During the past decade significant research has been aimed at better understanding for human immunodeficiency virus type 1 (HIV-1), and the use of mathematical modeling to interpret experimental results has made a significant contribution. However, time-delays, which play a critical role in various biological models, are still not amenable to many traditional analysis methods. In this paper, we apply a recently developed approach using the Lambert W function to handle the time delay in a HIV-1 pathogenesis dynamic model. Dominant eigenvalues in the infinite eigenspectrum of these time-delay systems are obtained and used to understand the effects of the parameters of the model on the immune system. Also, the result is extended to analyze the sensitivity of the eigenvalues with respect to the parameters in the HIV-1 model. The research makes it possible to know which parameters are more influential than others, and the obtained information is used to investigate the HIV-1 dynamic system analytically.

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