数学物理学报, 2019, 39(4): 909-917 doi:

论文

考虑部分免疫和环境传播的麻疹传染病模型的全局稳定性

靖晓洁, 赵爱民,, 刘桂荣

Global Stability of a Measles Epidemic Model with Partial Immunity and Environmental Transmission

Jing Xiaojie, Zhao Aimin,, Liu Guirong

通讯作者: 刘桂荣,E-mail: lgr5791@sxu.edu.cn

收稿日期: 2018-04-12  

基金资助: 国家自然科学基金
山西省自然科学基金
the NSFC
the Natural Science Foundation Program of ShanxiProvince

Received: 2018-04-12  

Fund supported: 国家自然科学基金
山西省自然科学基金
the NSFC
the Natural Science Foundation Program of ShanxiProvince

摘要

该文考虑一个具有部分免疫和环境传播的麻疹传染病模型,得到基本再生数R0,并通过构造Lyapunov函数,研究了该模型的无病平衡点和地方病平衡点的全局稳定性.当R0 < 1时,无病平衡点是全局渐近稳定的,即麻疹不会传播开;当R0 > 1时,模型存在唯一的地方病平衡点,且是全局渐近稳定的,即麻疹的传播保持在一个稳定的状态.最后,通过数值分析说明了这些结果的合理性.该文工作对于预防和控制麻疹病毒的传播具有实际意义.

关键词: 部分免疫 ; 环境传播 ; 基本再生数 ; Lyapunov函数 ; 全局稳定性

Abstract

In this paper, a measles epidemic model with partial immunity and environmental transmission is considered, and the basic reproduction number R0 is obtained. By constructing Lyapunov functions, we prove the global asymptotic stability of the infection-free equilibrium and the endemic equilibrium. When R0 < 1, the infection-free equilibrium is globally asymptotically stable, which implies that measles dies out eventually; when R0 > 1, the model has a unique endemic equilibrium, which is globally asymptotically stable, that is the transmission of measles keeps a steady state. Finally, the simulations are carried to verify the rationality of the results. This work has practical significance for guiding us to prevent and control the measles spread.

Keywords: Partial immunity ; Environmental transmission ; The basic reproduction number ; Lyapunov function ; Global stability

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本文引用格式

靖晓洁, 赵爱民, 刘桂荣. 考虑部分免疫和环境传播的麻疹传染病模型的全局稳定性. 数学物理学报[J], 2019, 39(4): 909-917 doi:

Jing Xiaojie, Zhao Aimin, Liu Guirong. Global Stability of a Measles Epidemic Model with Partial Immunity and Environmental Transmission. Acta Mathematica Scientia[J], 2019, 39(4): 909-917 doi:

1引言

麻疹是儿童最常见的急性传染病,是一种传染性很强的呼吸道疾病,因此通过数学建模来了解它的动力学性态是非常有必要的[1-9].人群普遍为易感者,接种疫苗是预防麻疹的最有效措施.由于麻疹疫苗需接种两次且间隔时间较长,导致漏种的情况及接种疫苗后免疫失败,所以接种疫苗者是部分免疫.此外,麻疹病毒有9-14天的潜伏期,接触过麻疹患者并在潜伏期接受被动免疫者可延至3-4周.未接种过疫苗的人,尤其是幼儿,具有罹患麻疹及其并发症(包括死亡)的最高风险.高传染性的麻疹病毒是通过空气中的飞沫传播(由于咳嗽和打喷嚏),或者是与染病者密切接触,或者是直接接触到他们的鼻腔和口腔分泌物进行传播.病毒在空气中或受染物表体的活力和传染性可维持两个小时.因此环境因素对于麻疹的传播有着重要的影响.

麻疹是已知最具有传染性的疾病之一.已经有许多学者对麻疹进行了研究.文献[1-3]研究了麻疹疫苗的有效性,其中文献[1]考虑了部分免疫对麻疹传播的影响.文献[4]研究了麻疹的潜伏期.文献[5]讨论了部分免疫和潜伏期共同作用下的麻疹传播动力学. Garba[6]等人在Trottier和Philippe[7-8]建立的标准传染率的麻疹传染病模型下,对感染麻疹的人群进行细化,把潜伏者分为接种过疫苗的潜伏者和未接种过疫苗的潜伏者,把染病者也分为接种过疫苗的染病者和未接种过疫苗的染病者,以及加入治疗者,研究了一个更为复杂的模型.文献[7, 9]研究了麻疹的周期性.尽管我国提出2012年消除麻疹的目标,但全国麻疹疫情自2012年年底开始持续回升.文献[9]通过对我国麻疹月数据进行分析,得到麻疹随季节的周期性行为,进而进行数学建头治基于以上的各种研究,均没有考虑环境传播对麻疹的影响.基于此,本文考虑部分免疫和环境传播对麻疹流行性态的影响,构建数学模型,分析了模型的全局稳定性.

