Is this mold, mildew or just dirt?
That appears to be black mold affiliated with a water leak or a flood.  If a flood you need to have mold mitigation done by a qualified mold company.  Drywall removal will be required as the mold you see is coming from the mold on the back of the drywall.  Mold loves cellulose as a food so the paper backing on the back of the drywall will be completely covered in mold.  If a flood situation it is recommended to cut a minimum of 1 foot of drywall above the noticed mold.  However the paint can hide the evidence of mold and can be 4 foot or higher that needs to be removed.  After removal in a flood situation it is recommended to use a disinfectant grease removal soap and scrub all areas affected to include the entire home.  I clean a minimum of 1 foot higher than flood level. .  I use Pinal lavendar sold at home depot as it has a nice smell.  Then dry the home or area to a maximum of 50% of wood moisture measured with a moisture meter.  You can buy this at home depot for about $30.00  After reaching 50% moisture then spray all wood down with anti microbial mold killer.  Soak it down completely 3 times over 3 days no fans, no dehumidifiers.  After 3 days then turn on fans to create a circle of wind in each room leading to a dehumidifier in each room.  For entire home floods, you will need more than 1 dehumidifier.  Recommended is 1 per room.  So in many cases a mold control company is cheaper as we own equipment.  It is also recommended to have a Hepa Air scrubber to remove airborne mold spores.  After the wood reaches a level of 10% or less, you can install new drywall and insulation.  However I require that my customers continue to dry under normal AC or heat for 1 week after certification.  In cases of clean water damage like fire or water leak you can paint the studs with antimicrobial paint.  In cases of flood damages, I do not recommend the paint unless you allow the home to dry out 6 months or more prior to painting and closing the walls.  Flood bacteria takes more time to allow the antimicrobials to penetrate to core .  And if you paint to soon...you will have structural failure from dry-rot bacteria (Fusarium sambucinum) prematurely.