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  \Large\textbf {Math 4 - Proposal\\Applications of Calculus to
    Medicine and Biology}
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\textbf{Applications of Calculus to Medicine and Biology}.
In this course we will establish the relevance of calculus to
medicine.  We will develop mathematical tools extending the techniques
of introductory calculus, including some matrix algebra and solution
techniques for first order differential equations.  Then we will use
these methods to construct simple and elegant models of phenomena such
as the mutation of HIV, spread of infectious disease, biological
disposition of drugs and inorganic toxins, enzyme kinetics and
population growth.


\noindent\textbf{Prerequisite:  Mathematics 3. }
\textbf{Note:} This is a second-term calculus course, but it does not cover the
same material as Math 8, and does not serve as a prerequisite for Math
13.



\bigskip


\noindent\textbf{Textbooks:} \textit{An Introduction to the Mathematics of
Biology} by Yeargers, Shonkwiler and Herod; \textit{Lake Victoria}, by Abonyo,
Cornell, Fixman, von Rittman, and Wallace.

\medskip
\noindent\textbf{Sample Syllabus}
\smallskip
\noindent\textbf{Week 1:}
  \begin{itemize}
  \item Introduction to course and discussion of mathematical 
models in biology.
\item Estimation of model parameters via transformation 
and least squares fits.
\item Multiple regression.
  \end{itemize}

\smallskip
\noindent\textbf{Week 2:}
\begin{itemize}
\item Differential Equations - separable, first order linear, 
direction fields and numerical solutions.
\item Systems of linear differential equations, phase portraits, 
matrix eigenvalues and long-term behavior of systems.
\end{itemize}

\smallskip
\noindent\textbf{Week 3:}
\begin{itemize}
\item Statistical representation of data through histograms, 
mean, median, standard deviation.
\item Elementary probability with an introduction to the law 
of total probability and Baye's rule.
\end{itemize}

\smallskip
\noindent\textbf{Week 4:}  First half of ``Lake Victoria: A Mathematical
Ecohistory" Includes discussion of history and colonization of Kenya,
Uganda and Tanzania. Development of the Malthusian model of population
growth.

\smallskip
\noindent\textbf{Week 5:}
Second half of "Lake Victoria: A Mathematical Ecohistory"
Continued discussion of Africa together with models for
ecosystems with carrying capacity. Predator prey models.

\smallskip
\noindent\textbf{Week 6:}
Discussion of organ systems and how drugs and inorganic 
toxins move through the body. Clinical effects of lead 
poisioning and discussion of the prevalence of lead
in inner cities.

\smallskip
\noindent\textbf{Week 7:}
Development of the compartment model for lead in the body.
Pharacokinetics and serum concentration cycles.

\smallskip
\noindent\textbf{Week 8:}
Background information necessary to understand HIV and 
AIDS. Cell biology, dna, rna etc. Development of Perelson's 
model of AIDS dynamics within the human body.

\smallskip
\noindent\textbf{Week 9:}
\begin{itemize}
\item Solution of Perelson's model, determination of its parameters
and discussion.
\item Beginning information on genetics and probability through Mendel's pea
experiment.
\end{itemize}

\smallskip
\noindent\textbf{Week 10:}
Discussion of the Hardy-Weinberg principle (if mating is random 
and all genotypes are equally fit, the fractions of alleles remains 
the same from generation to generation).


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