人类嗜T细胞病毒I型(HTLV-I)是一种外源性逆转录病毒, 与成人T细胞白血病(ATL)和HTLV-I相关脊髓病(HAM/TSP)密切相关. 然而, 只有一小部分感染者发展为ATL或HAM/TSP患者, 大部分HTLV-I感染者成为终身的无症状携带者(ACs). 此外, HTLV-I相关疾病的机制仍未明确, 暂无有效的治疗方法或可用的疫苗来治疗HTLV-I相关疾病^[1].

与HIV-1病毒不同, 细胞外的HTLV-I病毒几乎不能被检测到. 因此, 适宜检测前病毒载量: 即外周血单核细胞携带HTLV-I整合前病毒的比例^[2-3]. 在外周血中, HTLV-I主要感染CD$ {4^+} $辅助T细胞. 对于被感染的CD$ {4^+} $ T细胞, Asquith和Bangham发现只有一小部分受感染的CD$ {4^+} $ T细胞表达病毒蛋白Tax, 该蛋白参与激活HTLV-I基因的转录并激发病毒的增殖^[4]. 根据存在Tax与否, 被感染T细胞可以分成活跃的和潜伏的两种. 活跃的感染细胞经历病毒的持续选择性复制, 而潜伏的感染细胞不会产生新的病毒粒子. 尽管有丝分裂发生在所有CD$ {4^+} $ T细胞中, 但活跃感染的CD$ {4^+} $ T细胞的选择性有丝分裂发生率高于正常的稳态增殖. 因此, 区分潜伏的感染细胞与活跃的感染细胞十分重要. 另一方面, HTLV-I特异性CD$ {8^+} $细胞毒性T淋巴细胞(CTL)可以起到调节前病毒载量和清除感染细胞的作用. 然而, CTL的细胞毒性也会导致出现HAM/TSP的症状^{[3, 5]}. 为了解HTLV-I相关疾病的发病机制, 有必要研究HTLV-I感染与其特异性CTL反应之间的相互作用^{[2, 6]}. Khajanchi等人提出以下模型^[7]

(1.1) $ \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} $

其中, $ T(t) $, $ {T_L}(t) $, $ {T_A}(t) $, $ H(t) $分别表示$ t $时刻未感染T细胞、潜伏感染T细胞、感染T细胞和HTLV-I特定CD$ {8^+} $ T细胞的浓度. $ \Lambda $表示未感染T细胞的产生率; $ \beta $是感染率系数; $ \mu_1 $, $ \mu_3 $, $ \delta_1 $, $ b $分别表示未感染T细胞、潜伏感染T细胞、感染T细胞、CTL细胞的自然死亡率; $ \delta_2 $表示感染细胞的因病死亡率; $ \mu_2 $表示潜伏感染细胞到感染细胞的转化率; $ p{T_A}H $表示CTL免疫反应清除感染细胞的速率; $ {\gamma_1}{T_A} $表示感染细胞的有丝分裂速率; $ c $表示CTL免疫反应强度. 然而, $ t $时刻CTL细胞的产生与$ t-\tau $时刻的CTL细胞和感染细胞的数量有关, 其中$ \tau > 0 $是免疫时滞. 因此, 有学者研究了具有免疫时滞的HTLV-I感染模型的动力学行为^[8]. 在文献[9]提出的模型中, CTL细胞的动力学满足以下方程

$ \dot H\left( t \right) = c{T_A}\left( {t - \tau } \right)H\left( {t - \tau } \right) - bH\left( t \right). $

实际上, CTL降低了前病毒载量, 但这种减少也意味着对CTL增殖的刺激较少^[10]. 因此, 考虑饱和免疫反应是合理的^[10-11]. 文献[11]使用了$ H /(1+\varepsilon H)$作为CTL反应函数, 其中$ \varepsilon $为非零常数. 此外, 当病原体负荷过高时, CTL免疫反应会受到抑制甚至破坏^[12-13]. 为了更好地了解病毒和免疫系统之间的相互作用, 在病毒模型中考虑免疫损害更加合理^[14]. Regoes等^[14]研究了具有免疫损害项$ n{T_A}H $的病毒动力学系统, 其中$ n $是CTL免疫反应损害率系数. 随着$ n $的增大, CTL反应逐渐减弱.

基于上述工作, 本文研究免疫时滞、饱和CTL免疫反应和免疫损害对HTLV-I感染动力学的影响, 为此考虑以下模型

(1.2) $ \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} $

在系统(1.2)中, 我们假设感染细胞的移除率大于有丝分裂的速率, 即$ {\mu _3}+{\delta_2}>{\gamma_1} $, 其余参数的生物学意义与模型(1.1)相同, 并假设$ c>n $.

在相空间$ {{\Bbb R}} \times {{\Bbb R}} \times C \times C $中研究系统(1.2), 其中$ C = C\left( {\left[ { - \tau , 0} \right], {{\Bbb R}} } \right) $是将连续函数从$ {\left[ { - \tau , 0} \right]} $映到$ {{\Bbb R}} $的Banach空间. 当$ \phi \in C $时, 其范数定义为$ \left\| \phi \right\| = {\sup \limits_{ - \tau \le \theta \le 0}}\left| {\phi \left( \theta \right)} \right| $. $ C $的非负锥定义为$ {C^ + } = C\left( {\left[ { - \tau , 0} \right], {{{\Bbb R}} _ + }} \right) $. 系统(1.2) 满足以下初始条件

(1.3) $ \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} $

由泛函微分方程的基本理论^[15]可知, 在初始条件(1.3)下, 系统(1.2)的解存在且唯一.

