基于信息干预的SIRS传染病模型稳定性分析

李小妮; 张启敏

doi:10.6054/j.jscnun.2019090

基于信息干预的SIRS传染病模型稳定性分析

李小妮,
张启敏^,

宁夏大学数学统计学院, 银川 750021

基金项目:

国家自然科学基金项目 11661064

详细信息

通讯作者:
张启敏, 教授, Email:zhangqimin64@sina.com

中图分类号: O175.1
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出版历程
- 收稿日期: 2018-09-20
- 网络出版日期: 2021-03-08
- 刊出日期: 2019-10-24

Stability Analysis of an SIRS Epidemic Model with Information Intervention

LI Xiaoni,
ZHANG Qimin^,

School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China

摘要

摘要: 建立了一类基于信息干预和疫苗接种的SIRS传染病模型, 研究了该模型的全局渐近稳定性, 给出了疾病持久和灭绝的基本再生数ℜ₀.研究结果表明：当ℜ₀ < 1时, 该模型存在全局渐近稳定的无病平衡点; 当ℜ₀>1时, 该模型存在全局渐近稳定的地方病平衡点.数值算例验证了理论分析结果.
- SIRS传染病模型 /
- 信息干预 /
- 全局渐近稳定性 /
- 阈值
Abstract: The asymptotic behavior of an SIRS epidemic model containing information intervention and vaccination has been studied. The results indicate that the basic reproduction number ℜ₀ is the threshold of disease persistence and extinction. If ℜ₀ < 1, the system has an disease-free equilibrium which is globally asymptotically stable, while if ℜ₀>1, there exists an epidemic equilibrium which is globally asymptotically stable. At last, some numerical exam-ples are given to illustrate the results.
- SIRS epidemic model /
- information intervention /
- globally asymptotically stable /
- threshold

HTML全文

研究流行病的传播规律极为重要^[1-3].目前, 许多学者利用数学工具描述动力学模型, 进一步预测流行病传播^[4-6].疫苗接种是控制流行病传播的重要手段之一^[7-9], 为了研究疫苗接种对流行病动力学行为的影响, LAHROUZ等^[8]提出了如下带有疫苗接种的SIRS传染病模型:

$\left\{ \begin{array}{l} {\rm{d}}S = \left( {(1 - p)b - \mu S - \frac{{\beta SI}}{{\varphi (I)}} + \gamma R} \right){\rm{d}}t,\\ {\rm{d}}I = \left( { - (\mu + c + \alpha )I + \frac{{\beta SI}}{{\varphi (I)}}} \right){\rm{d}}t,\\ {\rm{d}}R = (pb - (\mu + \gamma )R + \alpha I){\rm{d}}t, \end{array} \right.$

(1)

其中, S、I、R分别代表易感者、感染者、恢复者的人数；所有参数均非负, 其生物学意义如下:b为出生率, p (0≤p≤1)为疫苗接种率, μ为自然死亡率, c为因病死亡率, β为感染系数, α为恢复率, γ为恢复者免疫力丧失率, φ(I)为正函数, 且φ(0)=1, φ′(I)≥0.该模型给出了无病平衡点和地方病平衡点, 讨论了2个平衡点的全局渐近稳定性.

在疾病爆发期间, 信息干预(宣传、报道、教育等)会对人们的行为产生影响, 从而影响流行病的传播速度.近年来, 有许多学者研究了信息干预对流行病传播的影响^[10-12].文献[6]指出媒体报道虽然不能使感染者恢复, 但可以降低感染者人数的峰值.因此, 将媒体报道因素考虑到模型中是很有必要的.

本文将信息干预引入到模型(1)中, 得到了新的基于信息干预和疫苗接种的SIRS传染病模型, 研究了新模型平衡点的存在性及其渐近稳定性, 给出了基本再生数, 并通过数值模拟验证了理论结果.

1. 主要结果及证明

将信息干预引入到模型(1)中, 得到了新的基于信息干预和疫苗接种的SIRS传染病模型:

$\left\{ \begin{array}{l} {\rm{d}}S = \left( {\left( {1 - p} \right)b - \mu S - \frac{{\beta SI}}{{\varphi \left( I \right)}} + \gamma R - {\mu _1}mZS} \right){\rm{d}}t,\\ {\rm{d}}I = \left( { - \left( {\mu + c + \alpha } \right)I + \frac{{\beta SI}}{{\varphi \left( I \right)}}} \right){\rm{d}}t,\\ {\rm{d}}R = \left( {pb - (\mu + \gamma )R + \alpha I + {\mu _1}mZS} \right){\rm{d}}t,\\ {\rm{d}}Z = \left( {\frac{{aI}}{{1 + {a_1}I}} - {a_0}Z} \right){\rm{d}}t, \end{array} \right.$

(2)

其中, Z为信息密度, m、μ₁、a、a₁和a₀分别表示信息干预率、信息强度、信息增长率、饱和常数和信息自然消亡率.

