It's really too general to say, if your model contains data manipulation logic, then it's not necessary to test against a real db. The logic can be validated by configuring fake data, calling the manipulation method and verifying the output. Your model must be designed in a way that supports this, though. If the model method is responsible for pulling data from the db, manipulating the data, and then persisting data it needs to be refactored to isolate those 3 distinct operations.
Testing that your code works with an actual specific version of a database is necessary too, but most of your application logic should be able to validated at the unit test level.
Mike Cohn's test pyramid visually describes this concept. The majority of automated validation should occur at the lowest level possible. Integration tests (your app to mysql) are necessary but should only cover the actual integration and not excessively exercise application logic. This is because integration tests are more difficult to maintain and are slower because they use external resources. If you test on a shared environment (jenkins) then there are often operational issues in managing the resources.