容易证明, 系统(1.2)满足初始条件(1.3)的解是正的且最终有界, 集合

$ \Omega = \left\{ {\left( {T, {T_L}, {T_A}, H} \right) \in {{\Bbb R}} _ + ^4:0 \le T\left( t \right) + {T_L}\left( t \right) + {T_A}\left( t \right) \le \frac{\Lambda }{\sigma }, H\left( t \right) \le \frac{{c\Lambda }}{{b\varepsilon \sigma }}} \right\} $

是系统(1.2)的正向不变集.

系统(1.2)始终存在病毒未感染平衡点$E_0\left(\Lambda / \mu_1, 0,0,0\right)$利用下一代矩阵的方法^[16]计算可得系统(1.2)的免疫未激活再生率

$ {\mathfrak{R}_0} = \frac{{{\mu _2}\left( {\beta \Lambda + {\gamma _1}{\mu _1}} \right)}}{{{\mu _1}\left( {{\mu _2} + {\delta _1}} \right)\left( {{\mu _3} + {\delta _2}} \right)}}. $

若$ \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

$ \begin{array}{c} {T_1} = \frac{{\left( {{\mu _2} + {\delta _1}} \right)\left( {{\mu _3} + {\delta _2}} \right) - {\gamma _1}{\mu _2}}}{{{\mu _2}\beta }}, \;\;{T_{L1}} = \frac{{\left( {{\mu _3} + {\delta _2}} \right){T_{A1}}}}{{{\mu _2}}}, \\{T_{A1}} = \frac{{{\mu _1}\left( {{\mu _2} + {\delta _1}} \right)\left( {{\mu _3} + {\delta _2}} \right)\left( {{\mathfrak{R}_0} - 1} \right)}}{{\beta \left[ {\left( {{\mu _2} + {\delta _1}} \right)\left( {{\mu _3} + {\delta _2}} \right) - {\gamma _1}{\mu _2}} \right]}}. \end{array}$

定义

$ {\mathfrak{R}_1} = \frac{{c - n}}{b}{T_{A1}} = \frac{{\left( {c - n} \right){\mu _1}\left( {{\mu _2} + {\delta _1}} \right)\left( {{\mu _3} + {\delta _2}} \right)\left( {{\mathfrak{R}_0} - 1} \right)}}{{b\beta \left[ {\left( {{\mu _2} + {\delta _1}} \right)\left( {{\mu _3} + {\delta _2}} \right) - {\gamma _1}{\mu _2}} \right]}}, $

称$ {\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

(2.1) $ \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} $

由(2.1)式的第四个方程可得

(2.2) $ \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} $

根据(2.4)式的前三个方程和(2.2)式, 有

(2.3) $ \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} $

由(2.2)式和(2.3)式可知, $ H $是关于变量$ {T_A} $的增函数; $ {T_A} $是关于变量$ H $的减函数. 通过计算可得, $ f_1(b /(c-n))$, $ {f_2}\left( 0 \right) = {T_{A1}} $. 当$ {\mathfrak{R}_1}>1 $时, 有$ T_{A 1}>b /(c-n)$. 因此, 当$ H \in(0,(c-n) / \varepsilon n)$时, 函数$ {f_1} $和$ {f_2} $的图像在第一象限只有一个交点$ (H_2, {T_{A2}}) $ (见图 1). 由(2.1)式可知

$ {T_2} = \frac{\Lambda }{{{\mu _1} + \beta {T_{A2}}}}, \ {H_2} = \frac{1}{\varepsilon }\left( {\frac{{c{T_{A2}}}}{{b + n{T_{A2}}}} - 1} \right), \ {T_{L2}} = \frac{{\left( {{\mu _3} + {\delta _2} + p{H_2}} \right){T_{A2}}}}{{{\mu _2}}}, $

$ {{T_{A2}}} $是下列方程的唯一正解

(2.4) $ \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} $

解得$ {T_{A2}} = \frac{{ - Q + \sqrt \Delta }}{{2P}} $, 其中

$ \begin{array}{c} P = \frac{{\left[ {\left( {{\mu _2} + {\delta _1}} \right)\left( {{\mu _3} + {\delta _2}} \right) - {\mu _2}{\gamma _1}} \right]n\beta }}{{{\mu _2}}} + \frac{{\left( {{\mu _2} + {\delta _1}} \right)\left( {c - n} \right)p\beta }}{{\varepsilon {\mu _2}}} > 0, \\ Q = \frac{{\left[ {\left( {{\mu _2} + {\delta _1}} \right)\left( {{\mu _3} + {\delta _2}} \right) - {\mu _2}{\gamma _1}} \right]\left( {n{\mu _1} + b\beta } \right)}}{{{\mu _2}}} + \frac{{p\left( {{\mu _2} + {\delta _1}} \right)\left[ {{\mu _1}\left( {c - n} \right) - b\beta } \right]}}{{\varepsilon {\mu _2}}} - n\beta, \\ S = \frac{{b{\mu _1}\left( {{\mu _2} + {\delta _1}} \right)\left( {{\mu _3} + {\delta _2}} \right)\left( {1 - {R_0}} \right)}}{{{\mu _2}}} - \frac{{\left( {{\mu _2} + {\delta _1}} \right)pb{\mu _1}}}{{\varepsilon {\mu _2}}} < 0, \\ S = \frac{{b{\mu _1}\left( {{\mu _2} + {\delta _1}} \right)\left( {{\mu _3} + {\delta _2}} \right)\left( {1 - {R_0}} \right)}}{{{\mu _2}}} - \frac{{\left( {{\mu _2} + {\delta _1}} \right)pb{\mu _1}}}{{\varepsilon {\mu _2}}} < 0, \\ \Delta = {Q^2} - 4PS. \end{array} $

