西藏自治区包虫病传播的数学建模及动力学分析

图1 包虫病在人、犬、家畜中的传播示意图

(2.1)$\begin{matrix} \left\{ \begin{array}{ll}\displaystyle \frac{{\rm d}H_{s}}{{\rm d}t}=A_{h}+\mu_{2}H_{i}+\mu_{1}qH_{e}-(1-p)\beta_{1}D_{i}H_{s}-d_{h}H_{s},\\ [3mm]\displaystyle \frac{{\rm d}H_{e}}{{\rm d}t}=(1-p)\beta_{1}D_{i}H_{s}-\mu_{1}qH_{e}-d_{h}H_{e}-(1-q)\omega H_{e}, \\[3mm]\displaystyle \frac{{\rm d}H_{i}}{{\rm d}t}=(1-q)\omega H_{e}-\mu_{2}H_{i}-d_{h}H_{i}, \\[3mm]\displaystyle \frac{{\rm d}D_{s}}{{\rm d}t}=b_{d}N_{d}-(1-p)\beta_{2}\varepsilon L_{i}D_{s}-(d+\theta)D_{s}-(b_{d}-d)\frac{N_{d}}{K}D_{s}, \\[3mm]\displaystyle \frac{{\rm d}D_{i}}{{\rm d}t}=(1-p)\beta_{2}\varepsilon L_{i}D_{s}-(d+\theta)D_{i}-(b_{d}-d)\frac{N_{d}}{K}D_{i}, \\[3mm] \displaystyle \frac{{\rm d}L_{s}}{{\rm d}t}=(d_{l}+\varepsilon)M-(1-p)\beta_{3}D_{i}L_{s}-(d_{l}+\varepsilon)L_{s}, \\[3mm]\displaystyle \frac{{\rm d}L_{i}}{{\rm d}t}=(1-p)\beta_{3}D_{i}L_{s}-(d_{l}+\varepsilon)L_{i}. \end{array} \right.\end{matrix}$

模型中的参数均为正值, 其中 $A_{h}$ 表示人口每年新增数量; $d_{h}, d_{l}$ 分别表示人, 家畜的自然死亡率; $\beta_{1}, \beta_{2}, \beta_{3}$ 分别表示人, 犬, 家畜的感染系数; $\frac{1}{\omega}$ 表示潜伏期人群转化成发病人群的平均潜伏时间; $\mu_{1}, \mu_{2}$ 分别表示潜伏人群和发病人群的治愈率系数; $q$ 表示潜伏人群接受治疗的比例; $p$ 表示地区宣传教育的影响系数; $\varepsilon$ 表示家畜的屠宰比例.

根据问题的实际含义我们仅在区域

$\Gamma=\{(H_{s},H_{e},H_{i},D_{s},D_{i},L_{s},L_{i})| H_{s}\geq0, H_{e}\geq0, H_{i}\geq0, D_{s}\geq0, D_{i}\geq0, L_{s}\geq0, L_{i}\geq0$

$H_{s}+H_{e}+H_{i}\leq\frac{A_{h}}{d_{h}}, D_{s}+D_{i}\leq\frac{K(b_{d}-d-\theta)}{b_{d}-d}, L_{s}+L_{i}=M\}$

中讨论系统 (2.1) 解的性态. 并且容易验证 $\Gamma$ 是系统 (2.1) 的正向不变域.

显然, 系统 (2.1) 总存在一个平凡平衡点 $E_{*}$ 和一个无病平衡点 $E_{0}$

$E_{*}=(\frac{A_{h}}{d_{h}},0,0,0,0,M,0), E_{0}=(\frac{A_{h}}{d_{h}},0,0,\frac{K(b_{d}-d-\theta)}{b_{d}-d},0,M,0).$

利用下一代矩阵的方法^[16], 令 $F$ 和 $V$ 分别表示新增感染项和保留转移项, 则基本再生数 $R_{0}$ 可由 $\rho(FV^{\dashv})$ 计算得出

$R_{0}=\sqrt[2]{\frac{(1-p)\beta_{3}M}{b_{d}}\cdot\frac{(1-p)\varepsilon\beta_{2}K(b_{d}-d-\theta)}{(d_{l}+\varepsilon)(b_{d}-d)}}.$

