Hi - think you need to look at your access rights "model", I'm pretty sure that if a resource does not have the "View - Management" rights against a project then they can not see the "Task list" for a project (and thus can not see other user's assignment details where the actuals live).
They can still book time to the project of course - they would see the task in their timesheet OK and could click on the task name there, which will take them to the task details page read-only, but without the assignment details area below the task details.
So look at the "Access to This Project" / Full-View / Key icon information against your "Sickness project" to identify how users are getting the "View - Management" right to the project.... and remove it!