由

$ \sqrt \Delta - Q \ge \sqrt \Delta - \vert Q \vert = \frac{{ - 4PS}}{{\sqrt \Delta + \vert Q \vert}}, $

易知$ {T_{A2}} > 0 $, $ {T_2}>0 $, $ {T_{L2}}>0 $, $ {H_2}>0 $.

本节将研究系统(1.2)的局部动力学行为与$ {E_2} $处Hopf分支的存在性.

定理3.1   当$ \mathfrak{R}_0 < 1 $时, 系统(1.2)的病毒未感染平衡点$ {E_0} $是局部渐近稳定的; 当$ \mathfrak{R}_0 >1 $时, $ {E_0} $不稳定.

证   系统(1.2)在$ {E_0} $处的特征方程为

(3.1) $ \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} $

显然, (3.1)式有两个负实根$ \lambda _0^* = - {\mu _1} $和$ \lambda _0^{**} = - b $, 其余的根由以下方程确定

(3.2) $ \begin{equation} \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). \end{equation} $

下证当$ \mathfrak{R}_0 < 1 $时, 方程(3.2)的根均具有负实部. 若否, 则方程(3.2)存在根$ \mathop \lambda \nolimits_0 = {\mathop{\rm Re}\nolimits} \mathop \lambda \nolimits_0 + {\rm i}{\mathop{\rm Im}\nolimits} \mathop \lambda \nolimits_0 $, $ {\mathop{\rm Re}\nolimits} \mathop \lambda \nolimits_0 \ge 0 $, 此时有

(3.3) $ \begin{equation} \vert {\left( {{\lambda _0} + {\mu _2} + {\delta _1}} \right)\left( {{\lambda _0} + {\mu _3} + {\delta _2}} \right)} \vert \ge \vert {\left( {{\mu _2} + {\delta _1}} \right)\left( {{\mu _3} + {\delta _2}} \right)} \vert. \end{equation} $

注意到, 当$ \mathfrak{R}_0 < 1 $时, 有$ \left|\mu_2\left(\beta \Lambda / \mu_1+\gamma_1\right)\right| <\left|\left(\mu_2+\delta_1\right)\left(\mu_3+\delta_2\right)\right|$, 与(3.3)式矛盾. 因此, 当$ \mathfrak{R}_0 < 1 $时, 方程(3.1)的根均具有负实部, 从而$ {E_0} $是局部渐近稳定的.

定义

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

显然, $ f\left( \lambda \right) $是关于$ \lambda $的连续函数. 若$ \mathfrak{R}_0 > 1 $, 有

$ f(0)=\left(\mu_2+\delta_1\right)\left(\mu_3+\delta_1\right)-\mu_2\left(\beta \Lambda / \mu_1+\gamma_1\right) <0 ;$

当$ \lambda \to +\infty $时, 有$ f\left( \lambda \right) \to +\infty $. 因此, 当$ \lambda \to +\infty $时, 函数$ f(\lambda) $至少存在一个正零点，从而方程(3.2)至少存在一个正实根, 故$ {E_0} $是不稳定的.   证毕.

定理3.2   当$ \mathfrak{R}_1 < 1 < \mathfrak{R}_0 $时, 免疫未激活平衡点$ {E_1} $是局部渐近稳定的.

证   系统(1.2)在$ {E_1} $处的特征方程为

(3.5) $ \begin{equation} {g_1}\left( \lambda \right){g_2}\left( \lambda \right) = 0, \end{equation} $

其中

$ \begin{eqnarray*} {g_1}\left( \lambda \right) &=& \lambda + b + n{T_{A1}} - c{T_{A1}}{{\rm e}^{ - \lambda \tau }}, \\ {g_2}\left( \lambda \right) &=& \left( {\lambda + {\mu _1} + \beta {T_{A1}}} \right)\left( {\lambda + {\mu _2} + {\delta _1}} \right)\left( {\lambda + {\mu _3} + {\delta _2}} \right) + {\mu _2}{\beta ^2}{T_1}{T_{A1}} {\nonumber} \\ & & - {\mu _2}\left( {\beta {T_1} + {\gamma _1}} \right)\left( {\lambda + {\mu _1} + \beta {T_{A1}}} \right). \end{eqnarray*} $

以下证明方程

(3.6) $ \begin{equation} {g_1}\left( \lambda \right) =\lambda + b + n{T_{A1}} - c{T_{A1}}{{\rm e}^{ - \lambda \tau }}=0 \end{equation} $

