it is possible to make an initial estimate for $a$, the infection rate.  A total of three people have been removed from the susceptible pool on day one (the one original infection, patient zero, and the two new infections). From $S_1-S_0 = -a S_0 I_0$ we have $760-762 = -a \times 762\times 1$  and hence $a = 0.002625$. 

These assumptions also allow us to make an initial estimate for $b$, the rate of recovery. On days zero and one, no students have yet recovered from the flu (according to our assumptions) and without vaccination, immunity to the virus is achieved through contracting it. On day two, the original student infected recovers. From the difference equation we have: $R_2 - R_1 = b I_1$, hence $1 -0 = b \times 3$ and $b = 0.3333$.


\begin{figure}[h]
\centering
\includegraphics[width=8cm]{6-1-EXCELPoorModela.eps}

\vspace*{- 6.0 true cm}
\parbox{13.2 true cm}{\caption{\label{fig:6-1-EXCELPoorModela}  Model of infected patients at the English boarding school over time plotted on top of the actual data (blue diamonds) using difference equation model (\ref{eq:disc}) (red squares) with initial estimates of model parameters, $a = 0.002625$ and $b = 0.3333$, a poor fit.}}
\vspace*{- .5 true cm}
\end{figure}


A quick numerical analysis (comparing outputs of the model using these
parameters to the given data) or graphical comparison (depicted in Figure~\ref{fig:6-1-EXCELPoorModela}) of the given data and
the model's projected number of students infected on day $n$, using the initial estimates for parameters $a = 0.002625$ and
$b = 0.3333$, shows that these estimates are also poor,  with a sum of square errors of  260,281.


It is possible that a continuous model could give a better fit or more predictive power; we now explore an ordinary differential equation model.

\subsection*{Differential equation model}

A treatment of a differential equations model (\ref{eq:cont}) can be found in the file 6-001-T-Mma-Epidemic-TeacherVersion-Continuous.nb and .pdf.  The main result of that discussion is presented in Figure~\ref{fig:6-1-MmaModelAndData}. This is a better model than the discrete approach offered above.



\begin{figure}[h]
\centering
\includegraphics[width=8cm]{6-1-MmaModelAndData.eps}



\parbox{13.5 true cm}{\caption{\label{fig:6-1-MmaModelAndData}  Model of infected patients at the English boarding school plotted over the actual data using differential equations model (\ref{eq:cont}) with model parameters, $a = 0.002565$ and $b = 0.483$. The SSE for this model is 5,130. These are parameters are close to those offered in \cite{Murray1993}.  The continuous model SSE is considerably less than the SSE from the discrete model (99,445.52)  and the plot in the continuous model looks better as well.}}
\end{figure}


In light of the continuous model perhaps we can make the discrete model found in the Excel spreadsheet file 6-1-T-Excel-Epidemic-TeacherVersion.xls better by taking smaller step sizes. 

\begin{figure}[h]
\centering
\includegraphics[width=8cm]{6-1-EXCELSecondModel.eps}

\vspace*{- 6true cm}
\parbox{13.2 true cm}{\caption{\label{fig:6-1-EXCELSecondModel}  Model of number of infected patients at the English boarding school plotted over the actual data (blue diamonds), using SIR difference equation model (\ref{eq:disc}) (red squares) with model parameters, $a = 0.002661$ and $b = 0.4944$, determined by Excel's Solver command which minimizes the sum of square errors between the model and the data.  The sum of square errors (SSE) is 6,412.1. }}
\vspace*{- .2 true cm}
\end{figure}

\noindent We do that in a second spreadsheet in this file and  offer a result in Figure~\ref{fig:6-1-EXCELSecondModel} which uses Excel's Solver command  to determine the parameters $a$ and $b$ which minimize the sum of square errors between the results from the difference equation model 
(\ref{eq:disc}) and the data as computed in (\ref{eq:SSEMin}).  Here the iteration method is again simply Euler's Method, but with a smaller step size of   $\Delta t = 0.05$ and the differences between the model and the data taken only at integer values of time.






In this situation Solver  determines the two parameters $a$ and $b$ which minimize the SSE:

$$a = 0.002661, \qquad  b = 0.4944, \quad {\rm and} \quad   SSE = 6,412.1\, .$$
We see in Figure~\ref{fig:6-1-EXCELSecondModel} that this model, using a step size of $\Delta t = 0.05$ in Euler's Method   does  look better than the same approach using a step size of $\Delta t = 0.1$ in Euler's Method   (indeed quite good)  when plotted over the data.


Of course, this could be an opportunity to build better numerical methods in the spreadsheet, e.g., Improved Euler's Method of Runge-Kutta Method, each a better approximation of the numerical methods used by Mathematica to obtain the numerical solutions needed for this nonlinear system of differential equations.