进一步, 当 $R_{0}>1$ 时系统 $(2.1)$ 存在唯一的地方病平衡点 $E^{*}=(H_{s}^{*},H_{e}^{*},H_{i}^{*},D_{s}^{*},D_{i}^{*},L_{s}^{*},L_{i}^{*})$, 其中

$H_{s}^{*}=\frac{\mu_{1}q+d_{h}+(1-q)\omega}{(1-p)\beta_{1}D_{i}^{*}}H_{e}^{*},$

$H_{e}^{*}=\frac{A_{h}(\mu_{2}+d_{h})(1-p)\beta_{1}D_{i}^{*}}{d_{h}[(1-q)\omega(1-p)\beta_{1}D_{i}^{*}+(\mu_{2}+d_{h})(\mu_{1}q+d_{h}+(1-q)\omega)+(\mu_{2}+d_{h})(1-p)\beta_{1}D_{i}^{*}]},$

$H_{i}^{*}=\frac{(1-q)\omega}{\mu_{2}+d_{h}}H_{e}^{*}, D_{s}^{*}=\frac{K(b_{d}-d-\theta)}{b_{d}-d}-D_{i}^{*}, D_{i}^{*}=\frac{(d_{l}+\varepsilon)L_{i}^{*}}{\beta_{3}(1-p)(M-L_{i}^{*})}, L_{s}^{*}=M-L_{i}^{*},$

$L_{i}^{*}=\frac{b_{d}(d_{l}+\varepsilon)(b_{d}-d)(R_{0}^{2}-1)}{(1-p)^{2}\beta_{2}\beta_{3}\varepsilon K(b_{d}-d-\theta)+(1-p)\beta_{2}\varepsilon(d_{l}+\varepsilon)(b_{d}-d)}.$

3 平衡点的稳定性

定理3.1 平凡平衡点 $E_{*}$ 总是不稳定的.

证系统 (2.1) 在 $E_{*}$ 处线性化系统的特征方程为

$(\lambda+d_{h})(\lambda+d_{l}+\varepsilon)^{2}(\lambda+d+\theta)(\lambda+\mu_{2}+d_{h})(\lambda+\mu_{1}q+d_{h}+(1-q)\omega)(\lambda-(b_{d}-d-\theta))=0.$

而 $\lambda=b_{d}-d-\theta>0$ 是其正特征根, 所以 $E_{*}$ 总是不稳定的. 证毕.

定理3.2 当 $R_{0}<1$ 时, 无病平衡点 $E_{0}=(\frac{A_{h}}{d_{h}},0,0,\frac{K(b_{d}-d-\theta)}{b_{d}-d},0,M,0)$ 全局渐近稳定;当 $R_{0}>1$ 时, $E_{0}$ 不稳定.

证系统 (2.1) 在 $E_{0}$ 处线性化系统的特征方程为

$\begin{matrix}&& (\lambda+d_{h})(\lambda+\mu_{2}+d_{h})(\lambda+\mu_{1}q+d_{h}+(1-q)\omega)(\lambda+(b_{d}-d-\theta)) (\lambda+d_{l}+\varepsilon)\\ &&\cdot [\lambda^{2}+(d_{l}+b_{d}+\varepsilon)\lambda+b_{d}(d_{l}+\varepsilon)(1-R^{2}_{0})]=0.\end{matrix}$

显然, 该特征方程有五个实根且都小于零, 另外两个根为

$ \lambda=\frac{-(d_{l}+b_{d}+\varepsilon)}{2}\pm\frac{\sqrt{(d_{l}+b_{d}+\varepsilon)^{2}-4b_{d}(d_{l}+\varepsilon)(1-R^{2}_{0})}}{2}.$

所以当 $R_{0}<1$ 时, $E_{0}$ 是局部渐近稳定; 而当 $R_{0}>1$ 时, $E_{0}$ 不稳定.

