# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021238
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## Modeling the effect of activation of CD4$^+$ T cells on HIV dynamics

 1 School of Science, Nanjing University of Science and Technology, Nanjing 210094, China 2 Center for Basic Teaching and Experiment, Nanjing University of Science and Technology, Jiangyin 214443, China 3 Aliyun School of Big Data, Changzhou University, Changzhou, 213164, China

* Corresponding author: Zhipeng Qiu

Received  April 2021 Revised  August 2021 Early access October 2021

Fund Project: Zhipeng Qiu is supported by the National Natural Science Foundation of China (12071217). Ting Guo is supported by the Postgraduate Research and Practice Innovation Program of Jiangsu Province (KYCX20_0243)

HIV infects active uninfected CD4$^+$ T cells, and the active CD4$^+$ T cells are transformed from quiescent state in response to antigenic activation. Activation effect of the CD4$^+$ 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$^+$ T cells on HIV dynamics. In the model, the uninfected CD4$^+$ 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 $R_0$ 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 $R_0$ 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$^+$ 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.

Citation: Linghui Yu, Zhipeng Qiu, Ting Guo. Modeling the effect of activation of CD4$^+$ T cells on HIV dynamics. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021238
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Schematic diagram of model (2)
Forward and backward bifurcations of system (2). (a) Forward bifurcation; (b) Backward bifurcation. Blue solid line indicates that infection-free equilibrium $E_0$ is L.A.S., red dotted line indicates that the equilibrium $E_0$ is a saddle, black solid line represents the positive equilibrium $E_*^1$ is L.A.S., and red dashed line represents positive equilibrium $E_*^2$ is unstable
Stable positive equilibrium $E_*^1$ when $R_0>1$. (a) Time series of quiescent uninfected CD4$^+$ T cells $Q$; (b) Phase diagram of system (2). When $k_{1} = 0.5$, solution trajectory of the system converges to the positive equilibrium $E_*^1$.
Periodic orbit at $E_*^1$ when $R_0>1$. (a) Time series of quiescent uninfected CD4$^+$ T cells $Q$; (b) Phase diagram of system (2). When $k_1 = 0.6$, solution trajectory of the system presents periodic oscillation at the positive equilibrium $E_*^1$.
, and $k_1 = 1$, $k_s = 10^4$. Red dashed line represents $\alpha_Q = 0.5$, blue solid line represents $\alpha_Q = 0.3$, yellow dashed line represents $\alpha_Q = 0.15$ and green solid line represents $\alpha_Q = 0.01$.">Figure 5.  Numerical solutions of the model (2). Parameter values are from Table 2, and $k_1 = 1$, $k_s = 10^4$. Red dashed line represents $\alpha_Q = 0.5$, blue solid line represents $\alpha_Q = 0.3$, yellow dashed line represents $\alpha_Q = 0.15$ and green solid line represents $\alpha_Q = 0.01$.
, and $\alpha_Q = 0.15$, $k_s = 10^4$. Yellow solid line represents $k_1 = 10^3$, red dashed line represents $k_1 = 10^2$, black solid line represents $k_1 = 10$, blue dashed line represents $k_1 = 1$ and green solid line represents $k_1 = 10^{-1}$.">Figure 6.  Numerical solutions of the model (2). Parameter values are from Table 2, and $\alpha_Q = 0.15$, $k_s = 10^4$. Yellow solid line represents $k_1 = 10^3$, red dashed line represents $k_1 = 10^2$, black solid line represents $k_1 = 10$, blue dashed line represents $k_1 = 1$ and green solid line represents $k_1 = 10^{-1}$.