本文的主要结果如下:

定理1 (ⅰ)如果ℜ₀ < 1, 则无病平衡点E₀是局部渐近稳定的；如果ℜ₀>1，则E₀是不稳定的.

(ⅱ)模型(2)有1个地方病平衡点E^*, 且当ℜ₀>1并满足A₁B₁>C₁及A₁(B₁C₁-A₁C₁)>C₁²时, E^*是局部渐近稳定的.

定理2 当ℜ₀=1时, 模型(2)有1个前向分支.

定理3 当ℜ₀ < 1时, 模型(2)的无病平衡点U₀=(X₀, 0)是全局渐近稳定的.

定理4 若ℜ₀>1且

${a^2} < \frac{{{a_0}}}{3}\left( {c + \frac{{2\mu \beta {S^*}}}{{\gamma \varphi \left( {{I^*}} \right)}}} \right),$

$\begin{array}{l} \mu _1^2{m^2} < \min \left\{ {\frac{{{a_0}{\gamma ^2}}}{{12{\mu ^2}{S^{*2}}}}\left( {c + \frac{{2\mu \beta {S^*}}}{{\gamma \varphi \left( {{I^*}} \right)}}} \right),\frac{{{a_0}\alpha {\mu ^2}(\mu + \gamma )}}{{c{b^2}}},} \right.\\ \;\;\;\;\;\;\;\;\left. {\frac{{8{\mu ^2}(\mu + \gamma )\alpha }}{{3c\gamma {Z^{*2}}}},\frac{{{a_0}\gamma }}{{6{S^{*2}}}},\frac{{2\gamma a_0^2{{\left( {\mu + {a_1}b} \right)}^2}}}{{9{a^2}{b^2}}}\left( {c + \frac{{2\mu \beta {S^*}}}{{\gamma \varphi \left( {{I^*}} \right)}}} \right)} \right\}, \end{array}$

则模型(2)的地方病平衡点E^*是全局渐近稳定的.

1.1 平衡点和正解的存在性

由文献[13]的方法, 可以得到模型(2)的基本再生数:

${\Re _0} = \frac{{\beta (1 - p)(\mu + \gamma ) + \beta \gamma p}}{{(\mu + \gamma )(\mu + c + \alpha )}}.$

(3)

下面考虑模型(2)的平衡点的存在性.令模型(2)右端等于零, 即

$\left\{ \begin{array}{l} \left( {1 - p} \right)b - \mu S - \frac{{\beta SI}}{{\varphi \left( I \right)}} + \gamma R - {\mu _1}mZS = 0,\\ - \left( {\mu + c + \alpha } \right)I + \frac{{\beta SI}}{{\varphi \left( I \right)}} = 0,\\ pb - \left( {\mu + \gamma } \right)R + \alpha I + {\mu _1}mZS = 0,\\ \frac{{aI}}{{1 + {a_1}I}} - {a_0}Z = 0. \end{array} \right.$

(4)

求解方程组(4), 可得模型(2)存在2个平衡点:(1)无病平衡点 $E_{0}=\left(\frac{\mu(1-p) b+\gamma b}{\mu(\mu+\gamma)}, 0, \frac{p b}{\mu+\gamma}, 0\right)$ . (2)当ℜ₀>1时，存在地方病平衡点E^*=(S^*, I^*, R^*, Z^*), 其中

${S^*} = \frac{{\left( {\mu + c + \alpha } \right)\varphi \left( {{I^*}} \right)}}{\beta },$

${R^*} = \frac{1}{{\mu + \gamma }}\left[ {pb + \alpha {I^*} + \frac{{{\mu _1}ma\left( {\mu + c + \alpha } \right)\varphi \left( {{I^*}} \right){I^*}}}{{{a_0}\beta \left( {1 + {a_1}{I^*}} \right)}}} \right],$

${Z^*} = \frac{{a{I^*}}}{{{a_0}\left( {1 + {a_1}{I^*}} \right)}},$

且I^*是以下方程的唯一正根:

