Estimate prevalence of ʻOumuamua-style interstellar obects inside Jupiter orbit now that C/2019 Q4 (Borisov) is found?
The enhancement of a cross-section due to gravitational focusing is given by
$$ \sigma_{\rm eff} = \pi a_J^2 \left(1 + \frac{2GM_{\odot}}{a_J\ v^2}\right),$$
where  $a_J$ is the semi-major axis of Jupiter's orbit (assumed circular), $v$ is the relative velocity (at infinity) and I have ignored the mass of Jupiter.

Thus, using $v=30$ km/s (as specified in the question) the term in brackets is the gravitational focusing enhancement of the cross-section and equals 1.38.

We now assume an isotropic velocity distribution and a homogeneous density of such objects of $n$ per unit volume. The flux of such particles is $f =n v \sigma_{\rm eff}$.

If $f = 0.5$ yr$^{-1}$, then $n \sim 5 \times 10^{12}$ per cubic parsec.