Entry next sibling return a deactivated entry
The getNextSibling documentation says:


  getNextSibling() will return the next sibling whether or not it’s enabled. If you want to get the closest enabled sibling, you can do this instead:


{% set next = craft.entries.positionedAfter(entry).order('lft asc').first() %}


Similarly for getPrevSibling to get the previous enabled sibling you can do:

{% set prev = craft.entries.positionedBefore(entry).order('lft desc').first() %}