Chapter 7.P1, Problem 5P

In the SIR system (1), describe qualitatively the asymptotic behaviour of $S,I,\text{and}R$ as $t\to \infty$. In particular, answer the question “Does everyone get inflected?” Then explain the statements “The epidemic does not die out due to the lack of susceptible, but rather due to a lack of infectives.”

$\begin{array}{c}S\text{'}=-\beta IS\\ I\text{'}=\beta IS-vI\\ R\text{'}=vI\end{array}$ (1)