# STABILITY OF DELAY-DISTRIBUTED HIV INFECTION MODELS WITH MULTIPLE VIRAL PRODUCER CELLS

• ELAIW, A.M. (DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, KING ABDULAZIZ UNIVERSITY) ;
• ELNAHARY, E.KH. (DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE SOHAG UNIVERSITY) ;
• SHEHATA, A.M. (DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE AL-AZHAR UNIVERSITY) ;
• ABUL-EZ, M. (DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE SOHAG UNIVERSITY)
• Accepted : 2018.02.26
• Published : 2018.03.25

#### Abstract

We investigate a class of HIV infection models with two kinds of target cells: $CD4^+$ T cells and macrophages. We incorporate three distributed time delays into the models. Moreover, we consider the effect of humoral immunity on the dynamical behavior of the HIV. The viruses are produced from four types of infected cells: short-lived infected $CD4^+$T cells, long-lived chronically infected $CD4^+$T cells, short-lived infected macrophages and long-lived chronically infected macrophages. The drug efficacy is assumed to be different for the two types of target cells. The HIV-target incidence rate is given by bilinear and saturation functional response while, for the third model, both HIV-target incidence rate and neutralization rate of viruses are given by nonlinear general functions. We show that the solutions of the proposed models are nonnegative and ultimately bounded. We derive two threshold parameters which fully determine the positivity and stability of the three steady states of the models. Using Lyapunov functionals, we established the global stability of the steady states of the models. The theoretical results are confirmed by numerical simulations.

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