People getting healthy after they get infected

As far as I know, people can only become healthy after the cure is discovered.

I don't know if you're playing on a mobile device or desktop, but the interface is roughly the same. Take note of the red circle in the bottom right corner, this indicates the progress toward the cure. Once it reaches 100%, the cure is completed and will be quickly distributed among the living population. This will quickly turn those who are infected into healthy people.

There are a few strategies to fight the cure.

1: Prevent your disease from being discovered: A disease with no symptoms often goes unnoticed. Any evolved symptoms can be easily devolved before they become noticeable, also netting you an extra 2 DNA refund (except in some modes where it costs DNA to devolve a symptoms).

2: Make the disease harder to cure: This can be done by giving your disease medication resistance and gene hardening. Both raise the time required to cure your disease. Some symptoms also make a disease harder to cure; paranoia, narcolepsy, insanity, paralysis, and coma all make it harder for humans to work on a cure. If the cure is reaching 100% you can always do a genetic reshuffle, this will set the cure back a few points.

3: Fight the cure: Once priority is given to discovering a cure you'll see a little blue plane flying around the map, occasionally stopping at various countries, then moving on. When the plane moves on it leaves a blue balloon. Pop that balloon immediately, the longer that balloon is allowed to stay up the quicker the cure is discovered; I often notice a cure percentage jumping by 1 to 3% just for allowing the balloon to stay up for a few seconds.