本文结构安排如下:第二节建立了具有部分免疫和环境传播的麻疹传染病模型(2.1)并给出了模型(2.2)的正不变集.第三节研究了模型(2.2)的平衡点的存在性和基本再生数.第四节分别证明了模型(2.2)的无病平衡点和地方病平衡点的全局渐近稳定性.第五节对模型(2.2)的结论进行了数值仿真.最后,第六节是本文的结论.

2模型的建立

本文假设新生儿一出生就接种疫苗,忽略因病死亡,并加入了环境因素,用$ S(t) $, $ V(t) $, $ E(t) $, $ I(t) $, $ R(t) $, $ B(t) $分别表示$ t $时刻易感者、疫苗接种者、潜伏者、染病者和恢复者的数量,及$ t $时刻环境中麻疹病毒的数量,建立如下数学模型

$ \begin{equation} \left\{\begin{array}{ll} S' = \lambda(1-\eta)-\beta SI-\alpha SB-\mu S, \\ V' = \lambda\eta-\varepsilon\beta VI-\varepsilon\alpha VB-\mu V-\delta V, \\ E' = \beta SI+\alpha SB+\varepsilon\beta VI+\varepsilon\alpha VB-\mu E-pE, \\ I' = pE-\mu I-\gamma I, \\ R' = \gamma I-\mu R+\delta V, \\ B' = kI-\tau B. \end{array}\right. \end{equation} $

显然,模型(2.1)中前四个方程与最后一个方程是封闭的,故只分析如下模型

$ \begin{equation} \left\{\begin{array}{ll} S' = \lambda(1-\eta)-\beta SI-\alpha SB-\mu S, \\ V' = \lambda\eta-\varepsilon\beta VI-\varepsilon\alpha VB-\mu V-\delta V, \\ E' = \beta SI+\alpha SB+\varepsilon\beta VI+\varepsilon\alpha VB-\mu E-pE, \\ I' = pE-\mu I-\gamma I, \\ B' = kI-\tau B. \end{array}\right. \end{equation} $

有关参数的意义及取值见表 1.

表表 1   模型中参数的意义及取值(单位:年)

参数取值意义来源
$\lambda$$1.728\times 10^{7}$新生儿的出生率国家统计局(2017)
$\mu$0.0131自然死亡率国家统计局(2017)
$\eta$0.8216新生儿接种疫苗的比例文献[9]
$\beta$$9.6696\times 10^{-9}$人与人的感染率估计
$\alpha$$1.1352\times 10^{-9}$环境对人的感染率估计
$\varepsilon$0.15无效的接种率中国疾病预防控制中心(2017)
$\delta$0.95接种疫苗后获得免疫并成为恢复者的转移率中国疾病预防控制中心(2017)
$p$26潜伏者变为染病者的比例中国疾病预防控制中心(2017)
$\gamma$20染病者的恢复率中国疾病预防控制中心(2017)
$k$8染病者排放到环境中的病毒的速率假设
$\tau$1.6环境中病毒的失效率假设

新窗口打开| 下载CSV


我们假设染病者把病毒排放到环境中的速率为8,麻疹病毒的失效率为1.6,然后使用2004-2016年的麻疹发病人数和模型(2.2)进行数值拟合,得到人与人的感染率$ \beta = 9.6696\times 10^{-9} $,环境对人的感染率$ \alpha = 1.1352\times 10^{-9} $. 图 1是拟合结果.

