  The spectral energy distributions and infrared (IR) spectra of a sample of
obscured AGNs selected in the mid-IR are modeled with recent clumpy torus
models to investigate the nature of the sources, the properties of the
obscuring matter, and dependencies on luminosity. The sample contains 21
obscured AGNs at z=1.3-3 discovered in the largest Spitzer surveys (SWIRE,
NDWFS, & FLS) by means of their extremely red IR to optical colors. All sources
show the 9.7micron silicate feature in absorption and have extreme mid-IR
luminosities (L(6micron)~10^46 erg/s). The IR SEDs and spectra of 12 sources
are well reproduced with a simple torus model, while the remaining 9 sources
require foreground extinction from a cold dust component to reproduce both the
depth of the silicate feature and the near-IR emission from hot dust. The
best-fit torus models show a broad range of inclinations, with no preference
for the edge-on torus expected in obscured AGNs. Based on the unobscured QSO
mid-IR luminosity function, and on a color-selected sample of obscured and
unobscured IR sources, we estimate the surface densities of obscured and
unobscured QSOs at L(6micron)>10^12 Lsun, and z=1.3-3.0 to be about 17-22
deg^-2, and 11.7 deg^-2, respectively. Overall we find that ~35-41% of luminous
QSOs are unobscured, 37-40% are obscured by the torus, and 23-25% are obscured
by a cold absorber detached from the torus. These fractions constrain the torus
half opening angle to be ~67 deg. This value is significantly larger than found
for FIR selected samples of AGN at lower luminosity (~46 deg), supporting the
receding torus scenario. A far-IR component is observed in 8 objects. The
estimated far-IR luminosities associated with this component all exceed
3.3x10^12 Lsun, implying SFRs of 600-3000 Msun/yr. (Abridged)
