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 infection in 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).