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>