数学物理学报, 2024, 44(3): 771-782

一类基于游离病毒感染和细胞-细胞传播的宿主体内 HIV-1 感染动力学模型

徐瑞,1,2,*, 周凯娟1,2,3, 白宁1,2,3

1.山西大学复杂系统研究所 太原 030006

2.山西省疾病防控的数学技术与大数据分析重点实验室 太原 030006

3.山西大学数学科学学院 太原 030006

A with-in Host HIV-1 Infection Dynamics Model Based on Virus-to-cell Infection and Cell-to-cell Transmission

Xu Rui,1,2,*, Zhou Kaijuan1,2,3, Bai Ning1,2,3

1. Complex Systems Research Center, Shanxi University, Taiyuan 030006

2. Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and Prevention, Shanxi University, Taiyuan 030006

3. School of Mathematics Science, Shanxi University, Taiyuan 030006

通讯作者: *徐瑞，E-mail:rxu88@163.com;xurui@sxu.edu.cn

收稿日期: 2022-10-26   修回日期: 2023-11-1  

基金资助: 国家自然科学基金(12271317)
国家自然科学基金(11871316)

Received: 2022-10-26   Revised: 2023-11-1  

Fund supported: NSFC(12271317)
NSFC(11871316)

摘要

该文考虑一类具有细胞-细胞传播、胞内时滞、饱和 CTL 免疫反应和免疫损害的 HIV-1 感染动力学模型. 通过计算得到了免疫未激活和免疫激活再生率. 通过分析特征方程根的分布, 讨论了可行平衡点的局部渐近稳定性. 通过构造适当的 Lyapunov 泛函并应用 LaSalle 不变性原理, 证明了模型的全局动力学由免疫未激活和免疫激活再生率决定: 如果免疫未激活再生率小于 1, 则病毒未感染平衡点是全局渐近稳定的; 如果免疫未激活再生率大于 1 且免疫激活再生率小于 1, 则免疫未激活感染平衡点是全局渐近稳定的; 如果免疫激活再生率大于 1, 则慢性感染平衡点是全局渐近稳定的. 此外, 通过数值模拟说明了免疫损害和细胞-细胞传播对模型动力学的影响.

关键词： HIV-1 感染; 细胞-细胞传播; 胞内时滞; 饱和 CTL 免疫反应; 免疫损害; 稳定性

Abstract

In this paper, we consider an HIV-1 infection model with cell-to-cell transmission, intracellular delay, saturated CTL immune response and immune impairment. By calculation, we get immunity-inactivated and immunity-activated reproduction ratios. By analyzing the characteristic equations, the local stability of each of feasible equilibria is established. By means of suitable Lyapunov functional and LaSalle's invariance principle, it is proved that the global asymptotic stability of each of feasible equilibria is determined by immunity-inactivated and immunity-activated reproduction ratios: If the immunity-inactivated reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable; if the immunity-inactivated reproduction ratio is greater than unity and the immunity-activated reproduction ratio is less than unity, the immunity-inactivated infection equilibrium is globally asymptotically stable; if the immunity-activated reproduction ratio is greater than unity, the chronic infection equilibrium is globally asymptotically stable. In addition, numerical simulation is carried out to illustrate the effects of immune impairment and cell-to-cell transmission on dynamics of the model.

Keywords： HIV-1 infection; Cell-to-cell transmission; Intracellular delay; Saturated CTL immune response; Immune impairment; Stability

PDF (758KB) 元数据 多维度评价 相关文章 导出 EndNote| Ris| Bibtex  收藏本文

本文引用格式

徐瑞, 周凯娟, 白宁. 一类基于游离病毒感染和细胞-细胞传播的宿主体内 HIV-1 感染动力学模型[J]. 数学物理学报, 2024, 44(3): 771-782

Xu Rui, Zhou Kaijuan, Bai Ning. A with-in Host HIV-1 Infection Dynamics Model Based on Virus-to-cell Infection and Cell-to-cell Transmission[J]. Acta Mathematica Scientia, 2024, 44(3): 771-782

1 引言

HIV-1 是一种逆转录病毒, 主要攻击人体的 $ {\rm CD4^+\ T} $ 淋巴细胞, 可引起人类细胞免疫功能缺陷, 导致一系列病毒感染的发生, 具有传染快, 病死率高的特性. 近年来, 对宿主体内 HIV-1 感染的研究已经有了很大的进展, 很多学者通过建立和分析数学模型来研究 HIV-1 的感染与治疗[1⇓⇓⇓⇓⇓-7]. 研究表明, 细胞毒性 T 细胞 (CTLs) 会分泌各种细胞因子杀伤被感染靶细胞. 近年来, 对具有 CTL 免疫反应的病毒感染动力学模型的研究已经引起了学者的广泛关注[8⇓⇓-11]. 在文献[11]中, Ren 等人考虑了以下具有 CTL 免疫反应的 HIV-1 感染动力学模型

$\begin{equation} \begin{aligned} \dot x(t)&=\lambda-dx(t)-\frac{\beta x(t)v(t)}{1+\alpha v(t)},\\ \dot{y}(t)&=\frac{\beta {\rm e}^{-m\tau}x(t-\tau)v(t-\tau)}{1+\alpha v(t-\tau)}-ay(t)-py(t)z(t),\\ \dot{v}(t)&=ky(t)-uv(t),\\ \dot{z}(t)&=\frac{cy(t)z(t)}{h+z(t)}-bz(t), \end{aligned}\end{equation}$

其中 $ x(t) $, $ y(t) $, $ v(t) $, $ z(t) $ 分别表示 $ t $ 时刻未感染靶细胞、产生病毒的被感染靶细胞、游离病毒和CTL 免疫细胞的浓度. 未感染靶细胞以速率 $ \lambda $ 产生, 以速率 $ dx $ 死亡, 以速率 $ \beta xv/(1+\alpha v) $ 被游离病毒感染; 被感染靶细胞以速率 $ \beta {\rm e}^{-m\tau} x(t-\tau)v(t-\tau)/(1+\alpha v(t-\tau)) $ 产生, 以速率 $ ay $ 死亡, 以速率 $ pyz $ 被CTL 免疫细胞杀伤, 其中 $ \tau $ 表示未感染靶细胞被游离病毒感染后变为激活细胞所需的时间, $ {\rm e}^{-m\tau} $ 表示 $ t-\tau $ 时刻到 $ t $ 时刻, 被感染靶细胞存活的概率; 游离病毒以速率 $ ky $ 产生, 以速率 $ uv $ 被清除; CTL免疫细胞对抗原增殖反应速率为 $ cyz/(h+z) $, 以速率 $ bz $ 衰减.

