Collective motion in dense packings of cells occurs in wound healing, embryonic development, and cancerous tumor growth. Most current computational models of dense cell packings either treat the system as collections of spherical particles or assume that the system is confluent, with no extracellular space. We have developed a new model for dense cell packings in two spatial dimensions, where the cells are modeled as deformable particles that have a preferred area and perimeter. We measure the packing fraction φj at jamming onset as a function of the asphericity, α, which is the ratio of the perimeter square to the area of the particle. We find that the jammed packing fraction increases monotonically with α and the system becomes confluent with φj = 1 for α > 1.16. Using surface Voronoi analysis, we show that this value for α corresponds to the case when the cells completely fill their Voronoi-tessellated regions. We also demonstrate that the free area per cell obeys a k-gamma distribution, which has been found for jammed packings of non-deformable particles. Finally, we will also describe results from our model concerning the mobility of deformable particles subjected to applied forces, as well as diffusion of deformable particles subjected to active forces. We show the shape asphericity can be used to control the collective behavior of cells.