下面证明当 $R_{0}<1$ 时, $E_{0}$ 全局渐近稳定. 因为系统 (2.1) 的前三个方程独立于其余方程, 所以可将系统 (2.1) 分为两个子系统进行研究. 首先考虑后四个方程所构成的子系统

(3.1)$\begin{matrix} \left\{ \begin{array}{l} \frac{{\rm d}D_{s}}{{\rm d}t}=b_{d}N_{d}-(1-p)\beta_{2}\varepsilon L_{i}D_{s}-(d+\theta)D_{s}-(b_{d}-d)\frac{N_{d}}{K}D_{s}, \\[3mm] \frac{{\rm d}D_{i}}{{\rm d}t}=(1-p)\beta_{2}\varepsilon L_{i}D_{s}-(d+\theta)D_{i}-(b_{d}-d)\frac{N_{d}}{K}D_{i}, \\[3mm] \frac{{\rm d}L_{s}}{{\rm d}t}=(d_{l}+\varepsilon)M-(1-p)\beta_{3}D_{i}L_{s}-(d_{l}+\varepsilon)L_{s}, \\[3mm] \frac{{\rm d}L_{i}}{{\rm d}t}=(1-p)\beta_{3}D_{i}L_{s}-(d_{l}+\varepsilon)L_{i}. \end{array} \right.\end{matrix}$

由于 $\lim\limits_{ t\rightarrow \infty}N_{d}(t)=\frac{K(b_{d}-d-\theta)}{b_{d}-d}$, 所以将$ N_{d}(t)=\frac{K(b_{d}-d-\theta)}{b_{d}-d}$ 带入, 可得系统 (3.1) 的极限系统

(3.2)$\begin{matrix} \left\{ \begin{array}{l} \frac{{\rm d}D_{i}}{{\rm d}t}=(1-p)\beta_{2}\varepsilon L_{i}(\frac{K(b_{d}-d-\theta)}{b_{d}-d}-D_{i})-b_{d}D_{i}, \\[3mm] \frac{{\rm d}L_{i}}{{\rm d}t}=(1-p)\beta_{3}D_{i}(M-L_{i})-(d_{l}+\varepsilon)L_{i}. \end{array} \right.\end{matrix}$

由极限系统的理论可知, 系统 (3.1) 平衡点的稳定性等价于系统 (3.2) 平衡点的稳定性. 所以现在问题转化为讨论系统 (3.2) 的平衡点 (0,0) 的全局稳定性.

设 $(D_{i}(t),L_{i}(t))$ 为系统(3.2)在 $\Gamma$ 中的任意解, 定义 Lyapunov 函数为

$V(D_{i},L_{i})=\frac{(1-p)\beta_{3}M}{b_{d}}D_{i}+L_{i}.$

将函数 $V$ 沿系统 (3.2) 求导可得

$\begin{matrix}\frac{{\rm d}V}{{\rm d}t}&=&\frac{(1-p)\beta_{3}M}{b_{d}}((1-p)\beta_{2}\varepsilon L_{i}(\frac{K(b_{d}-d-\theta)}{b_{d}-d}-D_{i})-b_{d}D_{i})\\&&+(1-p)\beta_{3}D_{i}(M-L_{i})-(d_{l}+\varepsilon)L_{i}\\&\leq&\frac{(1-p)^{2}\beta_{2}\beta_{3}\varepsilon MK(b_{d}-d-\theta)}{b_{d}(b_{d}-d)}L_{i}-(d_{l}+\varepsilon)L_{i}\\&=&(R^{2}_{0}-1)(d_{l}+\varepsilon)L_{i}.\end{matrix}$

则当 $R_{0}<1$ 时 $\frac{{\rm d}V}{{\rm d}t}\leq0,$ 并且 $\frac{{\rm d}V}{{\rm d}t}=0$ 当且仅当 $L_{i}=0.$ 即

$ E=\{(D_{i},L_{i})|\frac{{\rm d}V}{{\rm d}t}=0\}=\{(D_{i},0)\},$

而 $(0,0)$ 是系统 (3.2) 在 $E$ 中的唯一正向不变集, 由 LaSalle's 不变集原理可知 $(0,0)$ 是全局渐近稳定的.

