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\begin{document}

\newpage



\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\ICU SPREAD}


\markboth{ICU Spread}{ICU Spread}







\author{Brian Winkel,
Director SIMIODE\\
Chardon ON 44024 USA\\
BrianWinkel@simiode.org
}



\makeStitlePDFLaTex




\begin{abstract}
We offer students the opportunity to model the percentage of voluntary nonprofit hospitals in the United States with Intensive Care Units during the period of 1958-1974.
\end{abstract}


\section*{SCENARIO DESCRIPTION}
Table~\ref{table:ICU} shows data on the percentage of voluntary nonprofit hospitals in the United States with Intensive Care Units during the period 1958 to 1974. These hospitals in this survey had between  200 and 299 beds.


\begin{table}[h]
\begin{center}

\small
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline
Year\hspace*{-.05 true cm} & \hspace*{-.15 true cm} 1958 \hspace*{-.15 true cm} & \hspace*{-.15 true cm}	1959\hspace*{-.15 true cm} & \hspace*{-.15 true cm} 1960\hspace*{-.15 true cm} & \hspace*{-.15 true cm} 1961\hspace*{-.15 true cm} & \hspace*{-.15 true cm} 1962\hspace*{-.15 true cm} & \hspace*{-.15 true cm}1963\hspace*{-.15 true cm} & \hspace*{-.15 true cm}1964\hspace*{-.15 true cm} & \hspace*{-.15 true cm} 1965 \hspace*{-.15 true cm} & \hspace*{-.15 true cm} 1968 \hspace*{-.15 true cm} & \hspace*{-.15 true cm} 1969\hspace*{-.15 true cm} & \hspace*{-.15 true cm} 1970 \hspace*{-.15 true cm} & \hspace*{-.15 true cm} 1971 \hspace*{-.15 true cm} & \hspace*{-.15 true cm} 1972\hspace*{-.15 true cm} & \hspace*{-.15 true cm} 1973\hspace*{-.15 true cm} & \hspace*{-.15 true cm} 1974\\ \hline
\%\hspace*{-.15 true cm} & \hspace*{-.15 true cm} 14\hspace*{-.15 true cm} & \hspace*{-.15 true cm} 16 \hspace*{-.15 true cm} & \hspace*{-.15 true cm} 22 \hspace*{-.15 true cm} & \hspace*{-.15 true cm} 33 \hspace*{-.15 true cm} & \hspace*{-.15 true cm}38 \hspace*{-.15 true cm} & \hspace*{-.15 true cm} 47 \hspace*{-.15 true cm} & \hspace*{-.15 true cm} 54\hspace*{-.15 true cm} & \hspace*{-.15 true cm} 65\hspace*{-.15 true cm} & \hspace*{-.15 true cm}86\hspace*{-.15 true cm} & \hspace*{-.15 true cm}90\hspace*{-.15 true cm} & \hspace*{-.15 true cm}91\hspace*{-.15 true cm} & \hspace*{-.15 true cm}92\hspace*{-.15 true cm} & \hspace*{-.15 true cm}93\hspace*{-.15 true cm} & \hspace*{-.15 true cm}95\hspace*{-.15 true cm} & \hspace*{-.15 true cm} 96\\ \hline
\end{tabular}

\parbox{12.5 true cm}{\caption{\label{table:ICU}  Data on the percentage of voluntary nonprofit hospitals in the United States with Intensive Care Units during the period 1958 to 1974. Gaps for 1966 and 1967 indicate years in which data is not available from the survey. Data is from \cite{Russell1977}}}
\end{center}
\end{table}

\vspace*{-.9 true cm}
\subsection*{Activities}

\begin{enumerate}
\item Graph the data in Table~\ref{table:ICU}. Hint:  Make $t=0$ correspond to year 1958.
\item Suppose we let $P(t)$ be the percentage of voluntary nonprofit hospitals in the United States with Intensive Care Units  at time $t = 0$. Assume the units for $t$ are in years and $t = 0$ corresponds to year 1958.
Give an  argument for why the logistic differential equation,


\begin{equation}\label{eq:log}
P'(t) = r P(t) \frac{100 - P(t)}{100} \, ,
\end{equation}
is a good candidate for modeling this data. Be sure to address the  meanings of the rate of change in $P(t)$, the value of 100,   the parameter $r$, and the algebraic nature of the right hand side of the differential equation.
\item Fit a logistic curve (the solution to (\ref{eq:log})) to the data in Table~\ref{table:ICU}.
\item Locate, using your eye only, the year and percent during which there was the greatest increase in	$P(t)$.
\item From your final model from (3)  determine  when $P'(t)$ is greatest,
\item How do your answers to (4) and (5) compare?
\item Suppose you only had the data for years 1958-1965, but there was a consensus that the percentage of voluntary nonprofit hospitals in the United States with Intensive Care Units  would grow according to (\ref{eq:log}). Fit a logistic curve (the solution to (\ref{eq:log})) to this smaller data set from  Table~\ref{table:ICU}.
\item Discuss how you could use your results from (7)  in a policy discussion for the progress of medical care in the country. 
\end{enumerate}

\teacher{
\section*{NOTES FOR TEACHERS}

This is a simple and direct application of the logistic differential equation to model the spread of a technology, in this case ICU units in a hospital.  We offer analysis (but not for the narrative questions 2, 6, and 8) in a Mathematica notebook 1-9-T-Mma-ICUSpread-TeacherVersion.nb and a .pdf form of the same file for those who do not use Mathematica.

Additional questions can be asked, e.g., when does the model tell us we are at 50\% level?  How does the smaller data set model compare to the actual data in the larger set? 

Of course, the 100 can be replaced by $K$, an unknown percent, but probably pretty high and close to 100, to be obtained where the sum of square errors would now be a function of the two parameters $r$ and $K$. This would be of interest if we used the shortened data set to see if the model would predict complete saturation and  when it would be at 99\% level.

}

\begin{thebibliography}{99}
\bibitem{Russell1977}  Russell, L. B. 1977. The Diffusion of New Hospital Technologies in the United States.   {\it International Journal of Health Services\/}. 6(4): 557-580.  Reprinted as {\it The Diffusion of New Hospital Technologies in the United States, General Series Reprint 322\/}.  Washington DC:  The Brookings Institute. 



\end{thebibliography}









\end{document}
 