$\begin{array}{l} H\left( I \right) = \left( {1 - p} \right)b - \frac{{\left( {\mu + c + \alpha } \right)\varphi \left( I \right)}}{\beta } - \left( {\mu + c + \alpha } \right)I + \frac{{\gamma pb}}{{\mu + \gamma }} + \\ \;\;\;\;\;\;\;\;\frac{{\alpha \gamma I}}{{\mu + \gamma }} + \frac{{{\mu _1}ma\gamma \left( {\mu + c + \alpha } \right)\varphi (I)I}}{{{a_0}\beta \left( {\mu + \gamma } \right)\left( {1 + {a_1}I} \right)}} = 0, \end{array}$

事实上, 如果ℜ₀>1, 则H(0)>0, H′(I) < 0, 那么lim_I→∞ H(I)=-∞, 即ℜ₀>1当且仅当H(I)=0有唯一的正解.

接下来讨论模型(2)的正解.由模型(2)可得

${\left. {\frac{{{\rm{d}}S}}{{{\rm{d}}t}}} \right|_{S = 0}} = \left( {1 - p} \right)b + \gamma R \ge 0,{\left. {\frac{{{\rm{d}}I}}{{{\rm{d}}t}}} \right|_{I = 0}} \ge 0,$

${\left. {\frac{{{\rm{d}}R}}{{{\rm{d}}t}}} \right|_{R = 0}} = pb + \alpha I + {\mu _1}mZS \ge 0,{\left. {\frac{{{\rm{d}}Z}}{{{\rm{d}}t}}} \right|_{Z = 0}} = \frac{{aI}}{{1 + {a_I}}} \ge 0,$

如果考虑该区域内部, 由模型(2)可知人口总数N=S+I+R满足如下微分方程:

$\frac{{{\rm{d}}N}}{{{\rm{d}}t}} = b - \mu N - cI.$

所以, $\frac{\mathrm{d} N}{\mathrm{d} t} \leqslant b-\mu N$ , 即sup_t→∞ $N \leqslant \frac{b}{\mu}$ .因此, $\frac{b}{\mu}$ 是S、I和R的上界.根据模型(2)的第4个方程和I的界, 有lim sup_t→∞ $Z \leqslant \frac{a b}{a_{0}\left(\mu+a_{1} b\right)}$ .而且, 有以下不变集集合:

$\begin{gathered} \mathit{\Gamma } = \left\{ {(S,I,R,Z) \in \mathbb{R}_ + ^4:S + I + R \leqslant \frac{b}{\mu },0 \leqslant Z \leqslant \frac{{ab}}{{{a_0}\left( {\mu + {a_1}b} \right)}},} \right. \hfill \\ \left. {S \geqslant 0,I \geqslant 0,R \geqslant 0,Z \geqslant 0} \right\}. \hfill \\ \end{gathered}$

综上, 在非负集 $\mathbb{R}_{+}^{4}$ 上, 解将保持在Γ中.

1.2 E₀和E^*的局部稳定性

为了方便, 给出了模型(2)的可变矩阵:

$\mathit{\boldsymbol{D}} = \left[ {\begin{array}{*{20}{c}} { - \mu - \frac{{\beta I}}{{\varphi \left( I \right)}} - c}&{ - \frac{{\beta S}}{{\varphi \left( I \right)}}}&\gamma &{ - \frac{{\beta I}}{{\varphi \left( I \right)}}}\\ {\frac{{\beta I}}{{\varphi \left( I \right)}}}&{ - \left( {pq} \right)}&0&0\\ {{\mu _1}mZ}&\alpha &{ - \left( {\mu + \gamma } \right)}&{{\mu _1}mS}\\ 0&{\frac{a}{{{{\left( {1 + {a_1}I} \right)}^2}}}}&0&{ - {a_0}} \end{array}} \right].$

(5)

对地方病平衡点E^*, 其特征方程为:

${\lambda ^4} + {A_1}{\lambda ^3} + {B_1}{\lambda ^2} + {C_1}\lambda + {D_1} = 0,$

其中,

${A_1} = {a_0} + 2\mu + \gamma + \frac{{\beta {I^*}}}{{\varphi \left( {{I^*}} \right)}} + {\mu _1}m{Z^*},$

$\begin{array}{l} {B_1} = {a_0}(\mu + \gamma ) + \left( {\mu + {a_0}} \right)\left( {\mu + \frac{{\beta {I^*}}}{{\varphi \left( {{I^*}} \right)}} + {\mu _1}m{Z^*}} \right) + \\ \;\;\;\;\;\;\;\;\gamma \left( {\mu + \frac{{\beta {I^*}}}{{\varphi \left( {{I^*}} \right)}}} \right) + \frac{{{\beta ^2}{S^*}{I^*}}}{{\varphi {{\left( {{I^*}} \right)}^2}}}, \end{array}$

