  HIV-1 affects tens of millions of people worldwide. Current treatments often
involve a cocktail of antiretroviral drugs, which are effective in reducing the
virus and extending life spans. However, there is currently no FDA-approved
HIV-1 transcription inhibitor. Furthermore, there have only been a few attempts
to model the transcription process in HIV-1. In this work, we extend a novel
three-state model of HIV-1 transcription introduced in DeMarino et al. (2020)
that has been developed and validated against experimental data. After fitting
this model to in vitro data, significant differences in the transcription
process of HIV-1 in T-cells and macrophages have been observed. In particular,
the activation of the HIV-1 promoter in T-cells appears to take place rapidly
as the Tat protein approaches a critical threshold. In contrast, the same
process occurs smoother in macrophages.
  In this work, we carry out systematic mathematical analyses of the model to
complement experimental data fitting and sensitivity analysis performed
earlier. We derive explicit solutions of the model to obtain exact
transcription process decay rates for the original model and then study the
effect of nonlinearity on the system behavior, including the existence and the
local and global stability of the positive equilibrium. We were able to show
the stability of the positive steady state in limiting cases, with the global
stability in the general case remaining an open question.
  By modeling the effect of transcription-inhibiting drug therapy, we provide a
nontrivial condition for it to be effective in reducing viral load. Moreover,
our numerical simulations and analysis point out that the effect of the
transcription-inhibitor can be enhanced by synchronizing with standard
treatments, such as combination antiretroviral therapy, to allow the reduction
of total dosages and toxicity.
