Most jetted tubs' manufacturers suggest monthly special cleaning, and few people do it. This involves some fairly strong chemicals, and costs, not counting the time. What you are seeing is the result of that not being done. The other big thing on a jetted tub is the restrictions on what should be used in them. Generally, no oils, bubbles, etc. The lines in a jetted tub do not drain when you drain the tub, and if you use the wrong things, stuff will start to grow in the stagnent water. Even if you are vigilant on that, the dead skin, hair, etc. can accumulate in there. Most do not have a filter. It would restrict flow too much, and not actually do all that much.
Have you checked their website? Most have the operator's manual available online and offer the cleaning chemicals. Often, there's a port where you pour the stuff in, let it sit for awhile, then run the pump to circulate it around, then flush it all out. Often there's a second step with a different chemical.