的根均具有负实部. 若否, 则方程(3.6)存在根$ \mathop \lambda \nolimits_1 = {\mathop{\rm Re}\nolimits} \mathop \lambda \nolimits_1 + {\rm i}{\mathop{\rm Im}\nolimits} \mathop \lambda \nolimits_1 $, $ {\mathop{\rm Re}\nolimits} \mathop \lambda \nolimits_1 \ge 0 $. 当$ \mathfrak{R}_1 < 1 < \mathfrak{R}_0 $时, 有

$ \vert {{\lambda _1} + b + n{T_{A1}}} \vert \ge b + n{T_{A1}}, \;\;\vert {c{T_{A1}}{{\rm e}^{ - {\lambda _1}\tau }}} \vert \le \vert {c{T_{A1}}} \vert < b + n{T_{A1}}, $

与(3.6)式矛盾. 因此, (3.6)式的所有根都具有负实部. (3.5)式的其余根由以下方程确定

(3.7) $ \begin{equation} {g_2}\left( \lambda \right) ={\lambda ^3} + {k_1}{\lambda ^2} + {k_2}\lambda + {k_3} = 0, \end{equation} $

其中

$ \begin{array}{c} {k_1} = {\mu _1} + {\mu _2} + {\delta _1} + {\mu _3} + {\delta _2} + \beta {T_{A1}}>0, \\{k_2} = \left( {{\mu _1} + \beta {T_{A1}}} \right)\left( {{\mu _3} + {\delta _2} + {\mu _2} + {\delta _1}} \right) > 0, \\ {k_3} = \beta {T_{A1}}\left[ {\left( {{\mu _3} + {\delta _2}} \right)\left( {{\mu _2} + {\delta _1}} \right) - {\mu _2}{\gamma _1}} \right] > 0. \end{array} $

通过计算得到

$ \begin{eqnarray*} {k_1}{k_2} - {k_3} &=& \left( {{\mu _1} + \beta {T_{A1}}} \right)\left( {{\mu _3} + {\delta _2} + {\mu _2} + {\delta _1}} \right)\left( {{\mu _2} + {\delta _1} + {\mu _1} + \beta {T_{A1}}} \right)\\ & &+ \left( {{\mu _3} + {\delta _2}} \right)\left[ {\left( {{\mu _1} + \beta {T_{A1}}} \right)\left( {{\mu _3} + {\delta _2}} \right) + {\mu _1}\left( {{\mu _2} + {\delta _1}} \right)} \right] + {\mu _2}{\gamma _1}\beta {T_{A1}}\\ & >& 0. \end{eqnarray*} $

由Routh-Hurwitz判据知, (3.7)式的所有根都有负实部. 因此, 当$ \mathfrak{R}_1 < 1 < \mathfrak{R}_0 $时, $ {E_1} $是局部渐近稳定的.  证毕.

系统(1.2)在$ {E_2} $处的特征方程为

(3.8) $ \begin{equation} {{\lambda ^4} + {l_3}{\lambda ^3} + {l_2}{\lambda ^2} + {l_1}\lambda + {l_0}} + \left( {{q_3}{\lambda ^3} + {q_2}{\lambda ^2} + {q_1}\lambda + {q_0}} \right){{\rm e}^{ - \lambda \tau }} = 0, \end{equation} $

其中,

$ \begin{eqnarray*} {l_0}& =& {\mu _2}{\beta ^2}{T_2} {T_{A2}}\left( {b + n{T_{A2}}} \right) - pn{T_{A2}}{H_2}\left( {{\mu _1} + \beta {T_{A2}}} \right)\left( {{\mu _2} + {\delta _1}} \right), \\ {l_1} &=& \left( {{\mu _1} + \beta {T_{A2}}} \right)\left( {b + n{T_{A2}}} \right)\left( {{\mu _2} + {\delta _1} + {\mu _3} + {\delta _2} + p{H_2}} \right)\\ & &- pn{T_{A2}}{H_2}\left( {{\mu _1} + \beta {T_{A2}} + {\mu _2} + {\delta _1}} \right) + \beta {T_2}{\mu _2}\beta {T_{A2}}, \\ {l_2} &=& \left( {{\mu _1} + \beta {T_{A2}} + {\mu _2} + {\delta _1} + {\mu _3} + {\delta _2} + p{H_2}} \right)\left( {b + n{T_{A2}}} \right)\\ & &+ \left( {{\mu _1} + \beta {T_{A2}}} \right)\left( {{\mu _2} + {\delta _1} + {\mu _3} + {\delta _2} + p{H_2}} \right) - pn{T_{A2}}{H_2}, \\ {l_3}& =& {\mu _1} + \beta {T_{A2}} + {\mu _2} + {\delta _1} + {\mu _3} + {\delta _2} + p{H_2} + b + n{T_{A2}}, \\ {q_0} &=& \frac{{cp{T_{A2}}{H_2}}}{{1 + \varepsilon {H_2}}}\left( {{\mu _1} + \beta {T_{A2}}} \right)\left( {{\mu _2} + {\delta _1}} \right) - \frac{{c{\mu _2}{\beta ^2}}}{{{{\left( {1 + \varepsilon {H_2}} \right)}^2}}}{T_2}{T_{A2} ^2}, \\ {q_1} &=& \frac{{cp{T_{A2}}{H_2}}}{{1 + \varepsilon {H_2}}}\left( {{\mu _1} + \beta {T_{A2}} + {\mu _2} + {\delta _1}} \right)\\ & &- \frac{{c{T_{A2}}}}{{{{\left( {1 + \varepsilon {H_2}} \right)}^2}}}\left( {{\mu _1} + \beta {T_{A2}}} \right)\left( {{\mu _2} + {\delta _1} + {\mu _3} + {\delta _2} + p{H_2}} \right), \\ {q_2}& =& \frac{{cp{T_{A2}}{H_2}}}{{1 + \varepsilon {H_2}}} - \frac{{c{T_{A2}}}}{{{{\left( {1 + \varepsilon {H_2}} \right)}^2}}}\left( {{\mu _1} + \beta {T_{A2}} + {\mu _2} + {\delta _1} + {\mu _3} + {\delta _2} + p{H_2}} \right), \\ {q_3} &=& - \frac{{c{T_{A2}}}}{{{{\left( {1 + \varepsilon {H_2}} \right)}^2}}}. \end{eqnarray*} $

