## Dynamics of an HTLV-I Infection Model with Delayed and Saturated CTL Immune Response and Immune Impairment

Complex Systems Research Center, Shanxi University, Taiyuan 030006

 基金资助: 国家自然科学基金.  11871316

 Fund supported: the NSFC.  11871316 Abstract

In this paper, an HTLV-I infection model with delayed and saturated CTL immune response and immune impairment is developed. By calculations, the existences of feasible equilibria are established, immunity-inactivated and immunity-activated reproduction ratios are also derived. Under the assistance of proper Lyapunov functionals and LaSalle's invariance principle, it is shown that the infection-free equilibrium is globally asymptotically stable if the immunity-inactivated reproduction ratio is less than unity; the immunity-inactivated equilibrium is globally asymptotically stable if the immunity-activated reproduction ratio is less than unity, while the immunity-inactivated reproduction ratio is greater than unity; the immunity-activated equilibrium is globally asymptotically stable (when the time delay equals to zero) if the immunity-activated reproduction ratio is greater than unity. A Hopf bifurcation at the immunity-activated equilibrium occurs as the time delay crosses a critical value. Finally, numerical simulations are performed to illustrate the theoretical results.

Keywords： HTLV-I infection ; Immune delay ; Saturated CTL immune response ; Immune impairment ; Hopf bifurcation

Xu Rui, Yang Yan. Dynamics of an HTLV-I Infection Model with Delayed and Saturated CTL Immune Response and Immune Impairment. Acta Mathematica Scientia[J], 2022, 42(6): 1836-1848 doi:

## 1 引言

$\begin{equation} \begin{array}{l} { } \frac{{{\rm d}T}}{{{\rm d}t}} = \Lambda - {\mu _1}T\left( t \right) - \beta T\left( t \right){T_A}\left( t \right) , \\ { } \frac{{{\rm d}{T_L}}}{{{\rm d}t}} = \beta T\left( t \right){T_A}\left( t \right) + {\gamma _1}{T_A}\left( t \right) - \left( {{\mu _2} + {\delta _1}} \right){T_L}\left( t \right) , \\ { } \frac{{{\rm d}{T_A}}}{{{\rm d}t}} = {\mu _2}{T_L}\left( t \right) - \left( {{\mu _3} + {\delta _2}} \right){T_A}\left( t \right) - p{T_A}\left( t \right)H\left( t \right) , \\ { } \frac{{{\rm d}H}}{{{\rm d}t}} = c{T_A}\left( t \right)H\left( t \right) - bH\left( t \right), \end{array} \end{equation}$

$\begin{equation} \begin{array}{l} { } \frac{{{\rm d}T}}{{{\rm d}t}} = \Lambda - {\mu _1}T(t) - \beta T\left( t \right){T_A}\left( t \right) , \\ { } \frac{{{\rm d}{T_L}}}{{{\rm d}t}} = \beta T\left( t \right){T_A}\left( t \right) + {\gamma _1}{T_A}\left( t \right) - \left( {{\mu _2} + {\delta _1}} \right){T_L}\left( t \right) , \\ { } \frac{{{\rm d}{T_A}}}{{{\rm d}t}} = {\mu _2}{T_L}\left( t \right) - \left( {{\mu _3} + {\delta _2}} \right){T_A}(t) - p{T_A}\left( t \right)H\left( t \right) , \\ { } \frac{{{\rm d}H}}{{{\rm d}t}} = \frac{{c{T_A}\left( {t - \tau } \right)H\left( {t - \tau } \right)}}{{1 + \varepsilon H\left( {t - \tau } \right)}} - bH\left( t \right) - n{T_A}\left( t \right)H\left( t \right). \end{array} \end{equation}$

$\begin{equation} \begin{array}{l} T\left( \theta \right) = {\phi _1}\left( \theta \right), {T_L}\left( \theta \right) = {\phi _2}\left( \theta \right), {T_A}\left( \theta \right) = {\phi _3}\left( \theta \right), H\left( \theta \right) = {\phi _4}\left( \theta \right), \\ {\phi _i}\left( \theta \right) \ge 0, \theta \in \left[ { - \tau , 0} \right), {\phi _i}\left( 0 \right) > 0 \left( {i = 1, 2, 3, 4} \right). \end{array} \end{equation}$

## 2 基本再生率和可行平衡点

$\mathfrak{R}_0 > 1$, 系统(1.2)存在免疫未激活平衡点${E_1}\left( {{T_1}, {T_{L1}}, {T_{A1}}, 0} \right)$, 其中

