That depends on many things, such as your soil, the weather, etc. Spend some time with this article.

Here is a link that might be useful: Plant disease suppression with Compost

Being a virus, it is not at all likely it will overwinter because virii require a host to survive.

Common vectors for infection are the seeds you purchase as well as aphids which sometimes carry the virus.

Since you have the problem in your area you may wish to purchase only seed that is certified free of disease or purchase varieties with genetic resistance to the virus where practical.

I suppose another question would be are you certain it was the mosaic virus?

In any event you can read up on the various management strategies by googling on 'bean mosaic virus', but the short answer to your question is that it is unlikely to be able to persist in your soil without a host.