$\begin{array}{*{20}{c}} {{C_1} = \frac{{{\beta ^2}{S^*}{I^*}}}{{\varphi {{\left( {{I^*}} \right)}^2}}}\left( {{a_0} + \mu + \gamma } \right) + \frac{{a\beta {I^*}{\mu _1}m{S^*}}}{{{{\left( {1 + {a_1}{I^*}} \right)}^2}}} - \frac{{\beta {I^*}}}{{\varphi \left( {{I^*}} \right)}}\alpha {\gamma ^ + }}\\ {{a_0}\mu \left( {\mu + \frac{{\beta {I^*}}}{{\varphi \left( {{I^*}} \right)}} + {\mu _1}m{Z^*}} \right) + {a_0}\gamma \left( {\mu + \frac{{\beta {I^*}}}{{\varphi \left( {{I^*}} \right)}}} \right),} \end{array}$

$\begin{array}{*{20}{c}} {{D_1} = \frac{{{a_0}\beta {I^*}}}{{\varphi \left( {{I^*}} \right)}}(\mu (\mu + c + \alpha ) + \gamma (\mu + c)) + }\\ {\frac{{a\beta {I^*}}}{{\varphi \left( {{I^*}} \right){{\left( {1 + {a_1}{I^*}} \right)}^2}}}\mu {\mu _1}m{S^*}.} \end{array}$

定理1的证明 (ⅰ)与文献[13]中定理2的证明类似, 此处略.

(ⅱ)显然A₁>0, D₁>0.由Routh-Hurwitz判据, 若A₁B₁>C₁和A₁(B₁C₁-A₁D₁)>C₁², 则J_E^*的特征方程的所有根要么非负, 要么有负实部.因此, 如果ℜ₀>1, A₁B₁>C₁, A₁(B₁C₁-A₁D₁)>C₁²，则由Hartman-Grobman定理^[14]可知E^*是局部渐近稳定的.

定理2的证明 当ℜ₀=1时, 令x₁=S, x₂=I, x₃=R, x₄=Z, 将φ=β作为分叉参数.于是, 对φ=φ^*=β^*, 由ℜ₀=1可得 $\beta^{*}=\frac{b((1-p) \mu+\gamma)}{\mu(\mu+\gamma)(\mu+c+\alpha)}$ .应用新的变换, 模型(2)可以改写为：

$\left\{ \begin{array}{l} \frac{{{\rm{d}}{x_1}}}{{{\rm{d}}t}} = \left( {1 - p} \right)b - \mu {x_1} - \frac{{\beta {x_1}{x_2}}}{{\varphi \left( {{x_2}} \right)}} + \gamma {x_3} - {\mu _1}m{x_1}{x_4}: = {f_1},\\ \frac{{{\rm{d}}{x_2}}}{{{\rm{d}}t}} = - (\mu + c + \alpha ){x_2} + \frac{{\beta {x_1}{x_2}}}{{\varphi \left( {{x_2}} \right)}}: = {f_2},\\ \frac{{{\rm{d}}{x_3}}}{{{\rm{d}}t}} = pb - (\mu + \gamma ){x_3} + \alpha {x_2} + {\mu _1}m{x_1}{x_4}: = {f_3},\\ \frac{{{\rm{d}}{x_4}}}{{{\rm{d}}t}} = \frac{{a{x_2}}}{{1 + {a_1}{x_2}}} - {a_0}{x_4}: = {f_4}. \end{array} \right.$

对于无病平衡点x^*, 可以得到(x^*, β^*)的雅可比矩阵:

$\begin{array}{l} {\mathit{\boldsymbol{D}}_{{x^ * }}}\left( {{\beta ^ * }} \right) = \\ \left[ {\begin{array}{*{20}{c}} { - \mu }&{ - \left( {\mu + c + \alpha } \right)}&\gamma &{\frac{{ - {\mu _1}m\left[ {\left( {1 - p} \right)\left( {\mu + \gamma } \right) + \gamma p} \right]}}{{\varphi \left( {x_2^*} \right)\left( {\mu + \gamma } \right)}}}\\ 0&0&0&0\\ 0&\alpha &{ - \left( {\mu + \gamma } \right)}&{\frac{{{\mu _1}m\left[ {\left( {1 - p} \right)\left( {\mu + \gamma } \right) + \gamma p} \right]}}{{\varphi \left( {x_2^*} \right)\left( {\mu + \gamma } \right)}}}\\ 0&a&0&{ - {a_0}} \end{array}} \right]. \end{array}$

