An integrated model is introduced that represents all basic physical interactions in a complex plasma-combustion system. Because of the many potentially important interactions in this system, the model is designed specifically not to provide the ultimate, highest-fidelity possible representations of any of the mechanisms. Instead, it is designed with three general criteria in mind: (1) that it be physics based, expressible as approximations of more foundational or more detailed physics; (2) that it represents all the principal anticipated phenomenology in the corresponding system; and (3) that its numerical evaluation does not fundamentally increase the computational intensity beyond that of the corresponding inert turbulent flow. The model is applied to the laser-induced-breakdown seeded ignition of a hydrogen jet in cross-flow, mediated by a dielectric-barrier discharge plasma. Both the plasmas are known to affect the ignition and combustion processes, though the relative importance of different mechanisms in this configuration is unclear a priori. Predictions with this integrated model are compared with corresponding measurements.