  We introduce a general class of branching Markov processes for the modelling
of a parasite infection in a cell population. Each cell contains a quantity of
parasites which evolves as a diffusion with positive jumps. The drift,
diffusive function and positive jump rate of this quantity of parasites depend
on its current value. The division rate of the cells also depends on the
quantity of parasites they contain. At division, a cell gives birth to two
daughter cells and shares its parasites between them. Cells may also die, at a
rate which may depend on the quantity of parasites they contain. We study the
long-time behaviour of the parasite infection.
