Adaptive Observer for Estimating the Parameters of an HIV Model with Mutants

Human immunodeficiency virus (HIV) causes acquired immune deficiency syndrome (AIDS). The process of infection and mutation by HIV can be described by fifth-order ordinary differential equations. These equations can be reduced to third-order differential equations through the singular perturbation theory. The objective of this paper is to present a parameter estimation algorithm for this third-order HIV model, using two (out of three) state variables. We first show that the parameters of the HIV model are identifiable with these measurements. The structure of the proposed estimator parallels that of the full state feedback estimator with the unavailable state replaced with an estimated variable. We then prove that the resulting adaptive observer equipped with the so-called sigma-modification can be tuned to be ultimately bounded under some conditions in terms of the concentration of uninfected CD4+ T cells. Furthermore, it is seen through computer simulations that an iterative application of the proposed algorithm is effective; the estimated parameters approach their true values, and the stability analysis of the ensuing HIV model leads to the results that are consistent with those obtained previously.