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\title{Math 1b.  Lecture 32 \\
Systems of Differential Equations (II)}

\author{T. Judson}

\date{Spring 2006}


\begin{document}

\maketitle


\section{Goals}

\begin{itemize}

\item
To understand and be able to model an epidemic in a closed population.

\item
To understand and to be able to apply phase plane analysis to 
\begin{eqnarray*}
\frac{dx}{dt} & = & f(x, y), \\
\frac{dy}{dt} & = & g(x, y).
\end{eqnarray*}


\item
To understand that the system 
\begin{eqnarray*}
\frac{dx}{dt} & = & f(x, y), \\
\frac{dy}{dt} & = & g(x, y),
\end{eqnarray*}
is completely predictive.  If you choose a starting point in the $xy$-plane, then there is exactly one solution that starts at your chosen point.

\item
To be able to use {\em pplane}.

\end{itemize}


\section{An Epidemic Model}

Consider the model of a viral epidemic that moves through an isolated population.  We make the following assumptions.
\begin{enumerate}

\item[(a)]
Individuals are infected at a rate proportional to the product of the number of infected and susceptible individuals.  We assume that the constant of proportionality is $\lambda = 0.05$ per day.

\item[(b)]
The length of the incubation period is negligible.  Infectious individuals are immediately infectious.

\item[(c)]
On the average, an infected individual dies or recovers after 10 days.


\item[(d)]
No one is sick initially.

\item[(e)]
Infected individuals do not give birth, but susceptible individuals have a birth rate of 0.0003 per individual per year.  Newborns are susceptible.

\end{enumerate}
If $x(t)$ is the number of susceptible and $y(t)$ is the number of infected people, then
\begin{eqnarray*}
\frac{dx}{dt} & = & -\lambda xy + 0.0003x \\
\frac{dy}{dt} & = & \lambda xy - 0.1y.
\end{eqnarray*}
The first equation asserts that in a unit time interval, any infected individual will infect any given susceptible individual with $\lambda = 0.05$ percent probability.  The constant $\lambda$ is a measure of the relative  infectivity of the disease.  If $\lambda$ is high relative to the birth rate then the disease will burn itself out.

\subsection*{The Case $\lambda = 0.05$}

The $x$ null clines are at $x = 0$ and $y = 0.006$.  The $y$ null clines are at $y = 0$ and $x = 2$.  The disease burns itself out.

\subsection*{The Case $\lambda = 10^{-6}$}

The $x$ null clines are at $x = 0$ and $y = 300$.  The $y$ null clines are at $y = 0$ and $x = 100,000$.  Note that the infection rate is much smaller compared to the birth rate.  If we take our initial  value of $x = 20,000$ and $y = 100$ $x$ will increase and $y$ will decrease (but will never become zero).  Eventually, the number of infected individuals will hit 100,000, and the epidemic will take off.  We will either get a situation where the disease is cyclic or it will spiral into an equilibrium point.



\section{Phase Plane Analysis}


\begin{itemize}

\item
{\em Step 1.}
Draw the curves where $f(x, y) = 0$.  These curves are called the $x$ {\em null clines}.  When ${\bf v}(t)$ lies on one of these curves, $dx/dt = 0$.  Draw vertical slash marks on the $x$ null clines to remind yourself that a trajectory crossing the null cline can only do so if it is moving in a vertical direction at the instant of crossing.

\item
{\em Step 2.}
Draw the curves where $y(x, y) = 0$.  These curves are called the $y$ {\em null clines}.  When ${\bf v}(t)$ lies on one of these curves, $dy/dt = 0$.  Draw horizontal slash marks on the $y$ null clines to remind yourself that a trajectory crossing the null cline can only do so if it is moving in a horizontal direction at the instant of crossing.

\item
{\em Step 3.}
Label the points where the $x$ and $y$ null clines intersect.  These intersections are called {\em equilibrium points}.  If ${\bf v}(t)$ is ever at one of these points , then both $dx/dt$ and $dy/dt$ vanish.  This means that the trajectory stays at the point for all time.  If the system is going to settle into a steady state, then ${\bf v}(t)$ will approach on of the equilibrium points as $t \rightarrow \infty$.

\item
{\em Step 4.}
Label the regions of the $xy$-plane where $dx/dt < 0$ and where $dx/dy > 0$.  These regions are always separated by $x$ null clines.  Likewise, label the regions where $dy/dt$ is positive and negative.

\item
{\em Step 5.}
Go back and put arrows on the vertical hash marks of the $x$ null clines.  These arrows indicate whether the motion across the null cline is up or down.  The arrows are up on the parts of the $x$ null cline that are in the $dy/dt > 0$ region, and down on those parts of the $x$  null cline in the $dy/dt < 0$ regions.  Likewise, draw arrows on the horizontal slash marks of the $y$ null clines.    These arrows are pointing right on the parts of the $y$ null cline  in the $dx/dt > 0$ regions and left point on the parts in the $dx/dt < 0$ regions.

\item
{\em Step 6.}
\begin{enumerate}

\item[(a)]
If $dx/dt > 0$ and $dy/dt > 0$, then both $x(t)$ and $y(t)$ are increasing and the trajectory moves up and right.


\item[(b)]
If $dx/dt > 0$ and $dy/dt <0$, the trajectory moves down and right.


\item[(c)]
If $dx/dt < 0$ and $dy/dt > 0$, the trajectory moves up and left.


\item[(d)]
If $dx/dt < 0$ and $dy/dt < 0$, the trajectory moves down and left.


\end{enumerate}


\end{itemize}


\section{Existence and Uniqueness}

The system of equations
\begin{eqnarray}
\frac{dx}{dt} & = & f(x, y) \label{pplane1}\\
\frac{dy}{dt} & = & g(x, y) \nonumber
\end{eqnarray}
is completely predictive.  If you choose a starting point in the $xy$-plane, then there is exactly one solution to the system that starts at your chosen point.
\begin{itemize}

\item
For any starting point in the $xy$-plane, there is a unique solution to (\ref{pplane1}).

\item
Think of ${\bf v}(t)$ as tracing out a trajectory in the $xy$-plane as $t$ increases.  The goal is to predict the behavior of this trajectory.

\item
The phase plane analysis is done to help predict the trajectory.

\end{itemize}








\section*{References}

\begin{itemize}

\item
\S 7.6 in 
James Stewart.
{\it Single Variable Calculus: Concepts \& Context}, third edition.
Brooks/Cole, Belmont CA, 2005.
ISBN 0--534--41022--7.

\item
\S 31.5 in
Robin J. Gottlieb. 
{\it Calculus: An Integrated Approach to Functions and Their Rates of Change}, preliminary edition. 
Addison Wesley, Boston, 2002.
ISBN 0--201--70929--5.


\end{itemize}



\section*{Notes}

\subsection*{May 1, 2006}



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