Shaban2025-11-222025-11-222023https://escholar.umt.edu.pk/handle/123456789/12072This thesis delves deeply into the intricate landscape of HIV/AIDS, providing a comprehensive overview spanning its precise definition, clinical manifestations, modes of transmission, diagnostic procedures, associated complications, and therapeutic interventions. Leveraging the potency of mathematical epidemiology, this research employs deterministic compartmental models, specifically a tailored SIR model, to analyze the complex dynamics governing the dissemination of HIV/AIDS. This tailored SIR model serves as the foundation for analyzing and modeling the distinctive attributes characterizing the transmission of the HIV/AIDS virus. A central theme of this research centers on the meticulous evaluation of various numerical techniques, including the Euler method, the RK-4 method, and the Non-Standard Finite Difference (NSFD) method, applied to HIV/AIDS epidemic models both with and without delay factors. These techniques are essential for understanding the intricacies of transmission dynamics in both scenarios. Consistently, the outcomes resoundingly favor the NSFD method for its unparalleled accuracy and efficacy, particularly when juxtaposed against conventional methodologies. In the realm of equilibrium points and reproductive numbers, the findings underscore the pivotal role played by the basic reproductive number R0 in ascertaining outbreak potential and devising containment strategies. Specifically, the study identifies distinct equilibriums — the disease-free and endemic equilibriums — and rigorously assesses their stability based on R0 values, offering invaluable insights into disease control and the potential for eradication. In summation, this thesis not only enriches our understanding of HIV/AIDS epidemiology but also bequeaths a robust mathematical toolkit for the meticulous assessment of its intricate dynamics. The insights garnered, particularly through the innovative NSFD method, hold profound promise for the formulation of optimal strategies in the relentless battle against this formidable disease.enThesis