We present a new axisymmetric model of the heliosphere that includes three principal particle species (plasma with magnetic field, interstellar neutral atoms, and galactic cosmic rays). Unlike previous modulation models, which necessarily included a modulation boundary, we study the entry of the galactic cosmic rays into the heliosphere directly from the local interstellar medium population. An important improvement over previous global heliospheric models is that a kinetic description for the cosmic rays is used instead of the usual fluid approach. We compute the cosmic-ray diffusion coefficients from quasi-linear theory, assuming a constant (but different) power spectrum of the fluctuations in both the solar wind and the interstellar medium. Our results show that low-energy particles are strongly attenuated by the magnetic wall in the inner heliosheath and do not reach the termination shock. The model predicts small cosmic rays gradients in most of the heliosphere, implying little modification to the termination and bow shocks. The exception is the inner heliosheath region, where cosmic-ray gradients are large and the energetic particles can, consequently, affect the plasma flow. Particle spectra in the inner heliosphere are found to be in agreement with the observations. We also find that the inner heliospheric galactic cosmic-ray population is not sensitive to the upwind-downwind asymmetry of the termination shock.