Imbalanced dataset - Positive majority class
There are two things that might help you:

1)
If you use a classifier $f$ that returns a value $f(x)\in[0,1]$ between 0 and 1 instead of a direct class assignment, then you can use a threshold $\theta\in[0,1]$: 

$f(x)=\begin{cases}
    1,&amp; \text{if } f(x)\geq\theta\\
    0, &amp; \text{else }
\end{cases}$

The higher the value of $\theta$, the more you want to be sure to classify somebody as positive, which means that there will be fewer false positives, but at the same time you will have a fewer sensitivity (recall) because there will be more false negatives.

2) if your classifier always predicts the majority class, you could try oversample the minority class (using SMOTE or other oversampling techniques) to make the training data set balanced.