Search

In this paper, we study a tumor-immune system interaction model with dendritic cell therapy and immune response delay. First, it is shown that the ODE version of the model has a Bogdanov--Takens (BT) singularity or a weak focus with multiplicity at most 1 for different parameter values. As the parameters vary, the ODE model undergoes supercritical Hopf bifurcation and supercritical BT bifurcation. Our analysis indicates that there exists a threshold value of the activation rate of T cells, below which tumor immune escape occurs, above or at which T cells and tumor cells coexist in the form of a stable periodic oscillation or steady state. Second, we study how the immune response delay affects the dynamics of the model. Our results reveal that the delay can destabilize the stable positive equilibrium through Hopf bifurcation. Furthermore, the direction and stability of Hopf bifurcation are derived. When there is a cusp, we show that it is a BT singularity for any delay and the delay model also undergoes BT bifurcation. Finally, numerical simulations are presented 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]

The SARS-CoV-2 leads to a worldwide COVID-19 pandemic, which has caused tremendous damage to the world. In this paper, we develop a dynamic model in vivo, fitting and estimating parameters for T lymphocytes and pro-inflammatory cytokines IL-6 in patients with mild and severe COVID-19 at Yale New Haven Hospital through the GWMCMC algorithm. Meanwhile, we also analyze the structural identifiability and practical identifiability of the model. Further, we add time-varying parameters to the model, using the least squares method to perform data fitting and parameter estimation on survivors and non-survivors of the Italian infectious disease hospital. Then analyze the similarities and differences in immune response mechanisms between the two countries. Finally, we demonstrate the existence and stability of the equilibrium state of the model and analyze the Hopf bifurcation at the positive equilibrium state by using the central manifold theory and normal form theory. This result may explain the recurrence of infection in some COVID-19 patients. [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]

The literature suggests that effective defence against tumour cells requires contributions from both Natural Killer (NK) cells and CD 8 + T cells. NK cells are spontaneously active against infected target cells, whereas CD 8 + T cells take some times to activate cell called as cell-specific targeting, to kill the virus. The interaction between NK cells and tumour cells has produced the other CD 8 + T cell, called tumour-specific CD 8 + T cells. We illustrate the tumour–immune interaction through mathematical modelling by considering the cell cycle. The interaction of the cells is described by a system of delay differential equations, and the delay, τ represent time taken for tumour cell reside interphase. The stability analysis and the bifurcation behaviour of the system are analysed. We established the stability of the model by analysing the characteristic equation to produce a stability region. The stability region is split into two regions, tumour decay and tumour growth. By applying the Routh–Hurwitz Criteria, the analysis of the trivial and interior equilibrium point of the model provides conditions for stability and is illustrated in the stability map. Numerical simulation is carried out to show oscillations through Hopf Bifurcation, and stability switching is found for the delay system. The result also showed that the interaction of NK cells with tumour cells could suppress tumour cells since it can increase the population of CD 8 + T cells. This concluded that the inclusion of delay and immune responses (NK-CD 8 + T cells) into consideration gives us a deep insight into the tumour growth and helps us understand how their interactions contribute to kill tumour cells. [ABSTRACT FROM AUTHOR]