${\mathfrak{R}_1}$为免疫激活再生率. 当${\mathfrak{R}_1} > 1$时, 系统(1.2) 存在免疫激活平衡点${E_2}\left( {{T_2}, {T_{L2}}, {T_{A2}}, {H_2}} \right)$, 其各分量满足以下方程组

$\begin{equation} \begin{array}{ll} \Lambda - {\mu _1}T - \beta T{T_A} = 0, \\ \beta T{T_A} + {\gamma _1}{T_A} - \left( {{\mu _2} + {\delta _1}} \right){T_L} = 0, \\ {\mu _2}{T_L} - \left( {{\mu _3} + {\delta _2}} \right){T_A} - p{T_A}H = 0, \\ { } \frac{{c{T_A}H}}{{1 + \varepsilon H}} - bH - n{T_A}H = 0. \end{array} \end{equation}$

$\begin{equation} {f_1}\left( {{T_A}} \right) = H = \frac{1}{\varepsilon }\left( {\frac{{c{T_A}}}{{b + n{T_A}}} - 1} \right). \end{equation}$

$\begin{equation} {T_A} = \frac{{{\mu _2}\left( {\beta \Lambda + {\mu _1}{\gamma _1}} \right) - {\mu _1}\left( {{\mu _2} + {\delta _1}} \right)\left( {{\mu _3} + {\delta _2}} \right) - p{\mu _1}\left( {{\mu _2} + {\delta _1}} \right)H}}{{\beta \left[ {p\left( {{\mu _2} + {\delta _1}} \right)H + \left( {{\mu _2} + {\delta _1}} \right)\left( {{\mu _3} + {\delta _2}} \right) - {\mu _2}{\gamma _1}} \right]}}:={f_2}\left( {{H}} \right). \end{equation}$

${{T_{A2}}}$是下列方程的唯一正解

$\begin{equation} \frac{{\Lambda \beta }}{{{\mu _1} + \beta {T_{A}}}} + {\gamma _1} - \frac{{{\mu _2} + {\delta _1}}}{{{\mu _2}}}\left[ {\left( {{\mu _3} + {\delta _2}} \right) + \frac{p}{\varepsilon }\left( {\frac{{c{T_{A}}}}{{b + n{T_{A}}}} - 1} \right)} \right] = 0. \end{equation}$

## 3 可行平衡点的局部稳定性和Hopf分支

系统(1.2)在${E_0}$处的特征方程为

$\begin{equation} \left( {\lambda + {\mu _1}} \right)\left( {\lambda + b} \right)\left[ {\left( {\lambda + {\mu _2} + {\delta _1}} \right)\left( {\lambda + {\mu _3} + {\delta _2}} \right) - {\mu _2}\left( {\frac{{\beta \Lambda }}{{{\mu _1}}} + {\gamma _1}} \right)} \right] = 0. \end{equation}$

$\begin{equation} {\tau _0} = \mathop {\min }\limits_{k \in \left\{ {1, 2, 3, 4} \right\}} \left\{ {\tau _k^{\left( 0 \right)}} \right\}, \;\;{\omega _0} = {\omega _{{k0}}}. \end{equation}$

$\begin{equation} {\left( {\frac{{{\rm d}\lambda }}{{{\rm d}\tau }}} \right)^{ - 1}} = \frac{{3{q_3}{\lambda ^2} + 2{q_2}\lambda + {q_1}}}{{\lambda \left( {{q_3}{\lambda ^3} + {q_2}{\lambda ^2} + {q_1}\lambda + {q_0}} \right)}} - \frac{{4{\lambda ^3} + 3{l_3}{\lambda ^2} + 2{l_2}\lambda + {l_1}}}{{\lambda \left( {{\lambda ^4} + {l_3}{\lambda ^3} + {l_2}{\lambda ^2} + {l_1}\lambda + {l_0}} \right)}} - \frac{\tau }{\lambda }. \end{equation}$

$\begin{eqnarray} {\rm sign}{\left\{ {\frac{{{\rm d}\left( {{\rm Re} \lambda } \right)}}{{ {\rm d}\tau }}} \right\}_{\tau = {\tau _0}}} &=&{\rm sign}{\left\{ {{\rm Re} {{\left( {\frac{{{\rm d}\lambda }}{{{\rm d}\tau }}} \right)}^{ - 1}}} \right\}_{\tau = {\tau _0}}} {}\\ &=&{\rm sign}\left[ {\frac{{f'\left( {\omega _0^2} \right)}}{{{{\left( {\omega _0^4 - {l_2}\omega _0^2 + {l_0}} \right)}^2} + {{\left( {{l_3}\omega _0^2 - {l_1}} \right)}^2}\omega _0^2}}} \right]. \end{eqnarray}$

(ⅰ) 若(3.12)没有正实根, 对所有$\tau > 0$, ${{E_2}}$是局部渐近稳定的;

(ⅱ) 若(3.12)至少存在一个正实根, 且$f'\left( {\omega _0^2} \right) \ne 0$, 则对$\tau \in \left[ {0, {\tau _0}} \right)$, ${E_2}$是局部渐近稳定的; 对$\tau >{\tau _0}$, ${E_2}$是不稳定的; 当$\tau = {\tau _0}$时, 在${E_2}$处出现Hopf分支.

