Has any work or research been done to eradicate herpes zoster virus that lies dormant in the nervous system?

Herpes zoster: Basic science research on varicella zoster virus (VZV, the cause of chickenpox and shingles) and on other herpes viruses may someday give answers on this, but don't hold your breath: you should assume your infection will be lifelong. Happily, recurrent zoster/shingles is fairly uncommon, can be largely prevented by the VZV vaccine, and easily treated. It's an easy problem to live with. Good luck!