当ℜ₀=1时, D_x^*(β^*)有1个零特征值, 且其他特征值都是非负的. D_x^*(β^*)相应于零特征值的右特征向量为y=(y₁, y₂, y₃, y₄)′，其中

$\left\{ \begin{array}{l} {y_1} = \frac{1}{\mu }\left\{ {\frac{{\gamma \alpha }}{{\left( {\mu + \gamma } \right)a}} - \frac{{\mu + c + \alpha }}{a} - \frac{{\mu {\mu _1}m[(1 - p)(\mu + \gamma ) + \gamma p]}}{{\varphi \left( {x_2^*} \right){a_0}{{(\mu + \gamma )}^2}}}} \right\},\\ {y_2} = \frac{1}{a},\\ {y_3} = \frac{1}{{(\mu + \gamma )}}\left\{ {\frac{\alpha }{a} + \frac{{{\mu _1}m[(1 - p)(\mu + \gamma ) + \gamma p]}}{{{a_0}\varphi \left( {x_2^*} \right)(\mu + \gamma )}}} \right\},\\ {y_4} = \frac{1}{{{a_0}}}. \end{array} \right.$

类似地, 可以得到D_x^*(β^*)相应于零特征值的左特征向量为z=(z₁, z₂, z₃, z₄), 其中z₁=0, z₂=1, z₃=0, z₄=0.由文献[12]可知，当ℜ₀=1时, 常数a₂和b₂可决定无病平衡点的稳定性.下面计算a₂和b₂. f=(f₁, f₂, f₃, f₄)在(x^*, β^*)处的非零二阶偏导为：

$\frac{{{\partial ^2}{f_2}}}{{\partial {x_2}\partial {x_1}}} = {\beta ^*},\frac{{{\partial ^2}{f_2}}}{{\partial {x_1}\partial {x_2}}} = {\beta ^*},\frac{{{\partial ^2}{f_2}}}{{\partial {x_2}\partial \beta }} = x_1^*,$

则

$\begin{array}{l} {a_2} = \frac{{ - 2((1 - p)(\mu + \gamma ) + \gamma p)}}{{{a_0}{a^2}\mu {{(\mu + \gamma )}^3}(\mu + c + \alpha )\varphi \left( {x_2^*} \right)}}\left\{ {{a_0}\varphi \left( {x_2^*} \right) \times } \right.\\ \;\;\;\;\;\;\;(\mu + c + \alpha )(\mu + \gamma ) + {a_0}\varphi \left( {x_2^*} \right)\gamma (\mu + c)(\mu + \gamma ) + \\ \;\;\;\;\;\;\;\left. {a\mu {\mu _1}m[(1 - p)(\mu + \gamma ) + \gamma p]} \right\}, \end{array}$

${b_2} = \frac{{(\mu + \gamma )(1 - p) + \gamma p}}{{a\varphi \left( {x_2^*} \right)(\mu + \gamma )}}.$

显然, a₂ < 0, b₂>0.得证.

1.3 E₀和E^*的全局稳定性

假设模型(2)可改写为^[15]:

$\frac{{{\rm{d}}\mathit{\boldsymbol{X}}}}{{{\rm{d}}t}} = F(\mathit{\boldsymbol{X}},Y),\frac{{{\rm{d}}Y}}{{{\rm{d}}t}} = G(\mathit{\boldsymbol{X}},Y),G(\mathit{\boldsymbol{X}},0) = 0,$

其中, $\boldsymbol{X} \in \mathbb{R}^{3}$ 、 $Y \in \mathbb{R}$ 分别代表未感染者和感染者的人数.令U₀=(X₀, 0)为无病平衡点.

引理1^[15] 若ℜ₀ < 1且以下条件满足：

(ⅰ)对 $\frac{\mathrm{d} \boldsymbol{X}}{\mathrm{d} t}=F(\boldsymbol{X}, 0), \boldsymbol{X}_{0}$ 是全局渐近稳定的;

(ⅱ)∀(X, Y)∈Γ, 有G(X, Y)=D_YG(X₀, 0)Y- $\hat{G}(\boldsymbol{X}, Y), \hat{G}(\boldsymbol{X}, Y) \geqslant 0$ , 其中, D_YG(X₀, 0)为M-矩阵, 则模型(2)的无病平衡点U₀=(X₀, 0)是全局渐近稳定的.

