 Solution

At an endemic equilibrium, $I_{\ast }>0$, so $\beta S_{\ast }=\mu +\nu$. Substitution into the susceptible equation givesThese values are physically positive exactly when $R_0>1$. The Jacobian matrix at this equilibrium isBoth eigenvalues have negative real parts, so the endemic equilibrium is locally asymptotically stable. Its decay exponents areThe 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 $\sqrt{\mu \nu (R_0-1)}$. HenceStrictly, oscillations require $\mu (\mu +\nu )(R_0-1)>{\mu }^2{R}_0^2/4$; $\nu \gg \mu$ 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, 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.