Is there any reason to keep someone’s sickness a secret?
Absolutely. Sometimes there are certain close relatives who might become too anxious if they were to find out that a relative is very sick. Some people can become sick from anxiety. So it depends on the circumstances.
And you can’t give a general rule about keeping it quiet because the Gemara says that when a person is sick he should be mo’dee’ah others so that those people will be mispallel for him. When people know about the sickness they will pray for the choleh. It’s a tremendous chesed to daven for a choleh. And not just by rote. With feeling. All the time. Whenever you can. And Hashem listens to your tefillah.
So the answer is that it depends on the circumstances. Certainly it depends on the situation.