The above response is not necessarily true. My step-mother was "latent" for 45 years. When she had to go on meds for her rheumatoid arthritis, it triggered the TB and almost killed her. So, while it remains latent, it is just that. Once exposed to a trigger, TB can go full course, and be deadly.
From what I understand, is that the longer your walk around with latent TB the less likely it will turn into the active form (symptoms present). You are not contagious with latent TB, however it still poses risks to others and to yourself. If it does turn symptomatic then you get become seriously ill (death included), especially because you also have asthma. Your doctor wants to prescribe antibiotics more as a preventative measure, which I think is prudent in your case. Like I said the longer you can keep it latent the better you are off in the future.