情形1   当$ \tau = 0 $时, (3.8)式可化为

(3.9) $ \begin{equation} {\lambda ^4} + {m_1}{\lambda ^3} + {m_2}{\lambda ^2} + {m_3}\lambda + {m_4} = 0, \end{equation} $

其中,

$ \begin{eqnarray*} {m_1}& =& {\mu _1} + \beta {T_{A2}} + {\mu _2} + {\delta _1} + {\mu _3} + {\delta _2} + p{H_2} + b + n{T_{A2}} - \frac{{c{T_{A2}}}}{{{{\left( {1 + \varepsilon {H_2}} \right)}^2}}}, \\ {m_2} &=& \left( {{\mu _1} + \beta {T_{A2}} + {\mu _2} + {\delta _1} + {\mu _3} + {\delta _2} + p{H_2}} \right) \Big( {b + n{T_{A2}} - \frac{{c{T_{A2}}}}{{{{\left( {1 + \varepsilon {H_2}} \right)}^2}}}} \Big)\\ & &+ \left( {{\mu _1} + \beta {T_{A2}}} \right)\left( {{\mu _2} + {\delta _1} + {\mu _3} + {\delta _2} + p{H_2}} \right) + pb{H_2}, \\ {m_3}& =& \left( {{\mu _1} + \beta {T_{A2}}} \right)\left( {{\mu _2} + {\delta _1} + {\mu _3} + {\delta _2} + p{H_2}} \right)\Big( {b + n{T_{A2}} - \frac{{c{T_{A2}}}}{{{{\left( {1 + \varepsilon {H_2}} \right)}^2}}}} \Big) \\ &&+ \left( {{\mu _1} + \beta {T_{A2}} + {\mu _2} + {\delta _1}} \right)pb{H_2}, \\ {m_4} &=& \left( {{\mu _1} + \beta {T_{A2}}} \right)\left( {{\mu _2} + {\delta _1}} \right)pb{H_2} + \beta {T_2}{\mu _2}\beta {T_{A2}}\Big( {b + n{T_{A2}} - \frac{{c{T_{A2}}}}{{{{\left( {1 + \varepsilon {H_2}} \right)}^2}}}} \Big). \end{eqnarray*} $

显然, $ {m_i} > 0 \left( {i = 1, 2, 3, 4} \right) $. 计算可得$ {m_1}{m_2} - {m_3}>0 $, $ \left( {{m_1}{m_2} - {m_3}} \right){m_3} - m_1^2{m_4} > 0 $. 故(3.9)式的所有根均具有负实部. 因此, 当$ \tau = 0 $时, $ {{E_2}} $是局部渐近稳定的.

情形2   当$ \tau > 0 $时, 假设$ \lambda = {\rm i}\omega \left( {\omega > 0} \right) $是(3.8)式的纯虚根, 分离实虚部可得

(3.10) $ \begin{equation} \begin{array}{l} \left( {{q_3}{\omega ^3} - {q_1}\omega } \right)\sin \left( {\omega \tau } \right) - \left( {{q_0} - {q_2}{\omega ^2}} \right)\cos \left( {\omega \tau } \right) = {\omega ^4} - {l_2}{\omega ^2} + {l_0}, \\ \left( {{q_1}\omega - {q_3}{\omega ^3}} \right)\cos \left( {\omega \tau } \right) - \left( {{q_0} - {q_2}{\omega ^2}} \right)\sin \left( {\omega \tau } \right) = {l_3}{\omega ^3} - {l_1}\omega. \end{array} \end{equation} $

将(3.10)式的两个方程两边同时平方再相加, 得到

(3.11) $ \begin{equation} {\omega ^8} + {h_3}{\omega ^6} + {h_2}{\omega ^4} + {h_1}{\omega ^2} + {h_0} = 0. \end{equation} $

其中,

$ \begin{array}{c} {h_0} = l_0^2 - q_0^2, \qquad {h_1} = l_1^2 - 2{l_0}{l_2} + 2{q_0}{q_2} - q_1^2, \\{h_2} = 2{l_0} + l_2^2 - 2{l_1}{l_3} + 2{q_1}{q_3} - q_2^2, \qquad {h_3} = l_3^2 - 2{l_2} - q_3^2.\end{array} $

令$ s = {\omega ^2} $, (3.11)可化为

(3.12) $ \begin{equation} f\left( s \right) \buildrel \Delta \over = {s^4} + {h_3}{s^3} + {h_2}{s^2} + {h_1}s + {h_0} = 0, \end{equation} $

