My suggestion would be to forget for a moment the terms sensitivity, specificity, PPV, and NPV because they can be quite confusing when dealing with more than one type of condition. Instead, focus on the value of the test that you want to measure, i.e. which test is preferable over another test with different test characteristics?

For that you need to put a value on each possible test result. You can approach it in either way: as positive contribution to a patient's health or as 'cost' due to an erroneous test results. Both these approaches should result in the same preference. It is probably easiest to approach is from the cost side:

- What is the disadvantage of getting a positive test result if you don't have meningitis?

- What are the respective disadvantages of getting a negative test result if you do have either of the three types of meningitis? I assume that those three disadvantages are not of equal value because otherwise you would not need to bother with the different types of meningitis

Generally the disadvantage of a false negative is much larger than of a false positive, therefore your evaluation should reflect that you are willing to tolerate more than 1 false positive if that is needed for finding 1 more true positive.

Health economists tend to estimate preference through QALYs. Maybe you have a better way to determine the relative value of the four different types of test errors in your example.