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>antibody based method (Westerns, ELISAs)
novel antibodies take a long time to develop and they're super expensive
>sequencing based methods
You need to RT a sample containing the RNA virus, then do Sanger sequencing. Sanger is cheap, takes five bucks a reaction, less in bulk. RT kits are kinda expensive if you want to do quantitative PCR, but i don't think that's necessary? a yes/no call is probably enough.
the big cost is work-hours, probably. Procedures like RT-PCR aren't that hard, but they take time, especially if you have a clinical sample that might not have a lot of virus present in it. You might need some dedicated extraction protocols to get a good enough starting RNA sample. also, it looks like some clinical samples might need additional cleanup, or else they interfere with the polymerase activity. you also have to be careful with contamination, to make sure that if you amplify something, you're amplifying COVID-19 specifically and not some other Coronavirus.
medical lab techs earn on average $20/hour, and i could see the above taking a couple hours at least. Add in reagents, consumables, overhead costs, ongoing costs to set aside money for repairs and equipment maintenance, shit like that, the price can build quickyl, i think