## 4 全局渐近稳定性

令$\left( {T\left( t \right), {T_L}\left( t \right), {T_A}\left( t \right), H\left( t \right)} \right)$是系统(1.2)满足初始条件(1.3)的任意正解. 定义

$\begin{eqnarray} {V_0}\left( t \right)& =& \left( {T\left( t \right) - {T_0} - {T_0}\ln \frac{{T\left( t \right)}}{{{T_0}}}} \right) + {T_L}\left( t \right) + \frac{{{\mu _2} + {\delta _1}}}{{{\mu _2}}}{T_A}\left( t \right) {}\\ & &+ \frac{{\left( {{\mu _2} + {\delta _1}} \right)p}}{{c{\mu _2}}}H\left( t \right) + \frac{{\left( {{\mu _2} + {\delta _1}} \right)p}}{{c{\mu _2}}}\int_{t - \tau }^t {\frac{{c{T_A}\left( \theta \right)H\left( \theta \right)}}{{1 + \varepsilon H\left( \theta \right)}}{\rm d}\theta } , \end{eqnarray}$

$\begin{eqnarray} {{\dot V}_0}\left( t \right)& = & - \frac{{{\mu _1}{{\left( {T - {T_0}} \right)}^2}}}{T} - \frac{{\left( {{\mu _2} + {\delta _1}} \right)\left( {{\mu _3} + {\delta _2}} \right)}}{{{\mu _2}}}\left( {1 - {\mathfrak{R}_0}} \right){T_A} {}\\ & &- \frac{{\left( {{\mu _2} + {\delta _1}} \right)pb}}{{c{\mu _2}}}H - \frac{{\left( {{\mu _2} + {\delta _1}} \right)np}}{{c{\mu _2}}}{T_A}H - \frac{{\left( {{\mu _2} + {\delta _1}} \right)p\varepsilon }}{{{\mu _2}}}{T_A}{H^2}. \end{eqnarray}$

