= Solution
Substitute $S=N-I$ and $\widehat S=\widehat N-\widehat I$:
$$
\dot I=\beta(N-I)\widehat I-\nu I,\qquad
\dot{\widehat I}=\widehat\beta(\widehat N-\widehat I)I-\widehat\nu\widehat I.
$$
Besides the origin, solve both nullcline equations. Eliminating one infected population gives
$$
I_*=\frac{\widehat NN-\widehat\nu\nu/(\widehat\beta\beta)}
{\nu/\beta+\widehat N},\qquad
\widehat I_*=\frac{\widehat NN-\widehat\nu\nu/(\widehat\beta\beta)}
{\widehat\nu/\widehat\beta+N}.
$$
The assumed <epidemic threshold> inequality makes both positive.
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