Human Immunodeficiency Virus (HIV) has long been a major worldwide health concern and hence it is necessary to have a deeper understanding of its transmission dynamics and control strategies. A study says that in most cases, if an HIV patient gets adequate treatment, the chances of transferring them to AIDS are reduced. In our paper, we have studied those less likely chances, where even after treatment, an HIV patient can have AIDS. We have constructed a noble mathematical model SITA to study the dynamics of the infectious virus HIV, which manifests into AIDS disease. We have devised an original AIDS epidemic model using a non-linear dynamical system, subdividing the human population into four compartments Susceptible (S), Infected with HIV (I), Treated (T), and AIDS patients (A). The aim of the study is to understand the intricate dynamics of disease progression and control the spread of HIV/AIDS. The study takes into account both horizontal and vertical modes of transmission. The basic reproduction number R0 is determined. The disease-free equilibrium is locally asymptotically stable if R0 < 1 and the infection gets cleared. Disease persists in the population if R0 > 1. Local stability is analyzed using the Jacobian matrix, and global stability is assessed with Lyapunov function. Numerical simulations performed using MATLAB validate our theoretical conclusions and provide guidance for controlling the spread of the infection.