= Solution
At the origin the Jacobian is
$$
J_0=\begin{pmatrix}-\nu&\beta N\\ \widehat\beta\widehat N&-\widehat\nu\end{pmatrix},
\qquad
\det J_0=\nu\widehat\nu-\beta\widehat\beta N\widehat N<0.
$$
Thus it is a saddle and unstable. At the positive equilibrium the trace is negative. The product of the off-diagonal entries is $\nu\widehat\nu$ by the nullcline identities, while the product of the magnitudes of the diagonal entries is strictly larger, so the <determinant> is positive. Both <eigenvalues> therefore have negative real part and the endemic equilibrium is stable.
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