The horses coming in for the Olympics will be STRICTLY quarantined at the venue after they are screened on arrival (basically, they stay in quarantine that begins even before their departure). They are monitored by both agricultural authorities and FEI stewards (these are on duty 24/7 in droves for all disciplines).
There are diseases that are endemic in certain areas of the world (glanders has been reported on a fairly regular basis in the Mid East recently). The protocols that have been developed to protect sport horses that travel around the globe for competition purposes are strictly adhered to by everyone as doing otherwise would endanger not only the elite equine athletes but also equine populations in their home countries whenever they return.
The two most recent horses put down were well away from any of the locations pertaining to the competition horses.
Wasn’t on board w/Sandron when ToC went to the Olympics in '84, but I began working shortly thereafter and so have been aware of the procedures from '86-the present (husband is an FEI steward and international judge, and I’m also a Lic. Official; I monitor the quarantine for international horses at the WIHS). Protocols are exceptionally tight, and while I would guess that confidence in the overall Olympics might be questionable, the individual work of the sport federations is what will govern the level of safety and success at each venue. Not that anything is fail safe, but the FEI is on it for sure. The top stewards from each discipline are already at the venue, and on the job.