  We describe the evolution of the quantity of parasites in a population of
cells which divide in continuous-time. The quantity of parasites in a cell
follows a Feller diffusion, which is splitted randomly between the two daughter
cells when a division occurs. The cell division rate may depend on the quantity
of parasites inside the cell and we are interested in the cases of constant or
monotone division rate. We first determine the asymptotic behavior of the
quantity of parasites in a cell line, which follows a Feller diffusion with
multiplicative jumps. We then consider the evolution of the infection of the
cell population and give criteria to determine whether the proportion of
infected cells goes to zero (recovery) or if a positive proportion of cells
becomes largely infected (proliferation of parasites inside the cells).
