Chapter 7.P1, Problem 4P

Find an equation of the form $H(S,I)=c$ satisfied by the solutions of eq.(2). then construct a phase portrait within $\Gamma$ for the system (2) consisting of trajectories emanating from points along the upper boundary $S+I=N$ of $\Gamma$ corresponding to initial states in which $R(0)=0$.

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