, and $\alpha_Q = 0.15$, $k_1 = 1$. Yellow solid line represents $k_s = 10^8$, red dashed line represents $k_s = 10^6$, black solid line represents $k_s = 10^5$, blue dashed line represents $k_s = 10^4$ and green solid line represents $k_s = 10^2$.">Figure 7.  Numerical solutions of the model (2). Parameter values are from Table 2, and $\alpha_Q = 0.15$, $k_1 = 1$. Yellow solid line represents $k_s = 10^8$, red dashed line represents $k_s = 10^6$, black solid line represents $k_s = 10^5$, blue dashed line represents $k_s = 10^4$ and green solid line represents $k_s = 10^2$.
, and $k_1 = 10$, $k_s = 10^4$ and $\alpha_Q = 0.1$. Red line represents $V_*^1$ and blue line represents $V_*^2$.">Figure 8.  HIV dynamics in the case of antiretroviral therapy $\epsilon$. (a)$R_0$ vs drug efficacy. (b)Virus load vs drug efficacy for the quiescent cell model. Parameters are fixed in the Table 2, and $k_1 = 10$, $k_s = 10^4$ and $\alpha_Q = 0.1$. Red line represents $V_*^1$ and blue line represents $V_*^2$.
Four typical patterns of dynamical behaviors of system.
 Pattern Range of R0(Range of p) Steady states of system (2) 1 0 < R0 < R00 < 1(0 < p < p00) E0 is G.A.S. 2 0 < R0 < Rc < 1 < R00(p00 < p < pc < p0) E0 is L.A.S. 3 0 < Rc < R0 < 1 < R00(p00 < p < p0 < pc) E0 is L.A.S.; E*2 is unstable; E*1 is either L.A.S. or unstable(when Re(ηi(E*1)) < 0, i = 1; 2; 3; 4, E*1 is L.A.S.). 4 R0 > 1(p > p0) E0 is unstable; E*1 is either L.A.S. or unstable(when Re(ηi(E*1)) < 0, i = 1; 2; 3; 4, E*1 is L.A.S.).
 Pattern Range of R0(Range of p) Steady states of system (2) 1 0 < R0 < R00 < 1(0 < p < p00) E0 is G.A.S. 2 0 < R0 < Rc < 1 < R00(p00 < p < pc < p0) E0 is L.A.S. 3 0 < Rc < R0 < 1 < R00(p00 < p < p0 < pc) E0 is L.A.S.; E*2 is unstable; E*1 is either L.A.S. or unstable(when Re(ηi(E*1)) < 0, i = 1; 2; 3; 4, E*1 is L.A.S.). 4 R0 > 1(p > p0) E0 is unstable; E*1 is either L.A.S. or unstable(when Re(ηi(E*1)) < 0, i = 1; 2; 3; 4, E*1 is L.A.S.).
Parameters notations and values used.
 Symbol Description Value References $\lambda$ Generation rate of quiescent uninfected CD4$^+$ T cells $4.8\times10^3$ $cells\cdot ml^{-1}\cdot day^{-1}$ see text $\alpha_{Q}$ Stimulation rate by other antigens varied see text $k_{1}$ Stimulation rate by HIV varied see text $k_{s}$ Half-maximal stimulation threshold of HIV varied see text $\xi$ Revert rate of active uninfected CD4$^+$ T cells $0.15$ $day^{-1}$ [11] $d_{Q}$ Death rate of quiescent uninfected CD4$^+$ T cells $0.001$ $day^{-1}$ [19,43] $d_{T}$ Death rate of active uninfected CD4$^+$ T cells $0.01$ $day^{-1}$ [47] $d_{I}$ Death rate of infected CD4$^+$ T cells $1$ $day^{-1}$ [47] p Virus production rate of CD4$^+$ T cells $2000$ $virions\cdot cells^{-1}\cdot day^{-1}$ [47] c Clearance rate of free viruses $23$ $day^{-1}$ [47] $\beta$ Infection rate of active uninfected CD4$^+$ T cells by HIV $7.4\times10^{-8}$ $ml\cdot virion^{-1}\cdot day^{-1}$ [22,47]
 Symbol Description Value References $\lambda$ Generation rate of quiescent uninfected CD4$^+$ T cells $4.8\times10^3$ $cells\cdot ml^{-1}\cdot day^{-1}$ see text $\alpha_{Q}$ Stimulation rate by other antigens varied see text $k_{1}$ Stimulation rate by HIV varied see text $k_{s}$ Half-maximal stimulation threshold of HIV varied see text $\xi$ Revert rate of active uninfected CD4$^+$ T cells $0.15$ $day^{-1}$ [11] $d_{Q}$ Death rate of quiescent uninfected CD4$^+$ T cells $0.001$ $day^{-1}$ [19,43] $d_{T}$ Death rate of active uninfected CD4$^+$ T cells $0.01$ $day^{-1}$ [47] $d_{I}$ Death rate of infected CD4$^+$ T cells $1$ $day^{-1}$ [47] p Virus production rate of CD4$^+$ T cells $2000$ $virions\cdot cells^{-1}\cdot day^{-1}$ [47] c Clearance rate of free viruses $23$ $day^{-1}$ [47] $\beta$ Infection rate of active uninfected CD4$^+$ T cells by HIV $7.4\times10^{-8}$ $ml\cdot virion^{-1}\cdot day^{-1}$ [22,47]
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