Unbalanced classes -- How to minimize false negatives?
Class imbalance is a very common problem.  You can either oversample the positive class (or undersample the negative) or add class weights.

Another thing to remember in this case is that accuracy is not a very useful metric here.  You might consider AUC or F1 score.

Changing your decision threshold may seem appealing, but will obviously lead to (in this case likely drastically) increased false positives (though perhaps FPs aren't as bad as FNs in the case of medical diagnosis, if tests aren't prohibitively expensive).

A more in-depth analysis of options in the case of class imbalance is provided here.