DCDS-B
Mathematical modeling on helper T cells in a tumor immune system
Yueping Dong Rinko Miyazaki Yasuhiro Takeuchi
Discrete & Continuous Dynamical Systems - B 2014, 19(1): 55-72 doi: 10.3934/dcdsb.2014.19.55
Activation of CD$8^+$ cytotoxic T lymphocytes (CTLs) is naturally regarded as a major antitumor mechanism of the immune system. In contrast, CD$4^+$ T cells are commonly classified as helper T cells (HTCs) on the basis of their roles in providing help to the generation and maintenance of effective CD$8^+$ cytotoxic and memory T cells. In order to get a better insight on the role of HTCs in a tumor immune system, we incorporate the third population of HTCs into a previous two dimensional ordinary differential equations (ODEs) model. Further we introduce the adoptive cellular immunotherapy (ACI) as the treatment to boost the immune system to fight against tumors. Compared tumor cells (TCs) and effector cells (ECs), the recruitment of HTCs changes the dynamics of the system substantially, by the effects through particular parameters, i.e., the activation rate of ECs by HTCs, $p$ (scaled as $\rho$), and the HTCs stimulation rate by the presence of identified tumor antigens, $k_2$ (scaled as $\omega_2$). We describe the stability regions of the interior equilibria $E^*$ (no treatment case) and $E^+$ (treatment case) in the scaled $(\rho,\omega_2)$ parameter space respectively. Both $\rho$ and $\omega_2$ can destabilize $E^*$ and $E^+$ and cause Hopf bifurcations. Our results show that HTCs might play a crucial role in the long term periodic oscillation behaviors of tumor immune system interactions. They also show that TCs may be eradicated from the patient's body under the ACI treatment.
keywords: helper T cells periodic solutions Hopf bifurcation. Tumor immune system
DCDS-B
Global dynamics of a latent HIV infection model with general incidence function and multiple delays
Yu Yang Yueping Dong Yasuhiro Takeuchi
Discrete & Continuous Dynamical Systems - B 2019, 24(2): 783-800 doi: 10.3934/dcdsb.2018207

In this paper, we propose a latent HIV infection model with general incidence function and multiple delays. We derive the positivity and boundedness of solutions, as well as the existence and local stability of the infection-free and infected equilibria. By constructing Lyapunov functionals, we establish the global stability of the equilibria based on the basic reproduction number. We further study the global dynamics of this model with Holling type-Ⅱ incidence function through numerical simulations. Our results improve and generalize some existing ones. The results show that the prolonged time delay period of the maturation of the newly produced viruses may lead to the elimination of the viruses.

keywords: Virus dynamics general incidence multiple delays stability analysis Lyapunov functional

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