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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]

A mathematical model for the quantitative analysis of the interaction between cancer cells and cell-mediated immune system with two discrete-time delays, considering the role of antibodies, is studied in this paper. The model is analyzed both analytically and numerically to understand the dynamics of interaction delay and proliferation enhancement effect delay in the eradication of cancer. Hopf bifurcation occurs when one-time delay crosses the critical value in the absence of the other. It is also observed that as we increase any of the delays, it will increase the cancer burden. The analysis shows that a cancer-free state cannot be obtained solely by the killing rate of cytotoxic T-lymphocytes, but humoral response in combination with cell-mediated immune responses plays an important role to eradicate cancer. [ABSTRACT FROM AUTHOR]

A two‐dimensional tumor‐immune model with the time delay of the adaptive immune response is considered in this paper. The model is designed to account for the interaction between cytotoxic T lymphocytes (CTLs) and cancer cells on the surface of a solid tumor. The model considers the surface growth as a major growth pattern of solid tumors in order to describe the existence of necrotic kernels. The qualitative analysis shows that the immune‐free equilibrium is unstable, and the behavior of positive equilibrium is closely related to the ratio of the immune killing rate to tumor volume growth rate. The positive equilibrium is locally asymptotically stable when the ratio is smaller than a critical value. Otherwise, the occurrence of the delay‐driven Hopf bifurcation at the positive equilibrium is proved. Applying the center manifold reduction and normal form method, we obtain explicit formulas to determine the properties of the Hopf bifurcation. The global continuation of a local Hopf bifurcation is investigated based on the coincidence degree theory. The results reveal that the time of the adaptive immune system taken to response to tumors can lead to oscillation dynamics. We also carry out detailed numerical analysis for parameters and numerical simulations to illustrate our qualitative analysis. Numerically, we find that shorter immune response time can lead to longer patient survival time, and the period and amplitude of a stable periodic solution increase with the increasing immune response time. When CTLs recruitment rate and death rate vary, we show how the ratio of the immune killing rate to tumor volume growth rate and the first bifurcation value change numerically, which yields further insights to the tumor‐immune dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we consider the combined effects of cytotoxic T lymphocyte (CTL) responses on the competition dynamics of two Simian immunodeficiency virus (SIV) strains model. One of strains concerns a relatively slowly replicating and mildly cytopathic virus in the early infection (SIVMneCL8), the other is faster replicating and more cytopathic virus at later stages of the infection (SIVMne170). It is shown that the global dynamics of the ordinary differential equations can be determined by several threshold parameters, and we prove the global stability of the equilibria by rigorous mathematical analysis. To account for a series of infection mechanism leading to viral production, we incorporate time delays in the infection term. Using the methods of constructing suitable Lyapunov functionals and LaSalle’s invariance principle, we obtain the sufficient conditions for the global attractiveness of infection-free equilibrium with both virus strains going extinct, single-infection equilibrium with one of two virus strains out-competing the other one and the two strains coexisting infection equilibrium. We establish that the intracellular delays can destabilize the single-infection equilibrium leading to Hopf bifurcation and periodic oscillations. We show that introduction of immune responses is responsible for the coexistence of two virus strains and the intracellular delays may alter the two-strain competition results. Numerical simulations are presented to illustrate the theoretical conclusions. [ABSTRACT FROM AUTHOR]