## 一类具有抗原性的肿瘤-免疫系统的定性分析

1 陕西科技大学数学系 西安 710021

2 西安医学院医学免疫教研室 西安 710021

## A Qualitative Analysis of a Tumor-Immune System with Antigenicity

Xie Xin,1, Li Jianquan,1, Wang Yuping1, Zhang Dian2

1 Department of Mathematics, Shaanxi University of Science and Technology, Xi'an 710021

2 Department of Immunology, Xi'an Medical University, Xi'an 710021

 基金资助: 国家自然科学基金.  11971281国家自然科学基金.  12071268西安医学院科研基金.  2018GJFY05

 Fund supported: the NSFC.  11971281the NSFC.  12071268the Scientific Research Fund of Xi'an Medical University.  2018GJFY05

Abstract

In this paper, we propose and investigate a tumor-immune system interaction model with antigenicity. The existence of equilibria of the model is determined, and the local dynamics of each feasible equilibrium is analyzed. The global dynamics of the model is obtained by excluding the existence of periodic solutions. It is found that, under certain conditions, the saddle-node bifurcation and the bi-stability of strong equilibrium with tumor and equilibrium without tumor may occur for the model, which imply that the growth and development of the tumor will depend on its initial state. The obtained theoretical analysis results are verified by numerical simulations.

Keywords： Tumor-immune system ; Antigenicity ; Stability ; Saddle-node bifurcation

Xie Xin, Li Jianquan, Wang Yuping, Zhang Dian. A Qualitative Analysis of a Tumor-Immune System with Antigenicity. Acta Mathematica Scientia[J], 2021, 41(6): 1969-1979 doi:

## 1 引言

$$$\begin{array}{ll} { }\frac{{\rm d}E}{{\rm d}t} = s+\frac{cET}{\varepsilon+T}-\beta ET-\mu E, \\ { }\frac{{\rm d}T}{{\rm d}t} = rT\left(1-\frac{T}{K}\right)-\alpha ET, \end{array}$$$

$y\neq0$时, 将(3.2)式代入系统(3.1)的第一个方程得

$$$f(y)\triangleq(1+\eta)y-r(\delta y+1)(1-y) = \delta ry^2-(\delta r-r-\eta)y+1-r = 0.$$$

$r>1$时, 系统(2.1)存在唯一有瘤平衡点$P_1(x_1, y_1)$;

$r = 1$, $\delta>1 $$\eta<\delta-1 时, 系统(2.1)存在唯一有瘤平衡点 P_2(x_2, y_2) ; \frac{4\delta}{(\delta+1)^2}<r<1 , \delta>1$$ \eta<(\delta-1)r-2\sqrt{\delta r(1-r)}$时, 系统(2.1)存在两个不同的有瘤平衡点$P_3(x_3, y_3)$ (称为弱有瘤平衡点)和$P_4(x_4, y_4)$ (称为强有瘤平衡点);

### 4.1 无瘤平衡点的稳定性

$\eta\ne\delta$时, 将平衡点$P_0$平移到原点, 对系统(2.1)作平移变换, $u = x-1$, $v = y$, 得

$$$\left\{\begin{array}{ll} u' = (\eta-\delta)v-u-\delta uv, \\ v' = -quv-qrv^2. \end{array}\right.$$$

$$$w' = -q(1+\eta-\delta)w^2+q\delta(\eta-\delta)w^3+o(w^3).$$$

$\delta-\eta = 1$时, (4.3)式变为

$r = 1 $$\eta = \delta 时, 系统(2.1)变为 $$\left\{\begin{array}{ll} { }\frac{{\rm d}x}{{\rm d}t} = (1+\delta y)(1-x), \\ { }\frac{{\rm d}y}{{\rm d}t} = qy\left(1-x-y\right). \end{array}\right.$$ 定义函数 V = \frac{q(x-1)^2}{8}+y , 则 V 沿着系统(4.4)解的全导数为 则当 y\ge0 时, \frac{{\rm d}V}{{\rm d}t}\le 0 . 因此, 当 r = 1$$ \eta = \delta$时, $P_0$是稳定的.

### 4.2 有瘤平衡点的稳定性

$f'(y_3)<0$, 即det$J_1(P_3)<0$, 所以弱有瘤平衡点$P_3$是鞍点.

$f'(y_5) = 0$意味着det$J (P_5) = 0$, 故$P_5$是高阶平衡点. 此时, $J(P_5)$的特征值分别为$\lambda_1 = -(1+\delta y_5+qry_5) $$\lambda_2 = 0 . 其特征值对应的特征向量分别为 (1, A)^T$$ (-r, 1)^T$, 其中$A = \frac{-qry_5}{\eta-\delta x_5} = \frac{q(\delta r-r-\eta)}{\delta(\delta r+r-\eta)}>0$.

$$$\left\{\begin{array}{ll} u_1' = a_{11}u_1+a_{12}v_1+F_1(u_1, v_1), \\ v_1' = a_{21}u_1+a_{22}v_1+G_1(u_1, v_1), \end{array}\right.$$$

$$$\left\{\begin{array}{ll} z_1' = \lambda_1z_1+F_2(z_1, w_1), \\ w_1' = G_2(z_1, w_1), \end{array}\right.$$$

## 5 全局动力学性态

$\rm (1)$若无瘤平衡点存在, 无瘤平衡点$P_0$在区域$D$上是全局渐近稳定的(图 2(a)).

### 图 2

$\rm (2)$若存在唯一一个有瘤平衡点$P_1 $$P_2 时, 该平衡点在区域 D 内是全局渐近稳定的(图 2(b)(c)分别对应焦点和结点的情形). \rm (3) 若存在两个互异的有瘤平衡点 P_3$$ P_4$时, 则存在弱有瘤平衡点$P_3$的稳定流形将可行区域(第一象限)分为两部分(图 2(d)). 若初始值在下方区域, 则肿瘤最终消亡; 若初始值在上方区域, 则肿瘤最终向强有瘤平衡点$P_4$发展.

$\rm (4)$若存在有瘤平衡点$P_5$时, 则其稳定流形将可行区域(第一象限)分为两部分(图 2(e)), 使得当初始值在下方区域时, 肿瘤最终消亡; 当初始值在上方区域时, 肿瘤最终向有瘤平衡点$P_5$发展.

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