Analytical potential models are designed for simulations of water with excess protons. The potentials describe both intramolecular and intermolecular interactions, and allow dissociation and formation of the species (H2O)(n)H+. The potentials are parametrized in the form of interactions between H+ and O2- ions, with additional three-body (H-O-H) interaction terms and self-consistent treatment of the polarizability of the oxygen ions. The screening of electrostatic interactions caused by the overlap of the electron clouds in the real molecules is modeled by functions modifying the electric field at short distances. The model was derived by fitting to the potential surface of the H5O2+ ion and other species, as obtained from ab initio MP2 calculations employing an extensive basis set. Emphasis was put on modeling the potential-energy surface for the proton-transfer reaction. Potential-surface profiles, geometry-optimized structures and formation energies of H5O2+, protonated water clusters [H+(H2O)(n), n=2-4] and water clusters [(H2O)(n), n = 1-6] using these potentials are presented and compared to results using quantum-chemical calculations. The potential models can well reproduce ab initio results for the H5O2+ ion, and can provide formation energies and structures of both protonated-water and water-only clusters that agree favorably with ab initio MP2 calculations.