We derive an Ising Hamiltonian for kinetic simulations involving interstitial and vacancy defects in binary alloys. Our model, which we term 'ABVI', incorporates solute transport by both interstitial defects and vacancies into a mathematically-consistent framework, and thus represents a generalization to the widely-used ABV model for alloy evolution simulations. The Hamiltonian captures the three possible interstitial configurations in a binary alloy: A-A, A-B, and B-B, which makes it particularly useful for irradiation damage simulations. All the constants of the Hamiltonian are expressed in terms of bond energies that can be computed using first-principles calculations. We implement our ABVI model in kinetic Monte Carlo simulations and perform a verification exercise by comparing our results to published irradiation damage simulations in simple binary systems with Frenkel pair defect production and several microstructural scenarios, with matching agreement found.