Chapter 7.P1, Problem 2P

The triangular region $\Gamma =\{(S,I):0<S+I\le N\}$ in the SI-plane is depicted in Figure 7.P.2. Use an analysis based strictly on direction field to show that no solution of the system (2) can leave the set $\Gamma$. More precisely, show that each point on the boundary of $\Gamma$ is either a critical points of the system (2), or else the direction field vectors points toward the interior of $\Gamma$ or are parallel to the boundary of $\Gamma$.

$\begin{array}{c}S\text{'}=-\beta IS\\ I\text{'}=\beta IS-vI.\end{array}$ (2)