定理3的证明 类似文献[12]中定理2的证明, 有

$\begin{array}{l} F(\mathit{\boldsymbol{X}},Y) = \left( {(1 - p)b - \mu S - \frac{{\beta SI}}{{\varphi (I)}} + \gamma R - {\mu _1}mZS,} \right.\\ \left. {pb - (\mu + \gamma )R + \alpha I + {\mu _1}mZS,\frac{{aI}}{{1 + {a_1}I}} - {a_0}Z} \right), \end{array}$

$G(\mathit{\boldsymbol{X}},Y) = - (\mu + c + \alpha )I + \frac{{\beta SI}}{{\varphi (I)}},G(\mathit{\boldsymbol{X}},0) = 0,$

其中, X=(S, R, Z)′, Y=I.无病平衡点U₀=E₁=(X₀, 0)且 $\boldsymbol{X}_{0}=\left(\frac{\mu(1-p) b+\gamma b}{\mu(\mu+\gamma)}, \frac{p b}{\mu+\gamma}, 0\right)$ .取t→∞, 则有 $\boldsymbol{X} \rightarrow\left(\frac{\mu(1-p) b+\gamma b}{\mu(\mu+\gamma)}, \frac{p b}{\mu+\gamma}, 0\right)$ .显然X₀是全局渐近稳定的.更进一步地,

$\begin{array}{l} G(\mathit{\boldsymbol{X}},Y) = - (\mu + c + \alpha )\left( {1 - {\Re _0}} \right)I - \\ \;\;\;\;\;\;\;\frac{{\beta I}}{{\varphi (I)}}\left[ {\frac{{\varphi (I)(\mu (1 - p)b + \gamma b)}}{{\mu (\mu + \gamma )}} - S} \right]. \end{array}$

若 $S \leqslant \frac{\mu(1-p) b+\gamma b}{\mu(\mu+\gamma)}$ , 则

$\hat G(\mathit{\boldsymbol{X}},Y) = \frac{{\beta I}}{{\varphi (I)}}\left[ {\frac{{\varphi (I)(\mu (1 - p)b + \gamma b)}}{{\mu (\mu + \gamma )}} - S} \right] \ge 0.$

所以条件(ⅰ)、(ⅱ)满足.因此, 当ℜ₀ < 1时, 模型(2)的无病平衡点是全局渐近稳定的.

定理4的证明 在区间Γ上, 考虑函数V:

$\begin{array}{l} V(S,I,R,Z) = \frac{1}{2}{\left[ {\left( {S - {S^*}} \right) + \left( {I - {I^*}} \right) + \left( {R - {R^*}} \right)} \right]^2} + \\ \;\;\;\;{m_1}\left( {I - {I^*} - {I^*}\log \frac{I}{{{I^*}}}} \right) + \frac{{{m_2}}}{2}{\left( {S - {S^*} + I - {I^*}} \right)^2} + \frac{{{m_3}}}{2}{\left( {R - {R^*}} \right)^2} + \\ \;\;\;\;\frac{1}{2}{\left( {Z - {Z^*}} \right)^2} = {V_1} + {m_1}{V_2} + {m_2}{V_3} + {m_3}{V_4} + {V_5}, \end{array}$

(6)

其中，m₁、m₂、m₃都是正常数.可以得到V_i(i=1, 2, …, 5)的导数如下:

$\begin{array}{l} {{\dot V}_1} = - \mu {\left( {S - {S^*}} \right)^2} - (\mu + c){\left( {I - {I^*}} \right)^2} - \mu {\left( {R - {R^*}} \right)^2} - \\ \;\;\;\;\;\;(2\mu + c)\left( {S - {S^*}} \right)\left( {I - {I^*}} \right) - 2\mu \left( {S - {S^*}} \right)\left( {R - {R^*}} \right) - \\ \;\;\;\;\;\;(2\mu + c)\left( {I - {I^*}} \right)\left( {R - {R^*}} \right), \end{array}$

$\begin{array}{l} {{\dot V}_2} = - \frac{{\beta S}}{{\varphi (I)\varphi \left( {{I^*}} \right)}}\left( {\varphi (I) - \varphi \left( {{I^*}} \right)} \right)\left( {I - {I^*}} \right) + \\ \;\;\;\;\;\;\;\frac{\beta }{{\varphi \left( {{I^*}} \right)}}\left( {S - {S^*}} \right)\left( {I - {I^*}} \right), \end{array}$

