Any of the various filamentous elements that form the internal framework of host cells, and typically remain after treatment of the cells with mild detergent to remove membrane constituents and soluble components of the host cytoplasm. The term embraces intermediate filaments, microfilaments, microtubules, the microtrabecular lattice, and other structures characterized by a polymeric filamentous nature and long-range order within the host cell. The various elements of the host cytoskeleton not only serve in the maintenance of host cellular shape but also have roles in other host cellular functions, including cellular movement, cell division, endocytosis, and movement of organelles.