系统(1.1)只考虑了病毒对未感染靶细胞的感染. 事实上, 研究发现 HIV-1 可以通过病毒学突触的大分子粘附接触从被感染靶细胞转移到未感染靶细胞[12,13], 即被感染靶细胞可以通过直接接触感染未感染靶细胞. 在这种细胞间转移过程中, 许多病毒颗粒可以同时从被感染的靶细胞转移到未感染的靶细胞. 文献[14]指出细胞间HIV-1转移导致的感染比游离病毒颗粒导致的感染更有效. 因此, 细胞间的直接接触感染已经引起了许多学者关注[15⇓-17]. 文献[15]指出在仅考虑病毒传播的模型中, 基本再生数被低估了.

系统(1.1)只考虑了 CTL 免疫细胞可由被感染的细胞或淋巴因子诱导激活, 分化扩增后连续杀伤被感染细胞, 从而有效地发挥免疫反应. 然而, 实验表明抗原的出现不仅可以激发宿主的免疫调节, 还可能抑制免疫反应, 甚至损害免疫力, 即免疫损害. Iwami 等[18]报道了在 HIV-1 感染过程中可能引起 CTL 细胞的损伤. 目前有很多学者考虑免疫损害在病毒传播中的作用[19,20].

受文献[11,17,19]启发, 本文研究细胞-细胞传播、饱和 CTL 免疫反应和免疫损害对于 HIV-1 感染的影响. 为此, 我们考虑以下时滞微分方程模型

$\begin{equation} \begin{aligned} \dot x(t)&=\lambda-dx(t)-\frac{\beta x(t)v(t)}{1+\alpha v(t)}-\beta_1 x(t)y(t),\\ \dot{y}(t)&=\frac{\beta {\rm e}^{-m\tau}x(t-\tau)v(t-\tau)}{1+\alpha v(t-\tau)}+\beta_1 {\rm e}^{-m\tau} x(t-\tau)y(t-\tau)-ay(t)-py(t)z(t),\\ \dot{v}(t)&=ky(t)-uv(t),\\ \dot{z}(t)&=\frac{cy(t)z(t)}{1+\omega z(t)}-bz(t)-ny(t)z(t), \end{aligned} \end{equation}$

其中 $ nyz $ 表示免疫损害的速率, $ \beta_1 xy $ 表示被感染靶细胞对未感染靶细胞的感染率, 参数 $ \tau $ 表示未感染靶细胞被游离病毒或被感染靶细胞感染到变为激活细胞所需的时间. 系统(1.2)中的其余参数与系统(1.1)生物学意义相同. 在此,

假设系统(1.2)中所有参数都是正常数, 且 $ c>n $.

系统(1.2)满足初始条件

$\begin{equation*} \begin{array}{lll} x(\theta)=\varphi_1(\theta),\;\;y(\theta)=\varphi_2(\theta),\;\; v(\theta)=\varphi_3(\theta),\;\;z(\theta)=\varphi_4(\theta),\;\; \theta\in[-\tau,0),\\ \varphi_i(0)>0\ (i=1,2,3,4), \end{array} \end{equation*}$

其中 $ (\varphi_1(\theta),\varphi_2(\theta),\varphi_3(\theta),\varphi_4(\theta))\in C([-\tau, 0], R_+^4), $ 这里 $ C([-\tau, 0],R_+^4) $ 为区间 $ [-\tau, 0] $ 映射到 $ R_+^4 $ 的全体连续函数构成的 Banach 空间, $ R_+^4=\lbrace(x,y,v,z):x\geq 0,y\geq 0,v\geq 0,z\geq 0\rbrace $. 由泛函微分方程的基本理论[21]可知, 系统(1.2)满足初始条件(1.3)的解是唯一的.

2 解的正性和最终有界性

定理 2.1 对所有 $ t\geq0 $, 系统(1.2)满足初始条件(1.3)的任一解均是正的.

证 设 $ (x(t),y(t),v(t),z(t)) $ 是系统(1.2)满足初始条件(1.3)的任意一个解. 接下来, 我们利用反证法证明$ x(t) $ 的正性. 假设存在 $ t_1>0 $, 使得 $ x(t_1)=0 $, 且 $ x(t)>0 $ 对于任意 $ t\in [0,t_1) $ 成立.另一方面, 由系统(1.2)的第一个方程可得 $ \dot x(t_1)=\lambda>0 $, 这意味着存在充分小的 $ \epsilon $, 使得当 $ t\in(t_1-\epsilon,t_1) $ 时, $ x(t)<0 $, 这与假设矛盾. 因此, 对于任意 $ t\geq0 $ 都有 $ x(t)>0 $.

对于系统(1.2)的第二个方程, $ t\in [\tau] $ 意味着 $ t-\tau\in[-\tau,0] $, 根据初始条件有

$\begin{equation*} \dot{y}(t)\geq -ay-pyz, \end{equation*} $

求解可得 $ y(t)\geq y(0)\exp\lbrace-\int_{0}^{t}(a+pz(\theta)){\rm d\theta}\rbrace>0 $. 且当 $ t\in [\tau] $ 时, 由系统(1.2)的第三个方程可得 $ \dot{v}(t)\geq-uv(t), $ 直接求得 $ v(t)\geq v(0){\rm e}^{-ut}, $ 由 $ v(0)>0 $ 可知, $ v(t)>0 $. 类似地, $ t\in[\tau,2\tau] $ 时, 有 $ y(t)>0 $, $ v(t)>0 $. 由数学归纳法可以证明, 对于任意 $ t\ge0 $, 都有 $ y(t)>0 $, $ v(t)>0. $

对于系统(1.2)的第四个方程, 直接求解可得

$\begin{equation*} z(t)=z(0)\exp\left\lbrace\displaystyle\int_{0}^{t}\left(\frac{cy(\eta)}{1+\omega z(\eta)}-b-ny(\eta)\right){\rm d\eta}\right\rbrace. \end{equation*}$

假设存在 $ t_2>0 $, 使得 $ z(t_2)=0 $, 且 $ z(t)>0 $ 对于任意 $ t\in [0,t_2) $ 成立. 另一方面由 $ z(t) $ 表达式可得

$\begin{equation*} z(t_2)=z(0)\exp\left\lbrace\displaystyle\int_{0}^{t_2}\left(\frac{cy(\eta)}{1+\omega z(\eta)}-b-ny(\eta)\right){\rm d\eta}\right\rbrace. \end{equation*}$

由 $ z(0)>0 $ 可知, $ z(t_2)>0 $. 这与假设矛盾. 因此, 对于任意 $ t\ge0 $, 都有 $ z(t)>0 $.

定理 2.2 对于任意 $ t\geq0 $, 系统(1.2)满足初始条件(1.3)的任一解最终有界.

