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This paper explores the dynamics analysis of a human immunodeficiency virus (HIV) model with saturated cytotoxic T lymphocyte (CTL) immune response and Beddington–DeAngelis infection rate. There are two time delays in the model to describe the time needed for infection of cell and CTL immune response generation, respectively. We obtain two thresholds and three possible equilibria from the model. By analyzing the corresponding characteristic equations, we study the stabilities of equilibrium and the effect of delays on CTL immune response. The results indicate that when immune delay is present, the steady state of equilibrium is disrupted and leads to a Hopf bifurcation. Finally, we use sensitivity analyses to show the effect of parameters on thresholds and numerical simulations to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical behaviors of a multiple delayed HIV-1 infection model which describes the interactions of humoral, cytotoxic T lymphocyte (CTL) immune responses, and two modes of transmission that are the classical virus-to-cell infection and the direct cell-to-cell transmission are investigated. The model incorporates three delays, including the delays of cell infection, virus production and activation of immune response. We first prove the well-posedness of the model, and calculate the biological existence of equilibria and the reproduction numbers, which contain virus infection, humoral immune response, CTL immune response, CTL immune competition, and humoral immune competition. Further, the threshold conditions for the local and global stability of the equilibria for infection-free, immune-free, antibody response, CTL response, and interior are established by utilizing linearization method and the Lyapunov functionals. The existence of Hopf bifurcation with immune delay as a bifurcation parameter is investigated by using the bifurcation theory. Numerical simulations are carried out to illustrate the theoretical results and reveal the effects of some key parameters on viral dynamics. [ABSTRACT FROM AUTHOR]

In the course of reverse transcription, drug-sensitive strain can mutate to a drug-resistant one. Cell-to-cell transmission is proved to be more efficient than virus-to-cell infection. The population of Cytotoxic T Lymphocytes (CTLs) can be divided into CTL precursors and CTL effectors. Naive CTL precursors are those that never have contacted antigen, circulating around the body. Once the host is infected, CTL precursors (CTLp) begin to proliferate and differentiate into CTL effectors (CTLe), which have antiviral activity. In this paper, we study an HIV model with drug-sensitive and drug-resistant mixed strains, CTLp and CTLe, both viral and cellular infections. The basic reproduction ratios for each strain are derived and the stability of each feasible equilibrium is analyzed by using Routh–Hurwitz criteria. The periodic solutions may occur near the equilibrium with CTL immune response via Hopf bifurcation. By using normal form theory, we find more information about the limit cycles, such as the amplitude, phase and stability. [ABSTRACT FROM AUTHOR]

In this paper, a delayed human immunodeficiency virus (HIV) model with apoptosis of cells has been studied. Both immunological and intracellular delay have been incorporated to make the model more relevant. Firstly, the model has been investigated using local stability analysis. Next, the global stability analysis of steady states has been performed. The stability switch criteria taking the delay as the bifurcating parameter, leading to Hopf bifurcation has been studied. The transition of the system from order to chaos has been explored, and the analytical results have been verified by numerical simulations. The results thus can be used to describe the extensive dynamics exhibited by the model introduced in this article. The effects of apoptosis on viral load has been studied in the model numerically. [ABSTRACT FROM AUTHOR]

In this paper, we incorporate immune systems into an HIV model, which considers both logistic target-cell proliferation and viral cell-to-cell transmission. We study the dynamics of this model including the existence and stability of equilibria. Based on the existence of equilibria, we focus on the backward bifurcation and forward bifurcation. Considering the stability of equilibria, Hopf bifurcation is discussed by identifying the basic reproduction number R 0 as bifurcation parameter. The direction and stability of Hopf bifurcation are investigated by computing the first Lyapunov exponent. Specially, the effects of immune response on the basic reproduction number R 0 and viral dynamics are addressed by deriving the sensitivity analysis. As a result, we find that the removal rate of infected cells by cytotoxic T lymphocytes (CTLs), γ 1 , is the predominant factor of R 0 . However, we conclude from numerical results that it is unfeasible to decrease R 0 by increasing the value of γ 1 constantly. Numerical simulation is also presented to demonstrate the applicability of the theoretical predictions. These dynamics are investigated by the proposed model to point out the importance and complexity of immune responses in fighting HIV replication. [ABSTRACT FROM AUTHOR]