\vspace*{-.3 true cm}
\section*{FURTHER COMMENTS}

\vspace*{-.1 true cm}\subsection*{Implementation}

This project was given as an extra credit assignment to first-semester calculus
students who had previously taken a one-semester modeling course (fitting
data with polynomial, exponential, and trigonometric functions) that included
an introduction to difference equations. Students who elected to do the
assignment were allowed to ask questions, but did not receive any instruction
aside from the project itself. For students with experience modeling but not with
recurrence equations, a day or two spent in class familiarizing students with
the concepts and mechanics of recursive equations would likely be helpful.

Our goal was to design a project that could be used in a variety of courses and
by students with different levels of mathematical sophistication. For students
without a strong background in mathematical modeling and problem solving, the best
results will likely be achieved by placing this assignment at the end of a
two to ten lesson primer on difference equations and elementary data fitting.
Students should be familiar with basic tools used to assess the fit of a model
to a data set, such as sum of squared errors, coefficient of determination $R^2$,
and predictive power of models based on the nature of the data.

The level of difficulty of the assignment can be adjusted by the instructor. If the instructor would
like to see more of the problem solving process, he or she can give no additional
information; if more uniformity in solution sets is preferred, the instructor can
set initial parameter values or assumptions for the students. We envisioned this
assignment as a week-long homework assignment, a small project, or an in-class assignment to teach or review techniques of fitting models to data and assessing the fit of a model.

\subsection*{Student Challenges}


In an SIR model, very small adjustments to the parameters can
cause significant changes in the behavior of the model. The challenge presented by this feature is compounded by the fact that most students are accustomed to analytic functions so they struggle with understanding how changes in parameters influence the shape of their
models.  For students who lack well-developed
number sense, it might be difficult to find acceptable parameters.  
Furthermore, qualitative assessment of models can be a significant obstacle for students who prefer to solve all problems algorithmically.  Any mistakes when inputting the data, the difference equations to Excel, or the differential equations to a computer algebra system can be difficult to catch, particularly for students, and produce nonsensical results.


\subsection*{Benefits}

This project encourages students to reconcile themselves to the idea
that modeling is not only about following a recipe, e.g., minimizing SSE, but also
about critical thinking and problem solving. Because the data are not
contrived, they can see that the need for flexibility in their thinking is a
valuable real-world skill. Furthermore, there are many connections
between the study of epidemics and current events such as the H1N1 strain of
influenza (an avian flu), projections of infections in the event of a
global outbreak of any new airborne disease, and how to balance the rights
of the individual with those of the nation and world when such events
occur. Real-world connections and context help students feel invested in the material, and the import of modeling choices and analysis can foster a sense of ownership and personal connection.

Both discrete and continuous models are presented in the comments section.  While data collection on diseases is generally discrete, the most accurate way to model the progression of a disease
through a population arguably is not. This is a good opportunity to discuss
the difference between discrete and continuous functions and how to choose
a function type with which to model.

\subsection*{Improvements or Alternate Implementations}

Some students would benefit from assistance with estimating initial
parameters. This project was an extra-credit assignment, and we did not devote any class time to it. The models submitted by students might have been better if the assignment was discussed in class, which we recommend, but there were still many students who offered very good models.

Another possible enhancement is to have students discuss whether their model under-predicted or over-predicted the number of infections, and why it may have done so (for example, periods of latency before the appearance of symptoms, misdiagnoses, and modeling assumptions  such as ours that boys were not bedridden for any reason other than the influenza outbreak).  Such a conversation could also extend to a discussion of diseases with lethality, and how to model them.

}

\begin{thebibliography}{99}


\bibitem{CDSC1978} Communicable Disease Surveillance Center. 1978. News and Notes: Influenza in a Boarding School. {\it British Medical Journal\/}. 1(6112): 586-590. \href{http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1603269/pdf/brmedj00115-0064.pdf}{http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1603269/pdf/brmedj00115-0064.pdf}. Accessed 25 October 2014.

\teacher{\bibitem{MillerHelms2010} Miller, S. and J. Helms. 2010.  The Disease that Infected Half the World:  Mathematics, Biology, and History.  {\it PRIMUS - Problems, Resources, and Issues in Mathematics Undergraduate Studies\/}.  20(3): 245-260.}

\bibitem{Murray1993} Murray, J. 1993. {\it Mathematical Biology, 2nd, Corrected Edition\/}. New
York: Springer-Verlag.

\bibitem{Switkes2003} Switkes, J. 2003. A modified discrete SIR model. {\it The College
Mathematics Journal\/}. 34(5): 399-402.

\end{thebibliography}






\end{document}
