Modelling HIV dynamics with cell-to-cell transmission and CTL response

In most HIV models, the emergence of backward bifurcation means that the control for basic reproduction number less than one is no longer effective for HIV treatment. In this paper, we study an HIV model with CTL response and cell-to-cell transmission by using the dynamical approach. The local and global stability of equilibria is investigated, the relations of subcritical Hopf bifurcation and supercritical bifurcation points are revealed, especially, the so-called new type bifurcation is also found with two Hopf bifurcation curves meeting at the same Bogdanov-Takens bifurcation point. Forward and backward bifurcation, Hopf bifurcation, saddle-node bifurcation, Bogdanov-Takens bifurcation are investigated analytically and numerically. Two limit cycles are also found numerically, which indicates that the complex behavior of HIV dynamics. Interestingly, the role of cell-to-cell interaction is fully uncovered, it may cause the oscillations to disappear and keep the so-called new type bifurcation persist. Finally, some conclusions and discussions are also given.

Modeling the effect of activation of CD4$^+$ T cells on HIV dynamics

<p style='text-indent:20px;'>HIV infects active uninfected CD4<inline-formula><tex-math id="M1">\begin{document}$ ^+ $\end{document}</tex-math></inline-formula> T cells, and the active CD4<inline-formula><tex-math id="M2">\begin{document}$ ^+ $\end{document}</tex-math></inline-formula> T cells are transformed from quiescent state in response to antigenic activation. Activation effect of the CD4<inline-formula><tex-math id="M3">\begin{document}$ ^+ $\end{document}</tex-math></inline-formula> T cells may play an important role in HIV infection. In this paper, we formulate a mathematical model to investigate the activation effect of CD4<inline-formula><tex-math id="M4">\begin{document}$ ^+ $\end{document}</tex-math></inline-formula> T cells on HIV dynamics. In the model, the uninfected CD4<inline-formula><tex-math id="M5">\begin{document}$ ^+ $\end{document}</tex-math></inline-formula> T cells are divided into two pools: quiescent and active, and the stimuli rate of quiescent cells by HIV is described by saturated form function. We derive the basic reproduction number <inline-formula><tex-math id="M6">\begin{document}$ R_0 $\end{document}</tex-math></inline-formula> and analyze the existence and the stability of equilibria. Numerical simulations confirm that the system may have backward bifurcation and Hopf bifurcation. The results imply that <inline-formula><tex-math id="M7">\begin{document}$ R_0 $\end{document}</tex-math></inline-formula> cannot completely determine the dynamics of the system and the system may have complex dynamics, which are quite different from the models without the activation effect of CD4<inline-formula><tex-math id="M8">\begin{document}$ ^+ $\end{document}</tex-math></inline-formula> T cells. Some numerical results are further presented to assess the activation parameters on HIV dynamics. The simulation results show that the changes of the activation parameters can cause the system periodic oscillation, and activation rate by HIV may induce the supercritical Hopf bifurcation and subcritical Hopf bifurcation. Finally, we proceed to investigate the effect of activation on steady-state viral loads during antiretroviral therapy. The results indicate that, viral load may exist and remain high level even if antiretroviral therapy is effective to reduce the basic reproduction number below 1.</p>

Vectored Immunoprophylaxis and Cell-to-Cell Transmission in HIV Dynamics

We consider local and global bifurcations in a HIV model with cell-to-cell transmission and vectored immunoprophylaxis. Both theoretical and numerical analyses are conducted to explore various dynamical behaviors including backward bifurcation, Hopf bifurcation, homoclinic bifurcation, Bogdanov–Takens bifurcation, hysteresis and isola bifurcation. The isola bifurcation of periodic orbits was first detected numerically in HIV model, which means that there is a parameter interval with the same oscillations. It is shown that the effect of vectored immunoprophylaxis in this model is the main cause of the periodic symptoms of HIV disease. Moreover, it is shown that the increase of cell-to-cell transmission may be the main factor causing Hopf bifurcation to disappear, and thus eliminating oscillation behavior. Also, several patterns of dynamical behaviors are found in different parameter intervals including the bistability.