证 设 $ (x(t),y(t),v(t),z(t)) $ 是系统(1.2)满足初始条件(1.3)的任意一个正解. 定义

$G(t)=x(t-\tau)+{\rm e}^{m\tau}y(t)+\frac{{\rm e}^{m\tau}a}{1+k}v(t)+\frac{{\rm e}^{m\tau}p}{c}z(t).$

沿着系统(1.2)的解计算 $ G(t) $ 全导数可得

$\begin{equation*} \begin{split} \dot G(t)=\ &\lambda-dx(t-\tau)-\frac{{\rm e}^{m\tau}a}{1+k}y(t)-\frac{{\rm e}^{m\tau}pb}{c}z(t)-\frac{{\rm e}^{m\tau}au}{1+k}v(t)\\ &-{\rm e}^{m\tau}py(t)z(t)+\frac{{\rm e}^{m\tau}py(t)z(t)}{1+\omega z(t)}-\frac{{\rm e}^{m\tau}pn}{c}y(t)z(t)\\ <\ &\lambda-\sigma G(t). \end{split} \end{equation*}$

通过计算可得 $ \limsup\limits_ {t\rightarrow +\infty} G(t)\le \lambda/\sigma $, 其中 $ \sigma=\min\left\lbrace d,u,b,\frac{a}{1+k}\right\rbrace $.

因此, 集合

$\mathrm{X}=\left\lbrace (x(t),y(t),v(t),z(t)):x(t-\tau)+{\rm e}^{m\tau}y(t)+\frac{{\rm e}^{m\tau}a}{1+k}v(t)+\frac{{\rm e}^{m\tau}p}{c}z(t)\le\frac{\lambda}{\sigma}\right\rbrace$

是系统(1.2)的正不变集.

3 可行平衡点

显然, 系统(1.2)始终存在一个病毒未感染平衡点 $ E_{0}\left(\lambda/d,0,0,0\right) $. 下面, 根据文献[22]中的方法定义系统(1.2)的免疫未激活再生率.

易知, 系统(1.2)中感染仓室 $ U=(y,v) $ 对应的子系统在 $ E_{0} $ 处的线性化系统为

$\begin{equation*} \frac{{\rm d}U(t)}{{\rm d}t}=FU(t-\tau)-VU(t), \end{equation*}$

其中

$\begin{equation*} F=\left ( \begin{matrix} \frac{\beta_1\lambda {\rm e}^{-m\tau}}{d} & \frac{\beta\lambda {\rm e}^{-m\tau}}{d} \\ 0 & 0 \end{matrix} \right ), \qquad V=\left ( \begin{matrix} a & 0 \\ -k & u \end{matrix} \right ). \end{equation*}$

引入感染仓室的初始分布 $ \varphi(0)=(\varphi_2(0),\varphi_3(0))^T, $ 则感染仓室浓度变化为

$\frac{{\rm d}\varphi}{{\rm d}t}=V\varphi(t), \quad\varphi(t)={\rm e}^{-Vt}\varphi(0).$

当 $ 0<t<\tau $ 时, 没有新感染细胞产生. 当 $ t\geq \tau $ 时, $ t $ 时刻新感染的细胞为

$\begin{equation*} F\varphi(t-\tau)=F{\rm e}^{-V(t-\tau)}\varphi(0). \end{equation*}$

因此新感染的细胞为

$\begin{equation*} \displaystyle\int_{\tau}^{+\infty} F\varphi(t-\tau){\rm d}t=\displaystyle\int_{\tau}^{+\infty} F{\rm e}^{-V(t-\tau)}\varphi(0){\rm d}t=\displaystyle\int_{0}^{+\infty} F{\rm e}^{-V(t)}\varphi(0){\rm d}t=FV^{-1}\varphi(0). \end{equation*}$

故系统(1.2)的免疫未激活再生率为

$\mathfrak{R}_0 =\rho(FV^{-1})=\frac{\lambda(k\beta +u\beta_1) }{adu}{\rm e}^{-m\tau}.$

容易证明, 如果 $ \mathfrak{R}_0>1 $, 则系统(1.2)有唯一免疫未激活感染平衡点 $ E_1(x_1,y_1,v_1,0) $, 其中

$\begin{equation} x_1=\frac{\lambda(u+\alpha ky_1)}{(d+\beta_1y_1)(u+\alpha ky_1)+\beta ky_1},\quad v_1=\frac{ky_1}{u},\quad y_1=\frac{-B-\sqrt{B^2-4AC}}{2A}, \end{equation}$

这里

$\begin{equation*}A=-\frac{\beta_1ak\alpha}{{\rm d}u}{\rm e}^{m\tau},\quad B=\frac{\beta_1\lambda k\alpha}{{\rm d}u}-\frac{\beta ak}{{\rm d}u}{\rm e}^{m\tau}-\frac{\beta_1a}{d}{\rm e}^{m\tau}-\frac{ak\alpha}{u}{\rm e}^{m\tau}, \quad C=a{\rm e}^{m\tau}(\mathfrak{R}_0-1).\end{equation*}$

为定义系统(1.2)的免疫激活再生率, 我们考虑其免疫仓室 $ z $, 并在 $ E_1 $ 处线性化系统, 则有

$\begin{equation*} \dot{z}(t)=(c-n)y_1z(t)-bz(t). \end{equation*}$

因此, 系统(1.2)的免疫激活再生率定义为

$\mathfrak{R}_1 =\frac{c-n}{b}y_1.$

当 $ \mathfrak{R}_0>1 $ 且 $ \mathfrak{R}_1>1 $ 时, 如果系统(1.2)存在一个慢性感染平衡点 $ E^{*}(x^*,y^*,v^*,z^*) $, 则其必满足

$\begin{equation} \begin{aligned} &\lambda-dx-\frac{\beta xv}{1+\alpha v}-\beta_1 xy=0,\\ & \frac{\beta {\rm e}^{-m\tau}xv}{1+\alpha v}+\beta_1 {\rm e}^{-m\tau} xy-ay-pyz=0,\\ & ky-uv=0,\\ & \frac{cyz}{1+\omega z}-bz-nyz=0. \end{aligned} \end{equation}$

由(3.2)的前三个方程可得

$\begin{equation*}z=\frac{\lambda}{{\rm e}^{m\tau}p}\frac{\beta_1\alpha ky+\beta_1u+\beta k }{(d+\beta_1y)(u+\alpha ky)+\beta ky}-\frac{a}{p}:=f(y).\end{equation*}$

另一方面, 由(3.2)的第四个方程直接计算可得

$\begin{equation*}z=\frac{1}{\omega}\left(\frac{cy}{b+ny}-1\right):=g(y).\end{equation*}$

根据上述讨论, 为证系统(1.2)存在唯一慢性感染平衡点 $ E^{*} $, 只需证函数 $ z=f(y) $ 与 $ z=g(y) $ 的图像在第一象限存在唯一的交点.

