Since the TV series seems to be set in a parallel universe rather than the future the Covid19 pandemic could be happening at the same time as real life. So how would the Gileadean authorities handle it? On hand Gilead is an isolated pariah state, but so is North Korea. A totalitarian state can implement much more severe restrictions than a democratic one, even making extremely harsh decisions triaging medical care, but Gilead is also a theocracy. Their religious fanaticism already leads them to things like not allowing women to practice medicine, or even nursing. Even without a pandemic their healthcare system must be barely functional.
Also no writing is allowed in public lest women see, and Gilead doesn’t appear to have any kind of mass media (ie the Econopeople don’t even have TVs or radios showing propaganda) so the only way they could disseminate instructions to the general public is loudspeakers on trucks, or gathering them together to listed to speechs/sermons. Are they gone to stop public Salvangings and Prayvaganzas? Commanders’ households can self isolate pretty easy and I imagine extreme measures would be taken to isolate Handmaids (or any fertile women), but would that look like? Do they get confined to their households or the Red Centers? Do any kind of restrictions get put on Commanders?
One gets the impression that Gilead has precious little trade with its neighbors. I’m sure its borders are heavily fortified to keep the population from fleeing. North Korea & East Germany would probably be the models. North Korea trades with China & East Germany traded with the USSR. Gilead… Not so much. If you limit the mobility of the population, you limit these sort of diseases.
They’re not portrayed as having much population density… therefore the emphasis on reproduction.
The Elite would get the best of advice & personal protective equipment. The outcasts would be allowed to keep on suffering. One boot is as good as another when they’re standing on your head.
Things that might tend to exacerbate the spread of COVID-19 are, as mentioned the anti-science agenda of the authorities, lack of education, declining literacy and widespread medical skills, although they could press-gang those women who had once been medical professionals and probably a medical infrastructure - I assume the spartan surgeries seen in the TV series are only for the elite and obs/gyn for Handmaids.
Things that might tend to help them could be their well-rehearsed ability to enforce any sort of social control they want, including isolation and extreme separation, lots of home knitting of scarves and masks, and probably a whole-hearted embrace of any other clean-living behaviours that might be marginally preventative like stoning the infected.
The borders are patrolled but seem pretty porous for smugglers and contraband. Maybe the Council of elders will take an actively ignorant view of the disease elsewhere as being as something that happens to sinners in their own country. If the first cases happen among the proles who are in a gathering, a picnic or mass execution, then the there will be widespread community transmission.