Cell contamination and branching processes in a random environment immigration

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