图图 1

图图 1   </b></p> </div> </div> <br> <div class="paragraph"> <div class="content-zw-1"> <p id="C10"><a class="table-icon" style="color:#2150f9" href="#Fig1"; id="inline_content图 1">图 1</a>模拟了中国2004-2016年的麻疹发病人数和模型(2.2)的染病者人数.虚线是<span class="formulaText"><inline-formula><tex-math id="M25">$ 2004-2016 $</tex-math></inline-formula></span>年的麻疹年数据,实线是模型(2.2)的<span class="formulaText"><inline-formula><tex-math id="M26">$ I(t) $</tex-math></inline-formula></span><span class="formulaNumber">.</span>根据文献[<a class="demo-basic" href="javascript:;" onmouseover="jjaxxawwhah(this,'b9')">9</a>]和中国疾病预防控制中心(2017)中的数据,取</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label/><tex-math id="FE2"> $ S(0) = 1.29\times10^{9}, \ V(0) = 1.06\times10^{9}, \ E(0) = 18527, \ I(0) = 70549, \ B(0) = 564392. $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C11">事实上,由模型(2.2)的前四个方程有</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label/><tex-math id="FE4"> $ \frac{{\rm d}(S+V+E+I)}{{\rm d}t} = \lambda-\mu(S+V+E+I)-\delta V-\gamma I\leq\lambda-\mu(S+V+E+I). $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C12">于是可得</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label/><tex-math id="FE5"> $ \limsup\limits_{t\rightarrow\infty}(S+V+E+I)\leq\frac{\lambda}{\mu}, \quad \limsup\limits_{t\rightarrow\infty}B\leq\frac{k \lambda}{\tau\mu}. $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C13">此外,容易证明</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label/><tex-math id="FE6"> $ \Omega = \Big\{(S, V, E, I, B)\in R^{5}_{+}:S, V, E, I, B\geq0, S+V+E+I\leq\frac{\lambda}{\mu}, B\leq\frac{k\lambda}{\tau\mu}\Big\} $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C14">是模型(2.2)的正不变集.以下在<span class="formulaText"><inline-formula><tex-math id="M27">$ \Omega $</tex-math></inline-formula></span>上研究模型(2.2)的动力学行为.</p> </div> </div> <h2 class="title-biaoti outline_anchor" level="1" id="outline_anchor_1"> 3平衡点的存在性与基本再生数 </h2> <div class="paragraph"> <div class="content-zw-1"> <p id="C15">模型(2.2)存在无病平衡点<span class="formulaText"><inline-formula><tex-math id="M28">$ P_{0}(S_{0}, V_{0}, 0, 0, 0) $</tex-math></inline-formula></span>,其中<span class="formulaText"><inline-formula><tex-math id="M29">$ S_{0} = \frac{\lambda(1-\eta)}{\mu} $</tex-math></inline-formula></span>, <span class="formulaText"><inline-formula><tex-math id="M30">$ V_{0} = \frac{\lambda\eta}{\mu+\delta} $</tex-math></inline-formula></span><span class="formulaNumber">.</span>通过下一代矩阵方法<sup>[<a class="demo-basic" href="javascript:;" onmouseover="jjaxxawwhah(this,'b10')">10</a>]</sup>可得模型(2.2)的基本再生数为</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label/><tex-math id="FE7"> $ R_{0} = \frac{(\beta S_{0}+\varepsilon\beta V_{0})p \tau+(\alpha S_{0}+\varepsilon\alpha V_{0})pk}{\tau(\mu+p)(\mu+\gamma)} = :R^{I}_{0}+R^{B}_{0}, $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C16">其中</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label/><tex-math id="FE8"> $ R^{I}_{0} = \frac{p\beta(S_{0}+\varepsilon V_{0})}{(\mu+p)(\mu+\gamma)}, \quad R^{B}_{0} = \frac{kp\alpha(S_{0}+\varepsilon V_{0})}{\tau(\mu+p)(\mu+\gamma)}. $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C17"><strong>注 3.1</strong>    由文献[<a class="demo-basic" href="javascript:;" onmouseover="jjaxxawwhah(this,'b5')">5</a>]知, <span class="formulaText"><inline-formula><tex-math id="M31">$ R^{I}_{0} $</tex-math></inline-formula></span>为不考虑环境传播(即<span class="formulaText"><inline-formula><tex-math id="M32">$ \alpha = 0 $</tex-math></inline-formula></span><span class="formulaNumber">)</span>时模型<span class="formulaText"><inline-formula><tex-math id="M33">$ (2.2) $</tex-math></inline-formula></span>的基本再生数.