$\mathfrak{R}_0 < 1$, 则$\mathop {\dot V}\nolimits_0 \left( t \right) \le 0$, 当且仅当$T = {T_0}$, ${T_L}=0$, ${T_A} = 0$, $H = 0$时等号成立. 易证${M_0} = \left\{ {{E_0}} \right\} \subset \Omega $$\left\{ {\left( {T\left( t \right), {T_L}\left( t \right), {T_A}\left( t \right), H} \right):{{\dot V}_0} = 0} \right\} 的最大不变子集. 由定理3.1和LaSalle不变性原理知, 当 \mathfrak{R}_0 < 1 时, E_0 是全局渐近稳定的. 证毕. 定理4.2 若 {\mathfrak{R}_1} < 1 < {\mathfrak{R}_0} , 则 {E_1} 是全局渐近稳定的. 令 \left( {T\left( t \right), {T_L}\left( t \right), {T_A}\left( t \right), H\left( t \right)} \right) 是系统(1.2)满足初始条件(1.3)的任意正解. 定义 \begin{eqnarray} {V_1}\left( t \right) &=& \left( {T\left( t \right) - {T_1} - {T_1}\ln \frac{{T\left( t \right)}}{{{T_1}}}} \right) + \left( {{T_L}\left( t \right) - {T_{L1}} - {T_{L1}}\ln \frac{{{T_L}\left( t \right)}}{{{T_{L1}}}}} \right) {}\\ & &+ \frac{{{\mu _2} + {\delta _1}}}{{{\mu _2}}}\left( {{T_A}\left( t \right) - {T_{A1}} - {T_{A1}}\ln \frac{{{T_A}\left( t \right)}}{{{T_{A1}}}}} \right) + \frac{{\left( {{\mu _2} + {\delta _1}} \right)p{T_{A1}}}}{{b{\mu _2}}}H\left( t \right) {}\\ & &+ \frac{{\left( {{\mu _2} + {\delta _1}} \right)p{T_{A1}}}}{{b{\mu _2}}}\int_{t - \tau }^t {\frac{{c{T_A}\left( \theta \right)H\left( \theta \right)}}{{1 + \varepsilon H\left( \theta \right)}}{\rm d}\theta }. \end{eqnarray} 沿着系统(1.2)的解计算 \mathop V\nolimits_1 \left( t \right) 的全导数可得 \begin{eqnarray} {{\dot V}_1}\left( t \right)& =& - \frac{{{\mu _1}{{\left( {T - {T_1}} \right)}^2}}}{T} - \beta {T_1}{T_{A1}}\left( {\frac{{{T_1}}}{T} + \frac{{{T_L}{T_{A1}}}}{{{T_{L1}}{T_A}}} + \frac{{T{T_A}{T_{L1}}}}{{{T_1}{T_{A1}}{T_L}}} - 3} \right) {} \\ & &- {\gamma _1}{T_{A1}}\left( {\frac{{{T_{L1}}{T_A}}}{{{T_L}{T_{A1}}}} + \frac{{{T_L}{T_{A1}}}}{{{T_{L1}}{T_A}}} - 2} \right) - \frac{{\left( {{\mu _2} + {\delta _1}} \right)\left( {b + n{T_{A1}}} \right)\varepsilon p}}{{b{\mu _2}\left( {1 + \varepsilon H} \right)}}{T_A}{H^2} {}\\ & &- \frac{{\left( {{\mu _2} + {\delta _1}} \right)\left( {1 - {\mathfrak{R}_1}} \right)p}}{{{\mu _2}\left( {1 + \varepsilon H} \right)}}{T_A}H. \end{eqnarray} 由均值不等式知, {{\dot V}_1}\left( t \right) \le 0 , 当且仅当 T = {T_1} , {T_L} = {T_{L1}} , {T_A} = {T_{A1}} , H = 0 时等号成立. 易证 {M_1} = \left\{ {{E_1}} \right\}\subset \Omega$$ \left\{ {\left( {T\left( t \right), {T_L}\left( t \right), {T_A}\left( t \right), H\left( t \right)} \right):{{\dot V}_1}\left( t \right) = 0} \right\}$的最大不变子集. 由定理3.2和LaSalle不变性原理知, 当${\mathfrak{R}_1}<1<{\mathfrak{R}_0}$时, ${{E_1}}$是全局渐近稳定的. 证毕.

令$\left( {T\left( t \right), {T_L}\left( t \right), {T_A}\left( t \right), H\left( t \right)} \right)$是系统(1.2)满足初始条件(1.3)的任意正解. 定义

$\begin{eqnarray} {V_2}\left( t \right)& =& \left( {T\left( t \right) - {T_2} - {T_2}\ln \frac{{T\left( t \right)}}{{{T_2}}}} \right) + \left( {{T_L}\left( t \right) - {T_{L2}} - {T_{L2}}\ln \frac{{{T_L}\left( t \right)}}{{{T_{L2}}}}} \right) {}\\ & &+ \frac{{{\mu _2} + {\delta _1}}}{{{\mu _2}}}\left( {{T_A}\left( t \right) - {T_{A2}} - {T_{A2}}\ln \frac{{{T_A}\left( t \right)}}{{{T_{A2}}}}} \right) {}\\ & &+ \frac{{\left( {{\mu _2} + {\delta _1}} \right)p{T_{A2}}}}{{b{\mu _2}}}\left( {H\left( t \right) - {H_2} - {H_2}\ln \frac{{H\left( t \right)}}{{{H_2}}}} \right). \end{eqnarray}$

$\begin{eqnarray} {\dot V_2}\left( t \right) &=&- \frac{{{\mu _1}{{\left( {T - {T_2}} \right)}^2}}}{T} + \beta {T_2}{T_{A2}}\left( {3-\frac{{{T_2}}}{T} - \frac{{{T_{A2}}{T_L}}}{{{T_A}{T_{L2}}}} - \frac{{T{T_A}{T_{L2}}}}{{{T_2}{T_{A2}}{T_L}}} } \right) {}\\ & &+ {\gamma _1}{T_{A2}}\left( {2-\frac{{{T_{L2}}{T_A}}}{{{T_L}{T_{A2}}}} - \frac{{{T_L}{T_{A2}}}}{{{T_{L2}}{T_A}}} } \right) - \frac{{\left( {{\mu _2} + {\delta _1}} \right)\varepsilon cp{T_{A2}}{T_A}{{\left( {H - {H_2}} \right)}^2}}}{{b{\mu _2}\left( {1 + \varepsilon {H_2}} \right)\left( {1 + \varepsilon H} \right)}}. \end{eqnarray}$

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