Ability to Block Users From Viewing Sys Admin User Updates

[#1]

As the system admin for our instance, I am responsible for deactivating users when they are terminated. This can be confidential info, and at times, deactivation occurs prior to the news being announced to the company. So I need to be able to block other users from being able to see the updates on My Profile page. Is there a way to do this that I am not aware of?