此外, <span class="formulaText"><inline-formula><tex-math id="M34">$ R^{B}_{0} $</tex-math></inline-formula></span>为环境传播的基本再生数.从而,环境传播会促进疾病的传播.</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C18">由文献[<a class="demo-basic" href="javascript:;" onmouseover="jjaxxawwhah(this,'b10')">10</a>]知,下列结论成立.</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C19"><strong>定理 3.1</strong>    若<span class="formulaText"><inline-formula><tex-math id="M35">$ R_{0} < 1 $</tex-math></inline-formula></span>,则模型<span class="formulaText"><inline-formula><tex-math id="M36">$ (2.2) $</tex-math></inline-formula></span>的无病平衡点<span class="formulaText"><inline-formula><tex-math id="M37">$ P_{0} $</tex-math></inline-formula></span>是局部渐近稳定的;若<span class="formulaText"><inline-formula><tex-math id="M38">$ R_{0} > 1 $</tex-math></inline-formula></span>,则模型<span class="formulaText"><inline-formula><tex-math id="M39">$ (2.2) $</tex-math></inline-formula></span>的无病平衡点<span class="formulaText"><inline-formula><tex-math id="M40">$ P_{0} $</tex-math></inline-formula></span>是不稳定的.</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C20">下面考虑模型(2.2)的地方病平衡点<span class="formulaText"><inline-formula><tex-math id="M41">$ P^{*}(S^{*}, V^{*}, E^{*}, I^{*}, B^{*}) $</tex-math></inline-formula></span><span class="formulaText"><inline-formula><tex-math id="M42">$ (S^{*} > 0, V^{*} > 0, E^{*} > 0, $</tex-math></inline-formula></span><span class="formulaText"><inline-formula><tex-math id="M43">$ I^{*} > 0, $</tex-math></inline-formula></span><span class="formulaText"><inline-formula><tex-math id="M44">$ B^{*} > 0) $</tex-math></inline-formula></span>的存在性.</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C21">考虑代数方程组</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label>((3.1))</label><tex-math id="E3.1"> $ \begin{equation} \left\{\begin{array}{ll} \lambda(1-\eta)-\beta SI-\alpha SB-\mu S = 0, \\ \lambda\eta-\varepsilon\beta VI-\varepsilon\alpha VB-\mu V-\delta V = 0, \\ \beta SI+\alpha SB+\varepsilon\beta VI+\varepsilon\alpha VB-\mu E-pE = 0, \\ pE-\mu I-\gamma I = 0, \\ kI-\tau B = 0. \end{array}\right. \end{equation} $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C22">由方程组(3.1)的第四个,第五个方程可得</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label>((3.2))</label><tex-math id="E3.2"> $ \begin{equation} E = \frac{(\mu+\gamma)I}{p}, \quad B = \frac{k I}{\tau}. \end{equation} $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C23">对于<span class="formulaText"><inline-formula><tex-math id="M45">$ I\neq0 $</tex-math></inline-formula></span>,将(3.2)式代入第一个和第二个方程分别可得</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label/><tex-math id="FE9"> $ S = \frac{\tau\lambda(1-\eta)}{(\tau\beta+k\alpha)I+\mu\tau}, \quad V = \frac{\tau\lambda\eta}{\varepsilon(\tau\beta+k\alpha)I+\tau(\mu+\delta)}. $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C24">将(3.2)式代入第三个方程可得</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label/><tex-math id="FE10"> $ S+\varepsilon V = \frac{\tau(\mu+p)(\mu+\gamma)}{p(\tau\beta+k\alpha)} = \frac{S_{0}+\varepsilon V_{0}}{R_{0}}. $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C25">定义函数</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label/><tex-math id="FE11"> $ f(I): = S+\varepsilon V-\frac{S_{0}+\varepsilon V_{0}}{R_{0}}\\ = \frac{\tau\lambda(1-\eta)}{(\tau\beta+k\alpha)I+\mu\tau} +\frac{\varepsilon\tau\lambda\eta}{\varepsilon(\tau\beta+k\alpha)I+\tau(\mu+\delta)}-\frac{S_{0}+\varepsilon V_{0}}{R_{0}}. $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C26">由于对于<span class="formulaText"><inline-formula><tex-math id="M46">$ I\geq0 $</tex-math></inline-formula></span>,有</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label/><tex-math id="FE12"> $ f'(I) = -\frac{\tau\lambda(1-\eta)(\tau\beta+k\alpha)}{[(\tau\beta+k\alpha)I+\mu\tau]^{2}} -\frac{\varepsilon^{2}\tau\lambda\eta(\tau\beta+k\alpha)}{[\varepsilon(\tau\beta+k\alpha)I+\tau(\mu+\delta)]^{2}} <0, $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C27">故<span class="formulaText"><inline-formula><tex-math id="M47">$ f(I) $</tex-math></inline-formula></span>是关于<span class="formulaText"><inline-formula><tex-math id="M48">$ I $</tex-math></inline-formula></span>的单调递减函数.