则

(3.13) $ \begin{equation} f'\left( s \right) = 4{s^3} + 3{h_3}{s^2} + 2{h_2}s + {h_1}. \end{equation} $

定义

$ {P_0} = \frac{{8{h_2} - 3h_3^2}}{{16}}, \;\;{Q_0} = \frac{{h_3^3 - 4{h_2}{h_3} + 8{h_1}}}{{32}}, \;\;{D_0} = \frac{{{{Q_0}^2}}}{4} + \frac{{{{P_0}^3}}}{{27}}, \;\;\sigma = \frac{{ - 1 + \sqrt 3 {\rm i}}}{2}. $

由卡丹公式可知方程(3.13)的最大实根有以下情形

当$ {D_0} > 0 $时,

$ s_1^* = - \frac{{{h_3}}}{4} + \sqrt[3]{{ - \frac{Q_0}{2} + \sqrt {D_0} }} + \sqrt[3]{{ - \frac{Q_0}{2} - \sqrt {D_0} }}; $

当$ {D_0} = 0 $时,

$ s_2^* = \max \left\{ { - \frac{{{h_3}}}{4} - 2\sqrt[3]{{ \frac{Q_0}{2}}}, - \frac{{{h_3}}}{4} + \sqrt[3]{{ \frac{Q_0}{2}}}} \right\}; $

当$ {D_0} < 0 $时,

$ s_3^* = \max \left\{ { - \frac{{{h_3}}}{4} + 2{\mathop{\rm Re}\nolimits} \left\{ \xi \right\}, - \frac{{{h_3}}}{4} + 2{\mathop{\rm Re}\nolimits} \left\{ {\xi \sigma } \right\}, - \frac{{{h_3}}}{4} + 2{\mathop{\rm Re}\nolimits} \left\{ {\xi \bar \sigma } \right\}} \right\}, $

其中$ \xi = \sqrt[3]{{ - \frac{{{Q_0}}}{2} + \sqrt {{D_0}} }} $.

与文献[17]的讨论类似，得到下列引理.

引理3.1   对于方程(3.12), 下列结论成立

(ⅰ) 若$ {h_0} < 0 $, (3.12)式至少存在一个正实根.

(ⅱ) 若$ {h_0} \ge 0 $, 则当下列任一条件成立时, (3.12)式没有正实根

1) $ {D_0} > 0 $, $ s_1^* < 0; $

2) $ {D_0} = 0 $, $ s_2^* < 0; $

3) $ {D_0} < 0 $, $ s_3^* < 0. $

(ⅲ) 若$ {h_0} \ge 0 $, 则当下列任一条件成立时, (3.12)式至少存在一个正实根

1) $ {D_0} > 0 $, $ s_1^* > 0 $, $ f\left( {s_1^*} \right) < 0 $;

2) $ {D_0} = 0 $, $ s_2^* > 0 $, $ f\left( {s_2^*} \right) < 0 $;

3) $ {D_0} < 0 $, $ s_3^* > 0 $, $ f\left( {s_3^*} \right) < 0. $

假设(3.12)有四个正实根, 记为$ {s_k} \left( {k = 1, 2, 3, 4} \right) $, 那么(3.11)式存在四个正实根$ {\omega _k} = \sqrt {{s_k}} $$ \left( {k = 1, 2, 3, 4} \right) $. 由(3.10)式可得

$ \tau _k^{\left( j \right)} = \frac{1}{{{\omega _k}}}\arcsin \left[ {\frac{{\left( {\omega _k^4 - {l_2}\omega _k^2 + {l_0}} \right)\left( {{q_3}\omega _k^3 - {q_1}{\omega _k}} \right) + \left( {{l_3}\omega _k^3 - {l_1}{\omega _k}} \right)\left( {{q_2}\omega _k^2 - {q_0}} \right)}}{{{{\left( {{q_1}{\omega _k} - {q_3}\omega _k^3} \right)}^2} + {{\left( {{q_0} - {q_2}\omega _k^2} \right)}^2}}} + 2\pi j} \right], $

其中, $ k = 1, 2, 3, 4 $, $ j = 0, 1, 2, \cdots $. 因此, 当$ \tau = \tau _k^{\left( j \right)} $时, $ \pm {\rm i}{\omega _k} $是方程(3.11)的纯虚根. 假设(3.8)式存在根$ \lambda \left( \tau \right) = \eta \left( \tau \right) + {\rm i}\omega \left( \tau \right) $, 且满足$ \eta \left( {{\tau _0}} \right) = 0, \omega \left( {{\tau _0}} \right) = {\omega _0} $. 令

(3.14) $ \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} $

对方程(3.8)关于$ \tau $求导, 得到

(3.15) $ \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} $

由(3.10)和(3.15)式可知

(3.16) $ \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} $

基于上述讨论和文献[18–19], 得到下面的结论.

定理3.3   对于系统(1.2), 当$ {\mathfrak{R}_1}>1 $时,

(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.

本节将通过构造适当的Lyapunov泛函并利用LaSalle不变性原理^[15]研究各可行平衡点的全局渐近稳定性.

定理4.1   若$ \mathfrak{R}_0 < 1 $, 则$ {E_0} $是全局渐近稳定的.

