- Your neighbour doesn't treat, his hive dies, is robbed out by your hive, bringing back enough Pathogen A to pass the threshold. You get disease.
- Your neighbour treats, his hive doesn't die yet still caries the pathogen as the treatment doesn't eradicate it. Your hive still comes into contact with the pathogen at lower dosages, it multiplies inside your hive, passes the threshold and you get disease.
So in these two cases the result is the same. Now the interesting case, and this might be where you're coming from is this scenario:
- Your neighbour treats, his hives don't die yet still carry the pathogen as the treatment doesn't erradicate it. Your hive still comes into contact with the pathogen at lower dosages, it multiplies inside your hive but not enough to pass the threshold and cause disease. If at any point your neighbor's hive dies and your bees rob it, the threshold gets passed and you get disease.
Is this a good characterization of what you're thinking? I see where this could be the case if your hive has enough resistance to Pathogen A to keep a small concentration at bay but enough concentration overcomes that resistance and allows multiplication. I don't know if that kind of fragile equilibrium is common in our hives.