Others are more expert than I, but my first thought was: Does your bathroom have a working exhaust fan, and does your family use it when they shower? Removing the moisture as best you can from the room is the first step. If you use the fan, try running it longer, or maybe upgrading the fan if it's undersized.
You can add mold inhibiting products to paint, if you're willing to repaint the ceiling. Any paint store will have it, and will add it for you when they mix the paint. Apply a good primer/sealer before painting.
Beyond that, if simpler fixes don't work, I would think you'd be into 'replace the ceiling' territory, involving perhaps some of the new paperless products that resist mold better than regular drywall.
You can add mold inhibiting products to paint, if you're willing to repaint the ceiling. Any paint store will have it, and will add it for you when they mix the paint. Apply a good primer/sealer before painting.
Beyond that, if simpler fixes don't work, I would think you'd be into 'replace the ceiling' territory, involving perhaps some of the new paperless products that resist mold better than regular drywall.