Article Contents
Article Contents

Spatiotemporal dynamics for a diffusive HIV-1 infection model with distributed delays and CTL immune response

• * Corresponding author: Zhijun Liu

The work is supported by the National Natural Science Foundation of China (No.11871201), and Natural Science Foundation of Hubei Province, China (No.2019CFB241)

• In this study, we develop a diffusive HIV-1 infection model with intracellular invasion, production and latent infection distributed delays, nonlinear incidence rate and nonlinear CTL immune response. The well-posedness, local and global stability for the model proposed are carefully investigated in spite of its strong nonlinearity and high dimension. It is revealed that its threshold dynamics are fully determined by the viral infection reproduction number $\mathfrak{R}_0$ and the reproduction number of CTL immune response $\mathfrak{R}_1$. We also observe that the viral load at steady state (SS) fails to decrease even if $\mathfrak{R}_1$ increases through unit to lead to a stability switch from immune-inactivated infected SS to immune-activated infected SS. Finally, some simulations are performed to verify the analytical conclusions and we explore the significant impact of delays and CTL immune response on the spatiotemporal dynamics of HIV-1 infection.

Mathematics Subject Classification: Primary: 92D30; Secondary: 34D23, 34K20.

 Citation:

• Figure 1.  (1)-(10): Spatial distribution of each solution variable of model (28) in two time points $t = 5, 12$ in the case that $\mathfrak{R}_0 = 0.386<1$ and the parameter values are from Data1 in Table 2; (a)-(e): The time evolutions of the corresponding solution variables in three different positions $(x, y) = (1, 1.5)$, $(2.5, 2.5)$ and $(1.5, 2)$ illustrates that $\mathcal{E}^0 = (T^0, 0, 0, 0, 0) = (80, 0, 0, 0, 0)$ is GAS

Figure 2.  (1)-(15): Spatial distribution of each solution variable of model (28) in three time points $t = 14, 20, 25$ when $\mathfrak{R}_0 = 3.858>1$, $\mathfrak{R}_1 = 0.831<1$ and the parameter values are from Data2 in Table 2

Figure 3.  In the case of $\mathfrak{R}_0 = 3.858>1$ and $\mathfrak{R}_1 = 0.831<1$, the graph trajectory of the solution to model (28) versus $t$ is illustrated by (a)-(c) with three different positions $(x, y) = (1, 1.5)$, $(2.5, 2.5)$ and $(1.5, 2)$ in $\Omega$, respectively, such that $\mathcal{E}^* = (T^*, E^*, I^*, V^*, 0) = (26.122, 0.004, 0.523, 51.972, 0)$ is GAS, where the parameter values are from Data2 in Table 2

Figure 4.  (1)-(15): Spatial distribution of each solution variable of model (28) in three time points $t = 14, 20, 25$ when $\mathfrak{R}_0 = 3.858>1$, $\mathfrak{R}_1 = 1.843>1$ and the parameter values are from Data3 in Table 2

Figure 5.  In the case of $\mathfrak{R}_0 = 3.858>1$ and $\mathfrak{R}_1 = 1.843>1$, the graph trajectory of the solution to model (28) versus $t$ is illustrated by (a)-(c) with three different positions $(x, y) = (1, 1.5)$, $(2.5, 2.5)$ and $(1.5, 2)$ in $\Omega$, respectively, such that $\mathcal{E}^† = (T^†, E^†, I^†, V^†, Z^†) = (41.888, 0.003, 0.201, 20.019, 3.485)$ is GAS, where the parameter values are from Data3 in Table 2

Table 1.  List of parameters

 Parameter Description $\lambda$ Generation rate of uninfected target cells $\mu$ Death rate of uninfected target cells $\beta$ Infection rate of cells by free virus $\theta$ Activation rate of uninfected target cells $a$ The sum of activation and death rates of latently infected cells $b$ Death rate of productively infected cells $c$ Clearance rate of free virus $d$ Death rate of effector cell of CTLs $p$ CTL effectiveness $q_3$ Released rate for free viral particles $\gamma$ Proliferation rate of CTLs by productively infected cells $q_1, q_2$ Fraction of infection leading to latency and production, respectively $D_1, D_2, D_3$ The diffusion coefficients $\alpha, \sigma$ Inhibitory rate from target cells and effector cell of CTLs, respectively

Table 2.  Parameters and their values

 Para. Units Data1 Data2 Data3 Range Source $\lambda$ cells$\cdot$ml$^{-1}$day$^{-1}$ 0.8 0.8 0.8 $[0, 10]$ [24] $\mu$ day$^{-1}$ 0.01 0.01 0.01 $[10^{-4}, 0.2]$ [16] $\beta$ ml$\cdot$virion$^{-1}$day$^{-1}$ $5\times10^{-5}$ $5\times10^{-4}$ $5\times10^{-4}$ $[4.6\times10^{-8}, 0.5]$ [26] $\theta$ day$^{-1}$ 0.01 0.01 0.01 $[0.01, 0.3]$ [31,43,32] $a$ day$^{-1}$ 0.014 0.014 0.014 $[0.001, 0.2]$ [31,43,30] $b$ day$^{-1}$ 1 1 1 $[1.9\times10^{-4}, 1.4]$ [31,16] $c$ day$^{-1}$ 2 2 2 $[0.081, 36]$ [4] $d$ day$^{-1}$ 0.2 0.2 0.2 $[0.004, 8.087]$ [4] $p$ ml$\cdot$cell$^{-1}$day$^{-1}$ 0.24 0.24 0.24 $[10^{-4}, 4.048]$ [16,26] $q_3$ virion$\cdot$cell$^{-1}$day$^{-1}$ 200 200 200 $[0.38, 2800]$ [16] $\gamma$ ml$\cdot$cell$^{-1}$day$^{-1}$ 1 0.3 1 $[0.0051, 3.912]$ [16] $q_1$ – $10^{-4}$ $10^{-4}$ $10^{-4}$ $[0, 1]$ [43,32] $q_2$ – $1-10^{-4}$ $1-10^{-4}$ $1-10^{-4}$ $[0, 1]$ [43,32] $D_1$ mm$^{2}$day$^{-1}$ $10^{-4}$ $10^{-4}$ $10^{-4}$ $[0, 1]$ [29] $D_2$ mm$^{2}$day$^{-1}$ $5\times10^{-4}$ $5\times10^{-4}$ $5\times10^{-4}$ $[0, 1]$ [29] $D_3$ mm$^{2}$day$^{-1}$ $2\times10^{-4}$ $2\times10^{-4}$ $2\times10^{-4}$ $[0, 1]$ [29] $\alpha$ – 0.005 0.005 0.005 $[0, 1]$ [37,36] $\sigma$ – 0.002 0.002 0.002 $[0, 1]$ [46] $m_1$ day$^{-1}$ 0.05 0.05 0.05 $[0, 1]$ [43] $\tau_1$ day 0.3 0.3 0.3 $[0, 0.5]$ [43] $m_2$ day$^{-1}$ 0.05 0.05 0.05 $[0, 1]$ [43] $\tau_2$ day 0.6 0.6 0.6 $[0.5, 1]$ [43] $m_3$ day$^{-1}$ 0.01 0.01 0.01 $[0, 1]$ [54] $\tau_3$ day 0.6 0.6 0.6 $[0, 10]$ [49] $\mathfrak{R}_0$ 0.386 3.858 3.858 $\mathfrak{R}_1$ 0.322 0.831 1.843
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