  We consider a branching model for a population of dividing cells infected by
parasites. Each cell receives parasites by inheritance from its mother cell and
independent contamination from outside the population. Parasites multiply
randomly inside the cell and are shared randomly between the two daughter cells
when the cell divides. The law of the number of parasites which contaminate a
given cell depends only on whether the cell is already infected or not. We
determine the asymptotic behavior of the number of parasites in a cell line,
which follows a branching process in random environment with state dependent
immigration. We then derive a law of large numbers for the asymptotic
proportions of cells with a given number of parasites. The main tools are
branching processes in random environment and laws of large numbers for Markov
tree.
