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\centerline{\large \bf MA 573 --- PROJECT 6}

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{\bf Due: Monday, November 20}

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(1) Consider the Sinko--Streifer model
$$
\begin{array}{l}
  {\ds \frac{\d u}{\d t} + \frac{\d (gu)}{\d x} = -\mu u } \\
  \noalign{\medskip}
  {\ds g(t,x_0) u(t,x_0) = R(t) } \\
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  {\ds u(0,x) = \Phi(x)}
\end{array}
$$
with the growth function 
$$
  \frac{dx}{dt} = g(x) = \frac{x_1 - x}{x_1 - x_0}
$$
where $x_0$ and $x_1$ are the smallest and largest fish sizes.  Determine the solution $u(t,x)$ for the initial
condition regime $t \leq G(x)$ with the assumption that $\mu = \bar{\mu}$ is constant.  You can take $x_0=1$~cm, $x_1=10$~cm and
$$
  \Phi(x) = \sin\left(\frac{\pi(x-x_0)}{x_1-x_0}\right).
$$
Plot the curve $G(x)$ along with a representative trajectory for $u(t,x)$.  


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(2) Consider the SIR epidemic model
$$
\begin{array}{l}
  {\ds \frac{d S}{dt} = - \beta I S } \\
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  {\ds \frac{d I}{d t} = \beta I S - \nu I } \\
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  {\ds \frac{d R}{d t} = \nu I }
\end{array}
$$
where $\beta$ and $\nu$ denote contact and recovery rates.  You can use the values
$\beta = 4$ and $\nu = 1.2$.

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(a) Simulate the spread of the epidemic
using the initial conditions $S(0) = .99, I(0) = 0.01$ and $R(0) = 0$.  When does
it end?

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(b) Now modify your model to include 90\% vaccination of all individuals that provides life-time
immunity.  Simulate your results using the same initial conditions.  Does your vaccination
program work?

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