  A host-parasite model is considered for a population of cells that can be of
two types, A or B, and exhibits unilateral reproduction: while a B-cell always
splits into two cells of the same type, the two daughter cells of an A-cell can
be of any type. The random mechanism that describes how parasites within a cell
multiply and are then shared into the daughter cells is allowed to depend on
the hosting mother cell as well as its daughter cells. Focusing on the
subpopulation of A-cells and its parasites, the model differs from the
single-type model recently studied by Bansaye (2008) in that the sharing
mechanism may be biased towards one of the two types. Main results are
concerned with the nonextinctive case and provide information on the behavior,
as $n\to\infty$, of the number A-parasites in generation n and the relative
proportion of A- and B-cells in this generation which host a given number of
parasites. As in (Bansaye,2008), proofs will make use of a so-called random
cell line which, when conditioned to be of type A, behaves like a branching
process in random environment.
