A comprehensive model is presented to describe the physics of positive ion sources used to generate hydrogen isotope neutral beams for the heating of confined plasma in magnetic fusion energy programs. The model considers 11 important atomic and molecular reactions. A set of particle balance equations, together with those for the primary and secondary electrons in the sources, are solved simultaneously. Taking as input the source geometry, gas inlet flow rate, arc current and voltage, as well as the wall recombination coefficient and the measured temperatures of neutral atoms and molecules, the model gives as output the following information: the ion current densities available for extraction, primary electron density and energy distribution, and secondary electron density and temperature. These calculated results are compared with data of several Lawrence Berkeley Laboratory ion sources, both conventional and magnetic bucket types. Exploratory calculations are made for sources of different proposed designs with the aim of improving the source performance.