According to your latest test results, I don't see how your doctor can say you are positive for Hep B.
Because your HBcAb is positive, this indicated you HAD Hep B infection in the past, but have recovered (your HBsAg is now Non-Reactive) and you have also developed immunity (your HBcAb is > 10 miu/ml) against future Hep B infection.
It has been discovered recently, for people who had recovered from Hep B infection (HBcAb positive, HBsAg negative) or who have their chronic Hep B under control (not on medication, HBsAg positive, hbvdna(viral load) very low), their Hep B may become active again if they receive immnuo suppressive therapies due to other diseases, such as cancer and rheumatoid arthritis, then some of them should receive prophylaxis Hepatitis B antiviral treatment as a precaution.
The latest thinking is that for those who had recovered from HepB infection (HBcAb positive, HBsAg negative), there may be some residual HBV cccDNA, but it is under the profound control of our immune system.
I would disagree with your doctor that you are infectious. There maybe very tiny amount of virions in your blood, but I believe most of these will be coated with HBsAg anti-bodies that renders them non-infectious if someone else was exposed to it. Since we don't normally have blood to blood contact, so I don't see how it matters. I also don't believe there will be any virions in your bodily fluid.
As for having your liver functions monitored closely, I again disagree. Liver functions panel is usually part of routine health checks and that should suffice.
I am not a doctor, you may like to get a second opinion from a liver specialist, and not an infectious disease specialist.
I see that you are immunized for heb b.are you sure that 187 is antibody not antigen.if yes I see that your doc is making a mistake and you have no hepatitis