Davos is a major ski resort and mountain area, and is heaven for trail running. There are 700 km of trails in the Davos Klosters area, offering something for trail runners of all abilities. The views from the trails are spectacular. Note that some of the trails aren’t generally available for running during the winter season. The signage/waymarking is excellent.
One good option is to utilize the extensive lift system to save some of the more challenging hill climbing.
This is an excellent resource for trail running options in the area, including suggested routes and itineraries.