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

Branching Feller diffusion for cell division with parasite infection