通过计算可得, 当 $ y\in(0,\infty) $ 时, $ f(y) $ 是变量 $ y $ 的减函数, 且 $ f(0)=a(\mathfrak{R}_0-1)/p>0 $, $ f(y_1)=0 $ (这里, $ y_1 $ 由(3.1)定义); $ g(y) $ 是变量 $ y $ 的增函数, 且 $ g(0)=-1/\omega $, $ g(b/(c-n))=0 $. 根据 $ \mathfrak{R}_1>1 $ 可得 $ y_1>b/(c-n) $. 图1 描述了函数 $ z=f(y) $ 与 $ z=g(y) $ 的图像. 由图1 所示, 函数 $ z=f(y) $ 和 $ z=g(y) $ 的图像在第一象限存在唯一的交点 $ (y^*,z^*) $, 从而系统(1.2)存在唯一慢性感染平衡点 $ E^{*}(x^*,y^*,v^*,z^*) $, 其中

$ x^*=\frac{\lambda(u+\alpha ky^*)}{(d+\beta_1y^*)(u+\alpha ky^*)+\beta ky^*},\quad v^*=\frac{ky^*}{u}.$

图1

新窗口打开| 下载原图ZIP| 生成PPT

图1   函数 $ z=f(y) $ 与 $ z=g(y) $ 曲线图


4可行平衡点的稳定性

本节, 通过分析相应特征方程根的分布和构造适当的 Lyapunov 泛函, 我们讨论各可行平衡点的局部和全局渐近稳定性.

定理 4.1 若 $ \mathfrak{R}_0 <1 $, 则系统(1.2)的病毒未感染平衡点 $ E_{0}\left(\lambda/d,0,0,0\right) $ 局部渐近稳定, 若 $ \mathfrak{R}_0 >1 $, 则 $ E_{0} $ 不稳定.

证 系统(1.2)在病毒未感染平衡点 $ E_0 $ 处的特征方程为

$\begin{equation}(s+d)(s+b)\left[s^2+\left(a+u-\frac{\lambda \beta_1 {\rm e}^{-(m+s)\tau}}{d}\right)s+au-\frac{(k\beta +u\beta_1)\lambda {\rm e}^{-(m+s)\tau}}{d}\right]=0.\end{equation}$

显然, 方程(4.1)有两个负实根 $ s=-d $ 和 $ s=-b $, 其余根由下面的方程决定

$\begin{equation}f(s):=s^2+\left(a+u-\frac{\lambda \beta_1 {\rm e}^{-(m+s)\tau}}{d}\right)s+au-\frac{(k\beta +u\beta_1)\lambda {\rm e}^{-(m+s)\tau}}{d}=0.\end{equation} $

若 $ \mathfrak{R}_0 >1 $, 对实数 $ s $, 容易验证

$f(0)=au(1- \mathfrak{R}_0 )<0,\quad \lim\limits_{s\rightarrow +\infty}f(s)=+\infty.$

因此, 由零点定理知, 方程 $ f(s)=0 $ 至少有一个正实根, 从而 $ E_0 $ 不稳定.

若 $ \mathfrak{R}_0 <1 $, 我们证明病毒未感染平衡点 $ E_0 $ 是局部渐近稳定的.

当 $ \tau=0 $ 时, 方程(4.2)变为

$\begin{equation*}s^2+\left(a+u-\frac{\lambda \beta_1 }{d}\right)s+au-\frac{\lambda(k\beta +u\beta_1) }{d}=0.\end{equation*}$

若 $ \mathfrak{R}_0<1 $, 则有 $ a>\lambda\beta_1/d $ 和 $ au>\lambda(\beta_1u+\beta k) /d $. 从而, 当 $ \tau=0 $ 时, 病毒未感染平衡点 $ E_0 $ 是局部渐近稳定的.

当 $ \tau>0 $ 时, 设 i $ \omega\;(\omega>0) $ 是方程(4.2)的根, 将其代入(4.2)并分离实部和虚部可得

$\begin{equation} \begin{aligned} & (a+u)\omega=\frac{\lambda \beta_1\omega {\rm e}^{-m\tau}\cos{\omega \tau}}{d}-au\mathfrak{R}_0\sin{\omega \tau},\\ & \omega^2-au=-\frac{\lambda \beta_1\omega {\rm e}^{-m\tau}\sin{\omega \tau}}{d}-au\mathfrak{R}_0\cos{\omega \tau}. \end{aligned} \end{equation}$

将方程组(4.3)两个方程的两端分别平方并相加, 可得

$\begin{equation}\omega^4+\left[a^2+u^2-\left(\frac{\lambda \beta_1 {\rm e}^{-m\tau}}{d}\right)^2\right]\omega^2+(au)^2(1-\mathfrak{R}_0^{2})=0.\end{equation} $

显然, 若 $ \mathfrak{R}_0 <1 $, 则方程(4.4)无正实根. 注意到当 $ \tau=0 $ 时, $ E_0 $ 局部渐近稳定, 由时滞微分方程特征方程的一般理论可知, 若 $ \mathfrak{R}_0<1 $, 则病毒未感染平衡点 $ E_0 $ 局部渐近稳定.

定理 4.2 若 $ \mathfrak{R}_1<1<\mathfrak{R}_0 $, 则系统(1.2)的免疫未激活感染平衡点 $ E_1(x_1,y_1,v_1,0) $ 局部渐近稳定.

证 系统(1.2)在免疫未激活感染平衡点 $ E_1 $ 的特征方程为

$\begin{equation}(s+b+ny_1-cy_1)\left[(s+a)(s+u)-{\rm e}^{-(m+s)\tau}\left(\beta_1x_1(s+u)+\frac{k\beta x_1}{(1+\alpha v_1)^2}\right)\right]=0.\end{equation}$

显然, 方程(4.5)存在一个负实根 $ s=(c-n)y_1-b<0 $, 其余根由方程

$\begin{equation}(s+a)(s+u)={\rm e}^{-(m+s)\tau}\left(\beta_1x_1(s+u)+\frac{k\beta x_1}{(1+\alpha v_1)^2}\right)\end{equation}$

确定. 当 $ \mathfrak{R}_1<1<\mathfrak{R}_0 $ 时, 我们断言, 方程(4.6)的所有根均具有负实部. 若否, 方程(4.6)至少存在一个根 $ s_1= $ Re $ s_1 $ +iIm $ s_1 $, 其中 Re $ s_1\geq0 $. 在这种情况下, 通过计算得到

$\begin{equation*} |(s_1+a)(s_1+u)|>|a(s_1+u)|>{\rm e}^{-m\tau}\left|\beta_1x_1s_1+\beta_1x_1u+\frac{k\beta x_1}{1+\alpha v_1}\right| \end{equation*}$

和

$\begin{equation*} \left|{\rm e}^{-(m+s_1)\tau}\left(\beta_1x_1(s_1+u)+\frac{k\beta x_1}{(1+\alpha v_1)^2}\right)\right|< {\rm e}^{-m\tau}\left|\beta_1x_1s_1+\beta_1x_1u+\frac{k\beta x_1}{1+\alpha v_1}\right|, \end{equation*} $

这与方程(4.6)矛盾. 因此, 若 $ \mathfrak{R}_1<1<\mathfrak{R}_0 $, 则方程(4.6)的根均有负实部, 免疫未激活感染平衡点 $ E_1 $ 是局部渐近稳定的.

定理 4.3 若 $ \mathfrak{R}_0>1 $ 且 $ \mathfrak{R}_1 >1 $, 则系统(1.2)的慢性感染平衡点 $ E^{*}(x^*,y^*,v^*,z^*) $ 局部渐近稳定.

