The three major diseases are cancer, heart attack, and stroke.
A rapid sickness is an illness or a disease.
This is a jukugo word that uses the on'yomi readings of the kanji. You didn't learn this reading for 病 yet, so here's a mnemonic to help you:
If you don't treat an illness right away you'll end up paying (ぺい) with your life. And if you don't die, you'll still have to pay a lot for doctor and hospital fees. Yuck.