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This study utilizes mathematical models to assess progress toward achieving the UNAIDS 90-90-90 and 95-95-95 targets aimed at managing and eradicating HIV/AIDS. It contrasts stochastic and deterministic models, focusing on their utility in optimizing public health strategies. Stochastic models account for real-world unpredictability, offering more realistic insights compared to deterministic approaches. The 95-95-95 targets aim for 95\% of people living with HIV to know their status, 95\% of those diagnosed to receive antiretroviral therapy (ART), and 95\% of those on ART to achieve viral suppression. These benchmarks are critical for reducing transmission and improving health outcomes. This analysis establishes the basic reproduction number ($R_0$) to guide interventions and examines the stability of disease-free and endemic equilibria, providing a foundation for applying optimal control strategies to minimize HIV prevalence effectively and cost-efficiently. Moreover, the data for this study was sourced from the official UNAIDS website, focusing on North America. An innovative feature of this study is the application of the Stochastic method, which enhances model accuracy and oper
The control of opportunistic infections among HIV infected individuals should be one of the major public health concerns in reducing mortality rate of individuals living with HIV/AIDS. In this study a deterministic co-infection mathematical model is employed to provide a quantification of treatment at each contagious stage against Pneumocystis Pneumonia (PCP) among HIV infected individuals on ART. The disease-free equilibrium for the HIV/AIDS sub model, PCP sub model and the co-infection model are shown to be locally asymptotically stable when their associated disease threshold parameter is less than a unity. By use of suitable Lyapunov functions, the endemic equilibrium corresponding to HIV/AIDS and PCP sub models are globally asymptotically stable whenever $\mathcal{R}_{0H}>1$ and $\mathcal{R}_{0P}>1$ respectively. The sensitivity analysis results implicate that the effective contact rates are the main mechanisms fueling the proliferation of the two diseases and on the other hand treatment efforts play an important role in reducing the incidence. Numerical simulations show that treatment of PCP at all contagious stages reduces its burden on HIV/AIDS patients and dual treatm