$\begin{array}{l} {{\dot V}_3} = - \mu {\left( {S - {S^*}} \right)^2} - \frac{{\beta {S^*}}}{{\varphi \left( {{I^*}} \right)}}{\left( {I - {I^*}} \right)^2} + \gamma \left( {S - {S^*}} \right)\left( {R - {R^*}} \right) - \\ \;\;\;\;\;{\mu _1}m{S^*}\left( {S - {S^*}} \right)\left( {Z - {Z^*}} \right) - {\mu _1}mZ{\left( {S - {S^*}} \right)^2} - \\ \;\;\;\;\;\left( {\mu + \frac{{\beta {S^*}}}{{\varphi \left( {{I^*}} \right)}}} \right)\left( {S - {S^*}} \right)\left( {I - {I^*}} \right) + \gamma \left( {I - {I^*}} \right)\left( {R - {R^*}} \right) - \\ \;\;\;\;\;{\mu _1}mZ\left( {I - {I^*}} \right)\left( {S - {S^*}} \right) - {\mu _1}m{S^*}\left( {I - {I^*}} \right)\left( {Z - {Z^*}} \right), \end{array}$

$\begin{array}{l} {{\dot V}_4} = - (\mu + \gamma ){\left( {R - {R^*}} \right)^2} + \alpha \left( {R - {R^*}} \right)\left( {I - {I^*}} \right) + \\ \;\;\;\;\;{\mu _1}m{Z^*}\left( {R - {R^*}} \right)\left( {S - {S^*}} \right) + {\mu _1}mS\left( {R - {R^*}} \right)\left( {Z - {Z^*}} \right), \end{array}$

${{\dot V}_5} = \frac{{a\left( {I - {I^*}} \right)\left( {Z - {Z^*}} \right)}}{{\left( {1 + {a_1}I} \right)\left( {1 + {a_1}{I^*}} \right)}} - {a_0}{\left( {Z - {Z^*}} \right)^2}.$

取 $m_{1}=\frac{\varphi\left(I^{*}\right)}{\beta}\left((2 \mu+c)+\frac{2 \mu}{\gamma}\left(\mu+\frac{\beta S^{*}}{\varphi\left(I^{*}\right)}\right)\right)$ , $m_{2}=\frac{2 \mu}{\gamma}$ , $m_{3}=\frac{c}{\alpha}$ , 可得到：

$\begin{array}{l} \dot V \le - \mu {\left( {S - {S^*}} \right)^2} - \left( {\mu + c} \right){\left( {I - {I^*}} \right)^2} - \mu {\left( {R - {R^*}} \right)^2} - \\ \;\;\;\;\;\frac{{2\mu }}{\gamma }\left( {\mu {{\left( {S - {S^*}} \right)}^2} + \frac{{\beta {S^*}}}{{\varphi \left( {{I^*}} \right)}}{{\left( {I - {I^*}} \right)}^2} + {\mu _1}mZ{{\left( {S - {S^*}} \right)}^2}} \right) - \\ \;\;\;\;\;\frac{{2\mu }}{\gamma }\left( {{\mu _1}m{S^*}\left( {S - {S^*}} \right)\left( {Z - {Z^*}} \right) + {\mu _1}mZ\left( {I - {I^*}} \right)\left( {S - {S^*}} \right) + } \right.\\ \;\;\;\;\;\left. {{\mu _1}m{S^*}\left( {I - {I^*}} \right)\left( {Z - {Z^*}} \right)} \right) - \frac{c}{\alpha }\left( {\mu + \gamma } \right){\left( {R - {R^*}} \right)^2} + \\ \;\;\;\;\;\frac{c}{\alpha }\left( {{\mu _1}m{Z^*}\left( {R - {R^*}} \right)\left( {S - {S^*}} \right) + {\mu _1}mS\left( {R - {R^*}} \right)\left( {Z - {Z^*}} \right)} \right) + \\ \;\;\;\;\;\frac{{a\left( {I - {I^*}} \right)\left( {Z - {Z^*}} \right)}}{{\left( {1 + {a_1}I} \right)\left( {1 + {a_1}{I^*}} \right)}} - {a_0}{\left( {Z - {Z^*}} \right)^2}. \end{array}$

在区间Γ上, 取S和Z的上界, 若满足

${a^2} < \frac{{{a_0}}}{3}\left( {c + \frac{{2\mu \beta {S^*}}}{{\gamma \varphi \left( {{I^*}} \right)}}} \right),$

