How to handle the plague in Civ VII

The moment you notice the green smoke clouds that indicate the plague, send a Physician to the affected settlement to nip it in the bud. The sooner they arrive, the sooner these plague doctors can cure the entire city.
You can recruit a Physician from any city or buy one from a town. The unit will cost you quite a bit of gold, so get farming!
Once you have a Physician, keep rotating them throughout your empire to ensure they visit each settlement to reduce the risk of plague spreading rapidly.