证 系统(1.2)在慢性感染平衡点 $ E^{*} $ 处的特征方程为

$\begin{matrix} &&(s+d+\frac{\beta v^*}{1+\alpha v^*}+\beta_1y^*)(s+u)\left[(s+a+pz^*)\left(s+b+ny^*-\frac{cy^*}{(1+\omega z^*)^2}\right)+pbz^*\right] \\&=&{\rm e}^{-(m+s)\tau}(s+d)\left(s+b+ny^*-\frac{cy^*}{(1+\omega z^*)^2}\right)\left[\beta_1x^*(s+u)+\frac{k\beta x^*}{(1+\alpha v^*)^2}\right].\end{matrix}$

当 $ \mathfrak{R}_0 >1 $ 且 $ \mathfrak{R}_1 >1 $ 时, 我们断言, 方程(4.7)的所有根均具有负实部. 若否, 则方程(4.7)至少存在一个根 $ s_2= $ Re $ s_2+ $ iIm $ s_2 $, 其中Re $ s_2\geq0 $. 在这种情况下, 显然有

$\left|s_2+d+\frac{\beta v^*}{1+\alpha v^*}+\beta_1y^*\right|>|{\rm e}^{-s_2\tau}(s_2+d)|.$

通过计算可得

$\begin{eqnarray*} &&|s_2+u|\left|(s_2+a+pz^*)\left(s_2+b+ny^*-\frac{cy^*}{(1+\omega z^*)^2}\right)+pbz^*\right|\\ & > & \left|\left(s_2+b+ny^*-\frac{cy^*}{(1+\omega z^*)^2}\right)(a+pz^*)(s_2+u)\right| \end{eqnarray*}$

和

$\begin{eqnarray*} &&\left|{\rm e}^{-m\tau}\left(s_2+b+ny^*-\frac{cy^*}{(1+\omega z^*)^2}\right)\left[\beta_1x^*(s_2+u)+\frac{k\beta x^*}{(1+\alpha v^*)^2}\right]\right|\\ &<&\left|\left(s_2+b+ny^*-\frac{cy^*}{(1+\omega z^*)^2}\right) \left[\beta_1x^*{\rm e}^{-m\tau}s_2+(a+pz^*)u\right]\right|\\ & < & \left|\left(s_2+b+ny^*-\frac{cy^*}{(1+\omega z^*)^2}\right)(a+pz^*)(s_2+u)\right|, \end{eqnarray*}$

这与方程(4.7)矛盾. 因此, 若 $ \mathfrak{R}_0 >1 $ 且 $ \mathfrak{R}_1 >1 $, 则方程(4.7)的所有根均具有负实部, 慢性感染平衡点 $ E^{*} $ 是局部渐近稳定的.

定理 4.4 若 $ \mathfrak{R}_0<1 $, 则系统(1.2)的病毒未感染平衡点 $ E_{0}\left(\lambda/d,0,0,0\right) $ 全局渐近稳定.

证 设 $ (x(t),y(t),v(t),z(t)) $ 是系统(1.2)满足初始条件(1.3)的任意一个正解. 定义

$V_{1}(t)=x-x_0-x_0\ln\frac{x}{x_0}+{\rm e}^{m\tau}y+\frac{\beta x_0}{u}v+\frac{p{\rm e}^{m\tau}}{c}z+\displaystyle\int_{t-\tau}^{t}\left[\frac{\beta x(s)v(s)}{1+\alpha v(s)}+\beta_1 x(s)y(s)\right]{{\rm d}s}.$

沿着系统(1.2)的解计算 $ V_{1}(t) $ 的全导数

$\begin{align*} \dot V_1(t)=&-d\frac{(x(t)-x_0)^2}{x(t)}-\frac{\alpha\beta x_0}{1+\alpha v(t)}v^2(t)+a{\rm e}^{m\tau}(\mathfrak{R}_0 -1)y(t)\\ & -\frac{\omega p{\rm e}^{m\tau}}{1+\omega z(t)}y(t)z^2(t)-\frac{p{\rm e}^{m\tau}}{c}bz(t)-\frac{p{\rm e}^{m\tau}}{c}ny(t)z(t). \end{align*}$

因此 $ \dot V_1(t)\le0 $, 且 $ \dot V_1(t)=0 $ 当且仅当 $ x=x_0 $, $ y=v=z=0 $, 可以证明, $ \lbrace E_{0}\rbrace $ 是$ \lbrace (x(t),y(t),$$ v(t), z(t)):\dot V_1(t)=0\rbrace $ 的最大不变子集. 由定理 4.1 可知, $ E_{0} $ 是局部渐近稳定的. 因此, 由 LaSalle 不变性原理可知病毒未感染平衡点 $ E_{0} $ 是全局渐近稳定的.

定理 4.5 若 $ \mathfrak{R}_1<1<\mathfrak{R}_0 $, 则系统(1.2)的免疫未激活感染平衡点 $ E_1(x_1,y_1,v_1,0) $ 全局渐近稳定.

证 设 $ (x(t),y(t),v(t),z(t)) $ 是系统(1.2)满足初始条件(1.3)的任意一个正解. 定义

$\begin{align*} V_{2}(t)=\ & x-x_1-x_1\ln\frac{x}{x_1}+{\rm e}^{m\tau}\left(y-y_1-y_1\ln\frac{y}{y_1}\right)+\frac{\beta x^*}{1+\alpha v^*}\frac{1}{u}\left(v-v_1-v_1\ln\frac{v}{v_1}\right)\\ &+\frac{{\rm e}^{m\tau}py_1}{b} z+\beta\displaystyle\int_{t-\tau}^{t}\left[\frac{ x(s)v(s)}{1+\alpha v(s)}-\frac{ x_1v_1}{1+\alpha v_1}-\frac{ x_1v_1}{1+\alpha v_1}\ln\frac{(1+\alpha v_1)x(s)v(s)}{x_1v_1(1+\alpha v(s))}\right]{{\rm d}s}\\ &+\beta_1\displaystyle\int_{t-\tau}^{t}\left[x(s)y(s)- x_1y_1-x_1y_1\ln\frac{x(s)y(s)}{x_1y_1}\right]{{\rm d}s}. \end{align*}$

沿着系统(1.2)的解计算 $ V_2(t) $ 的全导数

$\begin{align*} \dot V_2(t)=&-\frac{\beta x_1v_1}{1+\alpha v_1}\left[g\left(\frac{x_1}{x(t)}\right)+g\left(\frac{v_1y(t)}{y_1v(t)}\right)+g\left(\frac{y_1(1+\alpha v_1)x(t-\tau)v(t-\tau)}{x_1v_1y(t)(1+\alpha v(t-\tau))}\right)\right]\\ &-\frac{\beta x_1v_1}{1+\alpha v_1}g\left(\frac{1+\alpha v(t)}{1+\alpha v_1}\right)-\beta_1 x_1y_1\left[g\left(\frac{x_1}{x(t)}\right)+g\left(\frac{x(t-\tau)y(t-\tau)y_1}{x_1y_1y(t)}\right)\right]\\ &-\frac{\beta x_1v_1}{(1+\alpha v_1)^2}\frac{\alpha (v(t)-v_1)^2}{(1+\alpha v(t))v_1}-d\frac{(x(t)-x_1)^2}{x(t)}+{\rm e}^{m\tau}p(\mathfrak{R}_1-1)y(t)z(t). \end{align*}$

