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You can run with a strain, but don't try to run on a sprain. A sprained ankle will be swollen and warm to the touch, and it will hurt much more than a strain. If you have a strain, you can run, but modify your training until it feels better. That may mean skipping speed work and sticking to short, easy runs, or trading the planned 17-mile weekend long run for a 13-mile run, but your training won't suffer. As always, listen to your body, and if the pain persists, visit a sports medicine professional to address the problem and take a day or two off from running.
Similarly, you can run with a mild head cold, and I have found that an easy run helps to relieve congestion. But if you are seriously ill, stay home. A good rule of thumb: If you are sick from the neck up, you can probably run, but if you are sick from the next down (a respiratory or gastrointestinal illness) stay home and recover, and see your doctor if necessary.