= Solution
At an <endemic equilibrium>, $I_*>0$, so $\beta S_*=\mu+\nu$. Substitution into the susceptible equation gives
$$
\boxed{S_*=\frac N{R_0},\qquad I_*=\frac{\mu N}{\mu+\nu}\left(1-\frac1{R_0}\right).}
$$
These values are physically positive exactly when $R_0>1$. The <Jacobian matrix> at this equilibrium is
$$
J=\begin{pmatrix}-\mu R_0&-(\mu+\nu)\\\mu(R_0-1)&0\end{pmatrix},\qquad
\operatorname{tr}J=-\mu R_0<0,\quad\det J=\mu(\mu+\nu)(R_0-1)>0.
$$
Both <eigenvalues> have negative real parts, so the <endemic equilibrium> is locally asymptotically stable. Its decay exponents are
$$
\lambda_\pm=-\frac{\mu R_0}{2}\pm\sqrt{\frac{\mu^2R_0^2}{4}-\mu(\mu+\nu)(R_0-1)}.
$$
The regime $\nu\gg\mu$ means recovery is rapid compared with demographic replacement. For fixed $R_0>1$ away from the threshold, the exponents form a <complex conjugate> pair with frequency approximately $\sqrt{\mu\nu(R_0-1)}$. Hence
$$
\boxed{T\simeq\frac{2\pi}{\sqrt{\mu\nu(R_0-1)}}.}
$$
Strictly, oscillations require $\mu(\mu+\nu)(R_0-1)>\mu^2R_0^2/4$; $\nu\gg\mu$ alone does not ensure this uniformly for every $R_0$ arbitrarily close to one or arbitrarily large.
Vaccinating a fraction $v$ of newborns replaces the susceptible input by $\mu N(1-v)$. The effective <basic reproduction number> is $R_v=(1-v)R_0$, and the same calculation replaces $R_0$ by $R_v$. Provided $R_v>1$ and the oscillatory approximation remains applicable, \b[vaccination lengthens the <oscillation period>], because $R_v-1$ is smaller. Extremely near eradication the exponents become real and there is no <oscillation period> to extrapolate indefinitely.
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