这里 $ g(x)=x-1-\ln x $, 显然 $ g(x)\geq0 $, 且 $ g(x)=0 $ 当且仅当 $ x=1 $. 因此 $ \dot V_2(t)\le0 $, 且 $ \dot V_2(t)=0 $ 当且仅当 $ x=x_1,y=y_1,v=v_1,z=0 $, 可以证明, $ \lbrace E_1\rbrace $ 是 $ \lbrace (x(t),y(t),v(t),z(t)):\dot V_2(t)=0\rbrace $ 的最大不变子集. 由定理 4.2 可知, $ E_1 $ 是局部渐近稳定的. 因此, 由 LaSalle 不变性原理可知免疫未激活感染平衡点 $ E_1 $ 是全局渐近稳定的.

定理 4.6 若 $ \mathfrak{R}_0 >1 $ 且 $ \mathfrak{R}_1 >1 $, 则系统(1.2)的慢性感染平衡点 $ E^{*}(x^*,y^*,v^*,z^*) $ 全局渐近稳定.

证 设 $ (x(t),y(t),v(t),z(t)) $ 是系统(1.2)满足初始条件(1.3)的任意一个正解. 定义

$\begin{align*} V_{3}(t)=\ & x-x^{*}-x^{*}\ln\frac{x}{x^{*}}+{\rm e}^{m\tau}\left(y-y^{*}-y^{*}\ln\frac{y}{y^{*}}\right)+\frac{\beta x^*}{1+\alpha v^*}\frac{1}{u}\left(v-v^{*}-v^{*}\ln\frac{v}{v^{*}}\right)\\ &+\frac{{\rm e}^{m\tau} py^*}{b}\left(z-z^{*}-z^{*}\ln\frac{z}{z^{*}}\right)+\beta_1\displaystyle\int_{t-\tau}^{t}\left[x(s)y(s)- x^{*}y^{*}-x^{*}y^{*}\ln\frac{x(s)y(s)}{x^{*}y^{*}}\right]{{\rm d}s}\\ &+\beta\displaystyle\int_{t-\tau}^{t}\left[\frac{ x(s)v(s)}{1+\alpha v(s)}-\frac{ x^{*}v^{*}}{1+\alpha v^{*}}-\frac{ x^{*}v^{*}}{1+\alpha v^{*}}\ln\frac{(1+\alpha v^{*})x(s)v(s)}{x^{*}v^{*}(1+\alpha v(s))}\right]{{\rm d}s}. \end{align*}$

沿着系统(1.2)的解计算 $ V_3(t) $ 的全导数

$\begin{align*} \dot V_3(t)=& -\frac{\beta x^*v^*}{1+\alpha v^*}\left[g\left(\frac{x^*}{x(t)}\right)+g\left(\frac{v^*y(t)}{y^*v(t)}\right)+g\left(\frac{y^*(1+\alpha v^*)x(t-\tau)v(t-\tau)}{x^*v^*y(t)(1+\alpha v(t-\tau))}\right)\right]\\ & -\frac{\beta x^*v^*}{1+\alpha v^*}g\left(\frac{1+\alpha v(t)}{1+\alpha v^*}\right)-\beta_1 x^*y^*\left[g\left(\frac{x^*}{x(t)}\right)+g\left(\frac{x(t-\tau)y(t-\tau)}{x^*y(t)}\right)\right]\\ &-\frac{\alpha\beta x^* (v(t)-v^*)^2}{(1+\alpha v^*)^2(1+\alpha v(t))}-d\frac{(x(t)-x^*)^2}{x(t)}-\frac{cp\omega y^*y(t)(z(t)-z^*)^2}{b(1+\omega z^*)(1+\omega z(t))}{\rm e}^{m\tau}. \end{align*}$

因此, $ \dot V_3(t)\le0 $, 且 $ \dot V_3(t)=0 $ 当且仅当 $ x=x^*,y=y^*,v=v^*,z=z^* $, 可以证明, $ \lbrace E^*\rbrace $ 是 $ \lbrace(x(t),y(t),v(t),z(t)):\dot V_3(t)=0\rbrace $ 的最大不变子集. 由定理 4.3 可知, $ E^* $ 是局部渐近稳定的. 因此, 由 LaSalle 不变性原理可知慢性感染平衡点 $ E^{*} $ 是全局渐近稳定的.

5 数值模拟

本节, 我们通过数值模拟来探究细胞-细胞传播机制和免疫损害对系统(1.2)动力学性态的影响. 我们选取初始值为 $ (\varphi_1,\varphi_2,\varphi_3,\varphi_4)=(10000,0,200,0) $, 参数取值见表1[11,23,24].

表1   系统 (1.2) 的参数取值

新窗口打开| 下载CSV


5.1 细胞-细胞传播对系统 (1.2) 动力学性态的影响

未感染细胞有两种感染方式: 一种是游离病毒感染未感染细胞, 另一种是被感染细胞通过直接接触感染未感染细胞. 两种感染方式对病毒载量的相对贡献率分别为

$C_1=\frac{\frac{\beta xv}{1+\alpha v}}{\frac{\beta xv}{1+\alpha v}+\beta_1 xy},\quad C_2=\frac{\beta_1 xy}{\frac{\beta xv}{1+\alpha v}+\beta_1 xy}.$

$C_1,\ C_2 $ 依赖于系统(1.2)的最终收敛性. 因此, 下面针对平衡点 $ E^{*} $, 研究细胞-细胞传播的相对贡献率.

我们选取表 1 中参数对系统(1.2)进行数值模拟. 通过计算可得 $ \mathfrak{R}_0=149.1150>1 $ 且 $ \mathfrak{R}_1=1.0217>1 $. 根据定理 4.6, 平衡点 $ E^{*} $ 全局渐近稳定 (见图2). 如图3 所示, 随着时间增加, 游离病毒对病毒载量的贡献率不断减小, 最终趋于 0.38; 细胞-细胞传播对病毒载量的贡献率不断增大, 最终趋于 0.62. 这表明在感染初期, 游离病毒是感染的主要来源, 随着感染的继续, 细胞-细胞传播成为病毒载量的主要贡献者.