接下来考虑系统 (2.1) 前三个方程所构成的子系统

(3.3)$\begin{matrix} \left\{ \begin{array}{l} \frac{{\rm d}H_{s}}{{\rm d}t}=A_{h}+\mu_{2}H_{i}+\mu_{1}qH_{e}-(1-p)\beta_{1}D_{i}H_{s}-d_{h}H_{s},\\ [3mm] \frac{{\rm d}H_{e}}{{\rm d}t}=(1-p)\beta_{1}D_{i}H_{s}-\mu_{1}qH_{e}-d_{h}H_{e}-(1-q)\omega H_{e}, \\[3mm] \frac{{\rm d}H_{i}}{{\rm d}t}=(1-q)\omega H_{e}-\mu_{2}H_{i}-d_{h}H_{i}. \end{array} \right.\end{matrix}$

当 $t\rightarrow\infty$ 时 $D_{i}\rightarrow0$. 计算可得, 当 $t\rightarrow\infty$ 时 $H_{e}\rightarrow0, H_{i}\rightarrow0, H_{s}\rightarrow \frac{A_{h}}{d_{h}}.$ 故系统 (3.3) 的平衡点 $(\frac{A_{h}}{d_{h}},0,0)$ 是全局渐近稳定的.

综上所述, 当 $R_{0}<1$ 时, 系统 (2.1) 的无病平衡点 $E_{0}$ 是全局渐近稳定的.证毕.

定理3.3 当 $R_{0}>1$ 时, 地方病平衡点 $E^{*}=(H_{s}^{*},H_{e}^{*},H_{i}^{*},D_{s}^{*},D_{i}^{*},L_{s}^{*},L_{i}^{*})$ 全局渐进稳定.

证类似于上面的证明方法, 先考虑极限系统 (3.2) 的平衡点 $(D_{i}^{*},L_{i}^{*})$. 设 Lyapunov 函数为

$L(D_{i},L_{i})=\omega_{1}(D_{i}-D^{*}_{i}-D^{*}_{i}\ln\frac{D_{i}}{D^{*}_{i}})+\omega_{2}(L_{i}-L^{*}_{i}-L^{*}_{i}\ln\frac{L_{i}}{L^{*}_{i}}).$

选取 $\omega_{1}>0, $且 $\omega_{1}, \omega_{2} $满足

$ \omega_{1}\frac{(1-p)\beta_{2}\varepsilon }{D^{*}_{i}}(\frac{K(b_{d}-d-\theta)}{b_{d}-d}-D^{*}_{i})=\omega_{2}\frac{(1-p)\beta_{3}}{L^{*}_{i}}(M-L^{*}_{i}).$

显然, $L(D_{i},L_{i})$ 是正定无穷大函数, 将函数 $L$ 沿系统 (3.2) 求导可得

$\begin{matrix}\frac{{\rm d}L}{{\rm d}t}&=&\omega_{1}\frac{D_{i}-D^{*}_{i}}{D_{i}}\frac{{\rm d}D_{i}}{{\rm d}t}+\omega_{2}\frac{L_{i}-L^{*}_{i}}{L_{i}}\frac{{\rm d}L_{i}}{{\rm d}t}\\&=&\left(\begin{array}{cc}D_{i}-D^{*}_{i}~&L_{i}-L^{*}_{i}\end{array}\right)\\&&\cdot \left[\begin{array}{cc} -\omega_{1}\frac{(1-p)\beta_{2}\varepsilon K(b_{d}-d-\theta)}{D_{i}D^{*}_{i}(b_{d}-d)}L_{i}~& \omega_{1}\frac{(1-p)\beta_{2}\varepsilon }{D^{*}_{i}}(\frac{K(b_{d}-d-\theta)}{b_{d}-d}-D^{*}_{i})\\[3mm] \omega_{2}\frac{(1-p)\beta_{3}}{L^{*}_{i}}(M-L^{*}_{i})~& -\omega_{2}\frac{(1-p)\beta_{3}M}{L_{i}L^{*}_{i}}D_{i}\end{array}\right]\left(\begin{array}{cc}D_{i}-D^{*}_{i}\\L_{i}-L^{*}_{i}\end{array}\right)\\&\leq& 0.\end{matrix}$

且等号仅在 $D_{i}=D_{i}^{*}, L_{i}=L_{i}^{*}$ 处成立, 故系统 (3.2) 的平衡点 $(D_{i}^{*},L_{i}^{*})$ 全局渐近稳定.