证   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

(4.1) $ \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} $

其中$T_0=\Lambda / \mu_1$. 沿着系统(1.2)的解计算$ \mathop V\nolimits_0 \left( t \right) $的全导数可得

(4.2) $ \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} $

若$ \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 $是$ \left\{ {\left( {T\left( t \right), {T_L}\left( t \right), {T_A}\left( t \right), H} \right):{{\dot V}_0} = 0} \right\} $的最大不变子集. 由定理3.1和LaSalle不变性原理^[15]知, 当$ \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.

证   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

(4.3) $ \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

(4.4) $ \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 $是$ \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\} $的最大不变子集. 由定理3.2和LaSalle不变性原理^[15]知, 当$ {\mathfrak{R}_1}<1<{\mathfrak{R}_0} $时, $ {{E_1}} $是全局渐近稳定的. 证毕.

定理4.3   若$ {\mathfrak{R}_1} > 1 $, 则当$ \tau = 0 $时, $ {E_2} $是全局渐近稳定的.

证   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

(4.5) $ \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} $

沿着系统(1.2)的解计算$ \mathop V\nolimits_2 \left( t \right) $的全导数可得

(4.6) $ \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} $

由均值不等式可知, $ {{\dot V}_2}\left( t \right) \le 0 $, 当且仅当$ T = {T_2} $, $ {T_L} = {T_{L2}} $, $ {T_A} = {T_{A2}} $, $ H = {H_2}\mbox{时等号成立.} $易证$ {M_2} = \left\{ {{E_2}} \right\} \subset \Omega $是$ \left\{ {\left( {T\left( t \right), {T_L}\left( t \right), {T_A}\left( t \right), H\left( t \right)} \right):{{\dot V}_2}\left( t \right) = 0} \right\} $的最大不变子集. 由定理3.3和LaSalle不变性原理^[15]知, 当$ {\mathfrak{R}_1}>1 $且$ \tau=0 $时, $ {{E_2}} $是全局渐近稳定的. 证毕.

本节利用Matlab软件对免疫激活平衡点$ {{E_2}} $的动力学行为进行数值模拟, 并对再生率进行参数敏感性分析.

选取参数值$ \Lambda=2 $, $ \beta=0.001 $, $ \mu_1=0.01 $, $ \mu_2=0.08 $, $ \mu_3=0.01 $, $ \delta_1=0.05 $, $ \delta_2=0.02 $, $ \gamma_1=0.03 $, $ p=0.029 $, $ b=0.04 $, $ c=0.036 $^[20], 且假设$ n = 0.005 $, $ \varepsilon = 0.01 $. 选取初值为$ \left( {160, 2, 1.5, 1} \right) $. 通过计算可得免疫激活再生率$ {\mathfrak{R}_1}=74.9167 >1 $, $ {{\tau}_0}=3.9161 $, 免疫激活平衡点

$ {E_2}=\left( {176.4186, 1.9926, 1.3367, 3.0778} \right). $

由定理3.3知, 若$ \tau <{{\tau}_0} $, $ E_2 $局部渐近稳定, 如图 2(a)所示; 若$ \tau >{{\tau}_0} $, $ E_2 $不稳定, 当时间$ t $趋于无穷时, 系统(1.2)的数值解趋于一个周期解, 如图 2(b)所示. 图 3表明Hopf分支的存在性.

由于参数的不确定性, 有必要研究再生率$ {\mathfrak{R}_0} $和$ {\mathfrak{R}_1} $对参数的敏感程度. 对量$ R $定义关于参数$ q $的弹性^[21]

$ \varepsilon _R^q = \frac{{\partial R}}{{\partial q}}\frac{q}{R}. $

根据$ {\mathfrak{R}_0} $和$ {\mathfrak{R}_1} $的表达式, 求得

$ \begin{array}{c} \varepsilon _{{\mathfrak{R}_0}}^\beta = \beta \Lambda \varsigma > 0, \qquad \varepsilon _{{\mathfrak{R}_1}}^\beta = \frac{{{\mu _1}\left( {c - n} \right)}}{{b\beta {\mathfrak{R}_1}}} > 0, \\ \varepsilon _{{\mathfrak{R}_0}}^{{\gamma _1}} = {\gamma _1}\varsigma > 0, \qquad\varepsilon _{{\mathfrak{R}_1}}^{{\gamma _1}} = {\mu _2}{\gamma _1}\vartheta > 0, \\ \varepsilon _{{\mathfrak{R}_0}}^{{\mu _1}} = - \beta \Lambda \varsigma < 0, \qquad \varepsilon _{{\mathfrak{R}_1}}^{{\mu _1}} = - \frac{{{\mu _1}\left( {c - n} \right)}}{{b\beta {\mathfrak{R}_1}}} < 0, \\ \varepsilon _{{\mathfrak{R}_0}}^{{\mu _2}} = \frac{{\varsigma {\delta _1}\left( {\beta \Lambda + {\gamma _1}{\mu _1}} \right)}}{{{\mu _2} + {\delta _1}}} > 0, \qquad \varepsilon _{{\mathfrak{R}_1}}^{{\mu _2}} = {\delta _1}\left( {{\mu _3} + {\delta _2}} \right)\vartheta > 0, \\ \varepsilon _{{\mathfrak{R}_0}}^{{\mu _3}} = \frac{{\varsigma {\mu _3}\left( {\beta \Lambda + {\gamma _1}{\mu _1}} \right)}}{{{\mu _3} + {\delta _2}}} < 0, \qquad \varepsilon _{{\mathfrak{R}_1}}^{{\mu _3}} = - {\mu _3}\left( {{\mu _2} + {\delta _1}} \right)\vartheta < 0, \\ \varepsilon _{{\mathfrak{R}_0}}^{{\delta _1}} = - \frac{{\varsigma {\delta _1}\left( {\beta \Lambda + {\gamma _1}{\mu _1}} \right)}}{{{\mu _2} + {\delta _1}}} < 0, \qquad \varepsilon _{{\mathfrak{R}_1}}^{{\delta _1}} = - {\delta _1}\left( {{\mu _3} + {\delta _2}} \right)\vartheta < 0, \\ \varepsilon _{{\mathfrak{R}_0}}^{{\delta _2}} = \frac{{\varsigma {\delta _2}\left( {\beta \Lambda + {\gamma _1}{\mu _1}} \right)}}{{{\mu _3} + {\delta _2}}} < 0, \qquad \varepsilon _{{\mathfrak{R}_1}}^{{\delta _2}} = - {\delta _2}\left( {{\mu _2} + {\delta _1}} \right)\vartheta < 0, \\ \end{array} $