$\begin{array}{l} \mu _1^2{m^2} < \min \left\{ {\frac{{{a_0}{\gamma ^2}}}{{12{\mu ^2}{S^{*2}}}}\left( {c + \frac{{2\mu \beta {S^*}}}{{\gamma \varphi \left( {{I^*}} \right)}}} \right),\frac{{{a_0}\alpha {\mu ^2}(\mu + \gamma )}}{{c{b^2}}},} \right.\\ \;\;\;\left. {\frac{{8{\mu ^2}(\mu + \gamma )\alpha }}{{3c\gamma {Z^{*2}}}},\frac{{{a_0}\gamma }}{{6{S^{*2}}}},\frac{{2\gamma a_0^2{{\left( {\mu + {a_1}b} \right)}^2}}}{{9{a^2}{b^2}}}\left( {c + \frac{{2\mu \beta {S^*}}}{{\gamma \varphi \left( {{I^*}} \right)}}} \right)} \right\}, \end{array}$

易得LV≤0.因此, 可知在Γ区间, 如果S=S^*, I=I^*, R=R^*, Z=Z^*, 则有LV < 0或LV=0.最后, 由LaSalle不变原理可知E^*在Γ区间是全局渐近稳定的.

2. 数值模拟

本节将通过一系列的数值例子来验证模型(2)的无病平衡点和地方病平衡点的稳定性.

例1 模型(2)的参数取值如下:

p=0.5, b=1, β=0.01, μ=0.01, μ₁=0.01, m=0.017, c=0.005, γ=0.001, α=0.8, a=0.01, a₀=0.045, a₁=1.令φ(I)=1+I², 可得ℜ₀=0.669 3 < 1, E₀=(54.545 5, 0, 45.454 5, 0)且E^*不存在.显然, 图 1A验证了定理3.

图 1 模型(2)中疾病的灭绝性分析

Figure 1. The extinction analysis of disease for model (2)

下载: 全尺寸图片幻灯片

此外, 给出了信息强度μ₁对传染病的影响.对于模型(2), 给出了关于I(t)的不同参数μ₁(μ₁=0, μ₁=0.10, μ₁=0.20)的时间序列(图 1B).可知在疾病爆发期间, 若增加信息强度μ₁, 则可以减少感染者的数量并加速疾病的灭绝.

例2 模型(2)的参数取值如下:

p=0.5, b=4, β=0.04, μ=0.02, μ₁=0.009, m=0.01, c=0.005, γ=0.01, α=0.7, a=0.02, a₀=0.045, a₁=1.令φ(I)=1+I², 可知ℜ₀=3.174 6>1, a²=4.000 0×10^-4 < 0.033 1, μ₁²m²=8.100 00×10^-9 < min{6.634 0×10^-7, 1.771 9×10^-5, 13.889 7, 2.701 6×10^-8, 0.050 5}且E₀=(93.333 3, 0, 40.000 0, 0), E^*=(74.512 9, 1.238 7, 57.371 4, 0.245 9).显然, 图 2A验证了定理4.

图 2 模型(2)中疾病的持久性分析

Figure 2. The persistence analysis of disease for model (2)

下载: 全尺寸图片幻灯片

类似地, 给出了在不同的信息强度μ₁(μ₁=0, μ₁=0.10, μ₁=0.20)的时间序列I(t)(图 2B).可知若增加信息强度μ₁, 则可降低感染者数量.

3. 结论

本文研究了基于信息干预和疫苗接种的SIRS传染病模型(2), 分析了该模型中无病平衡点和地方病平衡点的局部、全局稳定性.研究结果表明:

(1) 通过疫苗接种率ℜ₀, 可以进一步知道疫苗接种如何影响疾病传播.事实上, ℜ₀可以被写成如下形式:

${\Re _0} = - p\frac{{\beta \mu }}{{(\mu + \gamma )(\mu + c + \alpha )}} + \frac{\beta }{{\mu + c + \alpha }}.$

如果增加疫苗接种率p, 则基本再生数ℜ₀将会减少.换言之, 疫苗接种率的增加可以抑制疾病的传播.

(2) 疾病爆发时, 信息强度的增加可以降低感染者的数量并且加速疾病的灭亡.

图 1 模型(2)中疾病的灭绝性分析

Figure 1. The extinction analysis of disease for model (2)

下载: 全尺寸图片幻灯片

图 2 模型(2)中疾病的持久性分析

Figure 2. The persistence analysis of disease for model (2)

下载: 全尺寸图片幻灯片

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1. 主要结果及证明
1.1 平衡点和正解的存在性
1.2 E0和E*的局部稳定性
1.3 E0和E*的全局稳定性
2. 数值模拟
3. 结论

基于信息干预的SIRS传染病模型稳定性分析

通讯作者: 张启敏, 教授, Email:zhangqimin64@sina.com

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