接下来考虑子系统 (3.3). 当 $t\rightarrow\infty$ 时 $D_{i}\rightarrow D^{*}$. 故子系统 (3.3) 的极限系统为

(3.4)$\begin{matrix} \left\{ \begin{array}{l} \frac{{\rm d}H_{s}}{{\rm d}t}=A_{h}+\mu_{2}H_{i}+\mu_{1}qH_{e}-(1-p)\beta_{1}D^{*}_{i}H_{s}-d_{h}H_{s},\\[3mm] \frac{{\rm d}H_{e}}{{\rm d}t}=(1-p)\beta_{1}D^{*}_{i}H_{s}-\mu_{1}qH_{e}-d_{h}H_{e}-(1-q)\omega H_{e}, \\[3mm] \frac{{\rm d}H_{i}}{{\rm d}t}=(1-q)\omega H_{e}-\mu_{2}H_{i}-d_{h}H_{i}. \end{array} \right.\end{matrix}$

显然, 系统 (3.4) 为一个线性系统, 其特征方程为

$\lambda^{3}+a_{1}\lambda^{2}+a_{2}\lambda+a_{3}=0,$

其中

$\begin{matrix} a_{1}&=& \mu_{1}q+\mu_{2}+3d_{h}+(1-q)\omega+(1-p)\beta_{1}D_{i}^{*},\\ a_{2}&= &(\mu_{2}+d_{h})(\mu_{1}q+d_{h}+(1-q)\omega)+d_{h}\mu_{1}q+((1-p)\beta_{1}D_{i}^{*}+d_{h})(\mu_{2}+2d_{h}+(1-q)\omega), \\ a_{3}&=&d_{h}(1-p)\beta_{1}D_{i}^{*}(\mu_{2}+d_{h}+(1-q)\omega)+d_{h}(\mu_{2}+d_{h})(\mu_{1}q+d_{h}+(1-q)\omega). \end{matrix}$

由于 $a_{1}>0, a_{3}>0, a_{1}a_{2}-a_{3}>0,$ 故其特征根都具有负实部, 所以系统 (3.4) 的平衡点 $(H_{s}^{*},H_{e}^{*},H_{i}^{*}) $全局渐近稳定.

综上所述, 当 $R_{0}<1$ 时, 系统 (2.1) 的地方病平衡点 $E^{*}$ 全局渐近稳定.证毕.

如图2 所示, 当取定参数, 使得 $R_{0}<1$ 时, 取不同初值, 潜伏期人数与发病人数最终都会趋于零, 疾病消失; 当 $R_{0}>1$ 时, 取不同的初值, 潜伏期人数与发病人数最终都会稳定于同一个平衡点, 形成地方疾病.

图2

图2 潜伏期人数与发病人数随时间变化图

4 模型的应用

在这一部分, 我们首先利用数据对模型的合理性进行检验, 其次结合模型模拟的结果, 对防控策略提出合理的建议和优化措施.

4.1 数值模拟

为了进行数值模拟, 我们需要对模型中的参数做出合理的估计. 其中人口的增长量, 患者的治愈率以及家畜的总量、屠宰率等相关数据主要参考全国第七次人口普查的结果与西藏统计年鉴, 而感染系数 $\beta_{1}, \beta_{2}, \beta_{3}$ 主要参考文献[13]. 而宣传教育产生的影响系数 $p$, 流浪犬的环境最大容纳量 $K$ 以及归置率系数 $\theta$ 不易获得, 均为估计值. 具体数值如表2 所示.

根据公开发表的文献和公共卫生科学数据中心统计的包虫病病例可知, 西藏自治区从 2004-2020 年全区共报告病例数 1474 例, 其中 2017 年病例数最多为 273 例. 而 2012-2016 年完成的包虫病流行病学调查则显示西藏地区的包虫病患病率高达 1.66%, 之后完成的关于包虫病的普查工作, 进一步精确了西藏地区包虫病的患病率为 0.91%, 即我们模型中的潜伏期患者 $H_{e}$. 由于西藏地区报告的病例数与当地实际的患病人数存在不相符的情况^[4], 所以我们在具体分析模拟结果时, 不仅要观察 $H_{i}$ 更应该关注 $H_{e}$ 的变化趋势.

