Investigating Lie Group Symmetry and Analyzing Stability in Mathe-matical Modeling of the HIV Epidemic

Abstract

This study explores the dynamics of the Human Immunodeficiency Virus (HIV)/Acquired Immunodeficiency Syndrome (AIDS) pandemic using stability and Lie symmetry analysis. The stability analysis indicates a marked decline in the susceptible population in the absence of treatment, which ultimately leads to the stabilization of the AIDS population due to increased mortality among untreated AIDS patients. Furthermore, the study identified a direct relationship between the number of HIV-positive individuals and the progression to AIDS. Lie point symmetries are applied to reveal a three-dimensional Lie algebra, facilitating the derivation of model reductions and closed-form solutions. In particular, the death rate of AIDS patients is analyzed in conjunction with the HIV infection rate and the transmission probability per partner contact. The study employs the most general Lie symmetry generator to obtain both reductions and numerical solutions, improving the understanding of the model behavior.