The cabin air is replaced every 2-3 minutes, so it's not really recirculated over and over again is it? Even if it's half and half (which it's not), what does get recirculated is filtered quite heavily. The only "modern" plane that doesn't use a bleed air system is the 787 which uses electric powered compressors to basically do the same thing. Those do recirculate more air than others. They represent a pretty small percentage of the worldwide fleet though.
People get sick from travelling from touching things, lack of sleep, dry conditions in planes cause dryness and break down mucus protection, different levels of hygiene in other parts of the world and myriad other reasons but "recirculated over and over again air" isn't one of them. Those masks aren't protecting you from touching a door knob, or surface a sick person touched and then touching your mouth or eyes. So unless you hide in a plastic bubble you're at risk.