Unfortunately detection of negative strands is not definitive ... the mites are feeding on bees containing significant levels of negative strand RNA. You have to demonstrate new negative strands are produced in the mite.
One way to conduct these studies is to feed mites genetically-tagged virus, purified and free of negative strands due to treatment with RNAse. You can feed the mites using a honey bee-free feed packet system. Mites ingest the virus and you then use a sensitive negative strand assay to a) detect negative strands (and there will be quite a bit present) and then b) demonstrate it contains the genetic tag introduced in the purified input virus.
The input virus contains a genetic tag, unique to that virus alone. The input virus contains no detectable negative strands as the virus has been treated by RNAse and the virus does not package negative strands. If the genetic tag is found in negative strands in the mite after feeding it is good evidence that the virus replicates in the mite.
We have recently published this type of study for DWV (https://www.mdpi.com/1999-4915/12/5/532) in collaboration with Alan Bowman in Aberdeen who has developed a feed packet system in which the virus cannot replicate and which contains no honey bee-derived material.
Of course, to do this requires a reverse genetic system for the virus and I don't think one exists for BQCV yet (I don't have current access to the JAR BQCV study from my socially distanced location so may have misinterpreted how they reached their conclusions).
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