图2

新窗口打开| 下载原图ZIP| 生成PPT

图2   表1 参数下系统(1.2)的解


图3

新窗口打开| 下载原图ZIP| 生成PPT

图3   两种感染对系统(1.2)的相对贡献率


5.2 免疫损害对系统 (1.2) 动力学性态的影响}

本小节我们讨论不同免疫损害率对系统(1.2)的影响, 分别选取 $ n=0.001 $, $ n=0.002 $, $ n=0.003 $. 如图4 所示, 随着 $ n $ 的降低, 被感染细胞和游离病毒的峰值随之降低, CTLs 的峰值随之增加; 被感染细胞、游离病毒和 CTLs 到达峰值的时间随之缩短, 收敛速度随之加快, 收敛值随之降低, 这表明降低免疫损害率有助于加速病毒感染, 恢复CTL免疫反应, 降低病毒载量, 这也为抗病毒治疗提供一种可行的思路.

图4

新窗口打开| 下载原图ZIP| 生成PPT

图4   不同免疫损害率对系统(1.2)动力学性态的影响


6 结论

本文考虑了一个具有细胞-细胞传播、饱和 CTL 免疫反应和免疫损害的 HIV-1 感染模型. 通过分析和计算, 我们证明了系统的动力学由免疫未激活再生率 $ \mathfrak{R}_0 $ 和免疫激活再生率 $ \mathfrak{R}_1 $ 决定. 如果 $ \mathfrak{R}_0<1 $, 则病毒未感染平衡点全局渐近稳定; 如果 $ \mathfrak{R}_1<1<\mathfrak{R}_0 $, 免疫未激活平衡点全局渐近稳定; 如果 $ \mathfrak{R}_0>1 $ 且 $ \mathfrak{R}_1>1 $, 则慢性感染平衡点全局渐近稳定. 此外, 胞内时滞不影响平衡点的稳定性, 因此不会导致 Hopf 分支. 最后, 我们讨论了免疫损害和细胞-细胞传播对系统(1.2) 的影响, 通过减小免疫损害率和细胞-细胞传播率可以有效降低病毒载量, 更好地控制 HIV-1 的感染.

参考文献

Perelson A S, Neumann A U, Markowitz M, et al.

HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time

Science, 1996, 271(5255): 1582-1586

PMID:8599114      [本文引用: 1]

A new mathematical model was used to analyze a detailed set of human immunodeficiency virus-type 1 (HIV-1) viral load data collected from five infected individuals after the administration of a potent inhibitor of HIV-1 protease. Productively infected cells were estimated to have, on average, a life-span of 2.2 days (half-life t 1/2 = 1.6 days), and plasma virions were estimated to have a mean life-span of 0.3 days (t 1/2 = 0.24 days). The estimated average total HIV-1 production was 10.3 x 10(9) virions per day, which is substantially greater than previous minimum estimates. The results also suggest that the minimum duration of the HIV-1 life cycle in vivo is 1.2 days on average, and that the average HIV-1 generation time--defined as the time from release of a virion until it infects another cell and causes the release of a new generation of viral particles--is 2.6 days. These findings on viral dynamics provide not only a kinetic picture of HIV-1 pathogenesis, but also theoretical principles to guide the development of treatment strategies.

Egger M, Hirschel B, Francioli P, et al.

Impact of new antiretroviral combination therapies in HIV infected patients in Switzerland: Prospective multicentre study

The British Medical Journal, 1997, 315(7117): 1194-1199

[本文引用: 1]

Perelson A S, Nelson P W.

Mathematical analysis of HIV-1: Dynamics in vivo

SIAM Review, 1999, 41(1): 3-44

[本文引用: 1]

Wasserstein-Robbins F.

A mathematical model of HIV infection: Simulating T4, T8, macrophages, antibody, and virus via specific anti-HIV response in the presence of adaptation and tropism

Bulletin of Mathematical Biology, 2010, 72(5): 1208-1253

DOI:10.1007/s11538-009-9488-5      PMID:20151219      [本文引用: 1]

A mathematical model of the host's immune response to HIV infection is proposed. The model represents the dynamics of 13 subsets of T cells (HIV-specific and nonspecific, healthy and infected, T4 and T8 cells), infected macrophages, neutralizing antibodies, and virus. The results of simulation are in agreement with published data regarding T4 cell concentration and viral load, and exhibit the typical features of HIV infection, i.e. double viral peaks in the acute stage, sero conversion, inverted T cell ratio, establishment of set points, steady state, and decline into AIDS. This result is achieved by taking into account thymic aging, viral and infected cell stimulation of specific immune cells, background nonspecific antigens, infected cell proliferation, viral production by infected macrophages and T cells, tropism, viral, and immune adaptation. Starting from this paradigm, changes in the parameter values simulate observed differences in individual outcomes, and predict different scenarios, which can suggest new directions in therapy. In particular, large parameter changes highlight the potentially critical role of both very vigorous and extremely damped specific immune response, and of the elimination of virus release by macrophages. Finally, the time courses of virus, antibody and T cells production and removal are systematically investigated, and a comparison of T4 and T8 cell dynamics in a healthy and in a HIV infected host is offered.

Weusten J, Drimmelen H, Lelie P N.

Mathematic modeling of the risk of HBV, HCV, and HIV transmission by window phase donations not detected by NAT

Transfusion, 2010, 42(5): 537-548

[本文引用: 1]

Yan Y, Wang W.

Global stability of a five-dimensional model with immune responses and delay

Discrete and Continuous Dynamical Systems-B, 2011, 17(1): 401-416

[本文引用: 1]

Jiang C, Wang W.

Complete classification of global dynamics of a virus model with immune response

Discrete and Continuous Dynamical Systems-B, 2014, 19(4): 1087-1103

[本文引用: 1]

Willie R, Zheng P, Parumasur N, et al.

Asymptotic and stability dynamics of an HIV-1-Cytotoxic T Lymphocytes (CTL) chemotaxis model

Journal of Nonlinear Science, 2020, 30(3): 1055-1080

[本文引用: 1]

Wang Y, Zhou Y C, Brauer F, et al.

Viral dynamics model with CTL immune response incorporating antiretroviral therapy

Journal of Mathematical Biology, 2013, 67(4): 901-934

DOI:10.1007/s00285-012-0580-3      PMID:22930342      [本文引用: 1]

We present two HIV models that include the CTL immune response, antiretroviral therapy and a full logistic growth term for uninfected CD4+ T-cells. The difference between the two models lies in the inclusion or omission of a loss term in the free virus equation. We obtain critical conditions for the existence of one, two or three steady states, and analyze the stability of these steady states. Through numerical simulation we find substantial differences in the reproduction numbers and the behaviour at the infected steady state between the two models, for certain parameter sets. We explore the effect of varying the combination drug efficacy on model behaviour, and the possibility of reconstituting the CTL immune response through antiretroviral therapy. Furthermore, we employ Latin hypercube sampling to investigate the existence of multiple infected equilibria.

Kang C J, Miao H, Chen X, et al.

Global stability of a diffusive and delayed virus dynamics model with Crowley-Martin incidence function and CTL immune response

Advances in Difference Equations, 2017, 2017(1): 1-16

[本文引用: 1]

Ren J, Xu R, Li L C.

Global stability of an HIV infection model with saturated CTL immune response and intracellular delay

Mathematical Biosciences and Engineering, 2021, 18(1): 57-68

[本文引用: 4]

Sattentau Q.

