  In this paper we study a two-level branching model for virus populations
under cell division. We assume that the cells are carrying virus populations
which evolve as a branching particle system with competition, while the cells
split according to a Yule process thereby dividing their virus populations into
two independently evolving sub-populations.
  We then assume that sizes of the virus populations are huge and characterize
the fast branching rate and huge population density limit as the solution of a
well-posed martingale problem. While verifying tightness is quite standard, we
provide a Feynman-Kac duality relation to conclude uniqueness. Moreover, the
duality relation allows for a further study of the long term behavior of the
model.