其中

$ \varsigma = \frac{{{\mu _2}}}{{{\mu _1}\left( {{\mu _2} + {\delta _1}} \right)\left( {{\mu _3} + {\delta _2}} \right){\mathfrak{R}_0}}}, \qquad \vartheta = \frac{{\left( {c - n} \right){\mu _2}\Lambda }}{{b{{\left[ {\left( {{\mu _2} + {\delta _1}} \right)\left( {{\mu _3} + {\delta _2}} \right) - {\mu _2}{\gamma _1}} \right]}^2}{\mathfrak{R}_1}}}. $

由此得出结论: $ {\mathfrak{R}_0} $和$ {\mathfrak{R}_1} $均与参数$ \beta $, $ {{\gamma}_1} $和$ {{\mu}_2} $正相关, 与参数$ {{\mu}_1} $, $ {{\mu}_3} $, $ {{\delta}_1} $和$ {{\delta}_2} $负相关. 选取$ \Lambda=1 $, $ \beta=0.001 $, $ \mu_1=0.01 $, $ \mu_2=0.03 $, $ \mu_3=0.01 $, $ \delta_1=0.03 $, $ \delta_2=0.02 $, $ \gamma_1=0.02 $, $ \varepsilon=0.01 $, $ p=0.029 $, $ b=0.04 $, $ c=0.036 $, $ n=0.005 $, 有

$ \vert {\varepsilon _{{\mathfrak{R}_0}}^\beta } \vert = \vert {\varepsilon _{{\mathfrak{R}_0}}^{{\mu _1}}} \vert > \vert {\varepsilon _{{\mathfrak{R}_0}}^{{\delta _2}}} \vert > \vert {\varepsilon _{{\mathfrak{R}_0}}^{{\mu _2}}} \vert = \vert {\varepsilon _{{\mathfrak{R}_0}}^{{\delta _1}}} \vert > \vert {\varepsilon _{{\mathfrak{R}_0}}^{{\mu _3}}} \vert > \vert {\varepsilon _{{\mathfrak{R}_0}}^{{\gamma _1}}} \vert, \\ \vert {\varepsilon _{{\mathfrak{R}_1}}^{{\delta _2}}} \vert > \vert {\varepsilon _{{\mathfrak{R}_1}}^{{\mu _2}}} \vert = \vert {\varepsilon _{{\mathfrak{R}_1}}^{{\delta _1}}} \vert > \vert {\varepsilon _{{\mathfrak{R}_1}}^{{\mu _3}}} \vert = \vert {\varepsilon _{{\mathfrak{R}_1}}^{{\gamma _1}}} \vert > \vert {\varepsilon _{{\mathfrak{R}_1}}^\beta } \vert = \vert {\varepsilon _{{\mathfrak{R}_1}}^{{\mu _1}}} \vert, $

表明相比于其他参数, $ \beta $和$ {{\mu}_1} $对$ {\mathfrak{R}_0} $影响最大, 对$ {\mathfrak{R}_1} $影响最小. $ {\delta_2} $对$ {\mathfrak{R}_1} $影响最大, $ {\gamma_1} $对$ {\mathfrak{R}_0} $影响最小.

本文研究了一类具有饱和CTL免疫反应、免疫损害和免疫时滞的HTLV-I感染模型. 通过计算, 推导出系统(1.2)的可行平衡点, 免疫未激活再生率和免疫激活再生率. 此外, 给出了每个平衡点稳定的条件: 当$ {\mathfrak{R}_0}<1 $时, $ {E_0} $全局渐近稳定; 当$ {\mathfrak{R}_1}<1<{\mathfrak{R}_0} $时, $ {E_1} $全局渐近稳定; 当$ \tau=0 $, $ {\mathfrak{R}_1}>1 $时, $ {E_2} $全局渐近稳定. 对于免疫激活平衡点$ {E_2} $, 将免疫时滞$ {\tau} $作为分支参数, 计算得到临界值$ {\tau}_0 $. 当$ {\tau}>{\tau_0} $时, $ {E_2} $局部渐近稳定; 当$ {\tau}>{\tau_0} $时, $ {E_2} $不稳定; 当$ {\tau}={\tau_0} $时, $ {E_2} $处会出现Hopf分支. 这意味着免疫时滞会改变免疫激活平衡点的稳定性.