Avoiding the void: Cell-to-cell spread of human viruses

Nature Reviews Microbiology, 2008, 6(11): 815-826

DOI:10.1038/nrmicro1972      PMID:18923409      [本文引用: 1]

The initial stages of animal virus infection are generally described as the binding of free virions to permissive target cells followed by entry and replication. Although this route of infection is undoubtedly important, many viruses that are pathogenic for humans, including HIV-1, herpes simplex virus and measles, can also move between cells without diffusing through the extracellular environment. Cell-to-cell spread not only facilitates rapid viral dissemination, but may also promote immune evasion and influence disease. This Review discusses the various mechanisms by which viruses move directly between cells and the implications of this for viral dissemination and pathogenesis.

Sattentau Q.

The direct passage of animal viruses between cells

Current Opinion in Virology, 2011, 1(5): 396-402

DOI:10.1016/j.coviro.2011.09.004      PMID:22440841      [本文引用: 1]

The paradigm that viruses can move directly, and in some cases covertly, between contacting target cells is now well established for several virus families. The underlying mechanisms of cell-to-cell spread, however, remain to be fully elucidated and may differ substantially depending on the viral exit/entry route and the cellular tropism. Here, two divergent cell-to-cell spread mechanisms are exemplified: firstly by human retroviruses, which rely upon transient adhesive structures that form between polarized immune cells termed virological synapses, and secondly by herpesviruses that depend predominantly on pre-existing stable cellular contacts, but may also form virological synapses. Plant viruses can also spread directly between contacting cells, but are obliged by the rigid host cell wall to move across pore structures termed plasmodesmata. This review will focus primarily on recent advances in our understanding of animal virus cell-to-cell spread using examples from these two virus families, and will conclude by comparing and contrasting the cell-to-cell spread of animal and plant viruses.Copyright © 2011 Elsevier B.V. All rights reserved.

Gummuluru S, Kinsey C M, Emerman M.

An in vitro rapid-turnover assay for human immunodeficiency virus type 1 replication selects for cell-to-cell spread of virus

Journal of Virology, 2000, 74(23): 10882-10891

PMID:11069982      [本文引用: 1]

We have developed a rapid-turnover culture system where the life span of a human immunodeficiency virus type 1-infected cell is controlled by periodic addition of a cytotoxic agent, mitomycin C. These mitomycin C-exposed cells are cocultured with a constant number of uninfected cells as new targets for the virus. Passage of the virus-infected cells under these conditions led to the emergence of a viral variant that was able to replicate efficiently in this culture system. After biologic and molecular cloning, we were able to identify a single frameshift mutation in the vpu open reading frame that was sufficient for growth of the mutant virus in the rapid-turnover assay. This virus variant spread more efficiently by cell-to-cell transfer than the parental virus did. Electron micrographs of cells infected with the delta vpu virus revealed a large number of mature viral capsids attached to the plasma membrane. The presence of these mature virus particles on the cell surface led to enhanced fusion and formation of giant syncytia with uninfected cells. Enhanced cell-to-cell transfer of the delta vpu virus provides an explanation for the survival of this mutant virus in the rapid-turnover culture system. The in vitro rapid-turnover culture system is a good representation of the in vivo turnover kinetics of infected cells and their continual replacement by host lymphopoietic mechanisms.

Lai X L, Zou X F.

Modeling cell-to-cell spread of HIV-1 with logistic target cell growth

Journal of Mathematical Analysis and Applications, 2015, 426(1): 563-584

[本文引用: 2]

Pourbashash H, Pilyugin S S, Leenheer P D, et al.

Global analysis of within host virus models with cell-to-cell viral transmission

Discrete and Continuous Dynamical Systems, 2014, 19(10): 3341-3357

[本文引用: 1]

Akbari N, Asheghi R.

Optimal control of an HIV infection model with logistic growth, celluar and homural immune response, cure rate and cell-to-cell spread

Boundary Value Problems, 2022, Article number 5

[本文引用: 2]

Iwami S, Nakaoka S, Takeuchi Y, et al.

Immune impairment thresholds in HIV infection

Immunology Letters, 2009, 123(2): 149-154

DOI:10.1016/j.imlet.2009.03.007      PMID:19428563      [本文引用: 1]

Longitudinal studies of patients infected with HIV-1 reveal a long and variable length of asymptomatic phase between infection and development of AIDS. Some HIV infected patients are still asymptomatic after 15 or more years of infection but some patients develop AIDS within 2 years. The mechanistic basis of the disease progression has remained obscure but many researchers have been trying to explain it. For example, the possible importance of viral diversity for the disease progression and the development of AIDS has been very well worked out in the early-1990s, especially by some important works of Martin A. Nowak. These studies can give an elegant explanation for a variability of asymptomatic phase. Here, a simple mathematical model was used to propose a new explanation for a variable length of asymptomatic phase. The main idea is that the immune impairment rate increases over the HIV infection. Our model suggested the existence of so-called "Risky threshold" and "Immunodeficiency threshold" on the impairment rate. The former implies that immune system may collapse when the impairment rate of HIV exceeds the threshold value. The latter implies that immune system always collapses when the impairment rate exceeds the value. We found that the length of asymptomatic phase is determined stochastically between these threshold values depending on the virological and immunological states. Furthermore, we investigated a distribution of the length of asymptomatic phase and a survival rate of the immune responses in one HIV patient.

Elaiw A M, Raezah A A, Azoz S A.

Stability of delayed HIV dynamics models with two latent reservoirs and immune impairment

Advances in Difference Equations, 2018, Atricle number 414

[本文引用: 2]

Krishnapriya P, Pitchaimani M.

Modeling and bifurcation analysis of a viral infection with time delay and immune impairment

Japan Journal of Industrial and Applied Mathematics, 2017, 34(1): 99-139

[本文引用: 1]

Hale J, Lunel S V.

An Introduction to Functional Differential Equations

Hoboken: Mathematical Methods in the Applied Sciences, 1993

[本文引用: 1]

Zhao X Q.

Basic reproduction ratios for periodic compartmental models with time delay

Journal of Dynamics and Differential Equations, 2017, 29(1): 67-82

[本文引用: 1]

Wang J L, Guo M, Liu X N, et al.

Threshold dynamics of HIV-1 virus model with cell-to-cell transmission, cell-mediated immune responses and distributed delay

Applied Mathematics and Computation, 2016, 291: 149-161

[本文引用: 1]

Wang J L, Pang J M, Kuniya T, et al.

Global threshold dynamics in a five-dimensional virus model with cell-mediated, humoral immune responses and distributed delays

Applied Mathematics and Computation, 2014, 241: 298-316

[本文引用: 1]

/

〈 〉

版权所有 © 《数学物理学报》杂志社
地址： 武汉市71010号信箱(430071)　电话: 027-87199206(中文版)　027-87199087(英文版)
E-mail: actams@apm.ac.cn
本系统由北京玛格泰克科技发展有限公司设计开发