下面我们利用 2012 年的数据 $H_{s}$=3033666, $H_{e}$=42500, $H_{i}$=34, $D_{s}$=2.38$\times$10⁵, $D_{i}=7.2\times10^{4}$, $L_{s}=2.53\times10^{7}$, $L_{i}=3.683\times10^{6}$ 作为初始值条件进行计算. 其结果如图3-5 所示, 可以看出 $H_{i}$ 在 2016 年左右会达到峰值, 之后便会下降, 并且最大值不超过 450 例, 整体的变化趋势与实际情况是一致的. 此外将 $H_{e}$ 的模拟结果与文献[5]中西藏实际患病人数(见表1)比较可知, 除 2020 年的数据外, 另外四年的数据与模拟解之间匹配良好并且在稳步地减少. 这说明模型基本上符合当地的实际情况, 具有一定的合理性.

图3

图3 西藏地区 2016-2020 年实际患病人数与$H_{e}$ 的比较

图4

图4 西藏地区未来 $50$ 年 $H_{e}$ 的变化趋势

图5

图5 西藏地区未来 $50$ 年 $H_{i}$ 的变化趋势

表1 2016-2020西藏自治区人间包虫病患病率

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表2 参数及参数值(单位:年)

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4.2 防控策略讨论

下面我们研究 $p$ 和 $\theta$ 对基本再生数 $R_{0}$ 的影响, 进而讨论这两种防控措施. 令 $R_{0}=1, P, \theta$ 为参数, 可得

$p=1-\sqrt{\frac{bd(d_{l}+\varepsilon)(b_{d}-d)}{\beta_{2}\beta_{3}\varepsilon MK(b_{d}-d-\theta)}}:=p(\theta).$

即图6, 它刻画了宣传教育与流浪犬归置对基本再生数 $R_{0}$ 的影响, 当参数取值落在曲线的右上方区域时 $R_{0}<1$, 疾病可控; 相反, 落在曲线左下方区域时 $R_{0}>1$, 疾病得不到有效的控制.

图6

图6 $R_{0}=1$ 其中 $p, \theta$ 为参数的临界曲线

通过图像可知, 如果没有了平时的宣传教育, 即 $p=0$ 时, 要使 $R_{0}<1$ 则流浪犬的归置率 $\theta$ 需达到 75% 以上,这几乎是不可能的, 并且经济成本太大. 同理, 如果 $\theta=0$ 则需 $p>0.225$, 这需要当地群众对包虫病的传播及预防有一个很高的认知. 但根据文献[17] 的调查, 2020 年中国西部五省(区)农牧民对包虫病防控相关知识的总知晓率仅为 57%, 而对包虫病感染途径的知晓率只有 30.8%. 因此, 以现阶段藏区群众的认知能力还不能达到如此效果. 这从侧面也说明了, 虽然近几年藏区患病人数在不断下降, 但防治成果还是很脆弱的, 为了能够使疾病继续保持一种下行的趋势, 两种防治措施还得同时实施, 如图 7 所示. 并且在未来很长的一段时间内, 在保持对流浪犬高归置率的同时(35%以上)还需加大宣传力度, 特别是针对一些重点人群的宣传, 比如一些远离聚集地, 年龄较大的农牧民, 因为这一部分群众接受信息的渠道少, 文化水平低, 但同时又处在最容易被感染的第一线. 未来随着整个地区群众对包虫病的认知能力都提高到一定程度后, 则可以逐步降低对流浪犬的归置率.

图7

图7 犬的归置率 $\theta$ 与宣传教育因素 $p$ 对疾病流行的影响

5 结论

本文根据包虫病的传播机理以及西藏地区的政策法规, 构建了一类符合西藏地区实际情况的包虫病动力学模型. 通过对模型的分析和应用, 发现当基本再生数 $R_{0}<1$ 时, 模型存在一个无病平衡点和一个平凡平衡点, 不存在地方病平衡点, 并且平凡平衡点总是不稳定的, 无病平衡点全局渐进稳定; 当 $R_{0}>1$ 时, 模型存在地方病平衡点并且全局渐近稳定. 利用表 1 中的数据可得 $R_{0}=0.96904$, 即西藏地区的包虫病正在逐渐消亡, 模型模拟的结果与实际传播情况相符合. 最后讨论了宣传教育和对流浪犬归置两种防治措施对包虫病传播的影响, 结果表明, 两种措施必须同时施行才能达到控制疾病的目的, 并且由于现阶段藏区人民对包虫病的知晓率还比较低, 所以对流浪犬的归置率应一直保持高水平, 同时还要加大宣传力度, 当人们的预防意识有所提高后, 可逐步减少对流浪犬的归置.

参考文献

原文顺序

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