We consider a cell population subject to a parasite infection. Cells divide at a constant rate and, at division, share the parasites they contain between their two daughter cells. The sharing may be asymmetrical, and its law may depend on the quantity of parasites in the mother. Cells die at a rate which may depend on the quantity of parasites they carry, and are also killed when this quantity explodes. We study the survival of the cell population as well as the mean quantity of parasites in the cells, and focus on the role of the parasites partitioning kernel at division.