Well I think the answer to this is it depends. In my most recent episode of experiencing exactly this it was a bad idea for me to attempt to exercise through the sickness.
I rarely get sick but when I do I am normally quite convinced that I can exercise my way through it. There probably isn't any science behind this but in the past I have actually helped to expedite the passing through of a cold by going for a short yet still existent workout. I'm not a jerk so no, I do not do this in a gym where I could infect others. I'm talking about going for a small run or something.
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not me in the picture :)
According to real medical advice that I can find online the jury is still really out on this notion so most of them, perhaps to avoid litigation, state that you really shouldn't exercise if you are sick but you can if you are only mildly sick. They say to use temperature and heart rate as a guide. If you are more than 1 degree over the natural body temp or if your heart rate is elevated compared to what it would normally be while at rest, to not exercise. They say to pursue the usual rest and consume a lot of fluids thing - which I agree with.
However, since I am a stubborn person I will often push the limits an go for a run anyway and many times, this has expedited me getting over the illness. It could be because I believe that it WILL accomplish that though rather than there being any medical benefit to this activity. The mind is a powerful thing after all.
Well the other day I didn't get very far in my very weak run before I realized that I had made a terrible mistake. I thought this could be a possibility so instead of heading to the park I merely ran around the block that my house is on. I made it about 1km before my stomach started talking to me. Then I threw up in that horrible, nothing in my stomach yet still retching sort of way. Thankfully everyone was at work so I had privacy in this endeavor. That ended my attempt to exercise the sickness away.
I think that by attempting to exercise that I may have actually made my situation worse this time around because I was really achy the rest of the day and my fever was getting bad as well. I was sweating yet freezing at the same time. I knew that my issue was stomach oriented this time and not a head cold like I had played around with in the past so I guess it was pretty stupid for me to try this . I knew I had a "delicate" stomach situation already so why not go and shake the stomach around a ton while trudging on down the street. What could possibly go wrong?
So I am not trying to use this as any sort of advice for anyone else out there because for the wrong person this could be something that lands you in the hospital. I was already in a state where I couldn't keep fluids in my body so it was magnificently stupid to put myself into a situation to potentially further dehydrate myself. Here i am, two days later, and I think I might be back in the state where I can go on a WALK, not a run.
You do you. Running while sick in the past has worked for me in the past but then again, so has drinking heavily - which makes zero sense but we are all different!
I suppose the only good side of what has been going on with me and my stomach the past few days is that I have had about 15 zero-calorie electrolyte drinks as well as a couple bowls of soup and maybe 4 pieces of bread in the past 2.5 days. So I have achieved a very good caloric deficit that is probably greater than normal days where I would eat normally and also exercise!