</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C28">当<span class="formulaText"><inline-formula><tex-math id="M49">$ R_{0} > 1 $</tex-math></inline-formula></span>时, <span class="formulaText"><inline-formula><tex-math id="M50">$ f(0) = S_{0}+\varepsilon V_{0}-\frac{S_{0}+\varepsilon V_{0}}{R_{0}} > 0 $</tex-math></inline-formula></span>, <span class="formulaText"><inline-formula><tex-math id="M51">$ f(\frac{\lambda}{\mu}) < 0 $</tex-math></inline-formula></span><span class="formulaNumber">.</span>从而,存在唯一的正数<span class="formulaText"><inline-formula><tex-math id="M52">$ I^{*}\in(0, \frac{\lambda}{\mu}) $</tex-math></inline-formula></span>使得<span class="formulaText"><inline-formula><tex-math id="M53">$ f(I^{*}) = 0 $</tex-math></inline-formula></span><span class="formulaNumber">.</span>进而<span class="formulaText"><inline-formula><tex-math id="M54">$ (S^{*}, V^{*}, E^{*}, I^{*}, B^{*}) $</tex-math></inline-formula></span>满足方程组(3.1),其中<span class="formulaText"><inline-formula><tex-math id="M55">$ S^{*} = \frac{\tau\lambda(1-\eta)}{(\tau\beta+k\alpha)I^{*}+\mu\tau} $</tex-math></inline-formula></span>, <span class="formulaText"><inline-formula><tex-math id="M56">$ V^{*} = \frac{\tau\lambda\eta}{\varepsilon(\tau\beta+k\alpha)I^{*}+\tau(\mu+\delta)} $</tex-math></inline-formula></span>, <span class="formulaText"><inline-formula><tex-math id="M57">$ E^{*} = \frac{(\mu+\gamma)I^{*}}{p} $</tex-math></inline-formula></span>, <span class="formulaText"><inline-formula><tex-math id="M58">$ B^{*} = \frac{kI^{*}}{\tau} $</tex-math></inline-formula></span><span class="formulaNumber">.</span>此外,可以证明<span class="formulaText"><inline-formula><tex-math id="M59">$ P^{*}(S^{*}, V^{*}, E^{*}, I^{*}, B^{*})\in\Omega $</tex-math></inline-formula></span>,因此模型(2.2)存在唯一的地方病平衡点<span class="formulaText"><inline-formula><tex-math id="M60">$ P^{*}(S^{*}, V^{*}, E^{*}, I^{*}, B^{*}) $</tex-math></inline-formula></span><span class="formulaNumber">.</span></p> </div> </div> <h2 class="title-biaoti outline_anchor" level="1" id="outline_anchor_1"> 4平衡点的全局稳定性 </h2> <div class="paragraph"> <div class="content-zw-1"> <p id="C29">下面给出模型(2.2)的平衡点的全局渐近稳定性.</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C30"><strong>定理 4.1</strong>    当<span class="formulaText"><inline-formula><tex-math id="M61">$ R_{0} < 1 $</tex-math></inline-formula></span>时,模型<span class="formulaText"><inline-formula><tex-math id="M62">$ (2.2) $</tex-math></inline-formula></span>的无病平衡点<span class="formulaText"><inline-formula><tex-math id="M63">$ P_{0}(S_{0}, V_{0}, 0, 0, 0) $</tex-math></inline-formula></span>在<span class="formulaText"><inline-formula><tex-math id="M64">$ \Omega $</tex-math></inline-formula></span>上是全局渐近稳定的.</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C31"><strong>证</strong>    根据模型(2.2)的无病平衡点<span class="formulaText"><inline-formula><tex-math id="M65">$ P_{0}(S_{0}, V_{0}, 0, 0, 0) $</tex-math></inline-formula></span>,模型(2.2)可改写为</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label>((4.1))</label><tex-math id="E4.1"> $ \begin{equation} \left\{\begin{array}{ll} S' = S[\lambda(1-\eta)(\frac{1}{S}-\frac{1}{S_{0}})-\beta I-\alpha B], \\ V' = V[\lambda\eta(\frac{1}{V}-\frac{1}{V_{0}})-\varepsilon\beta I-\varepsilon\alpha B], \\ E' = (\beta I+\alpha B)[(S-S_{0})+\varepsilon(V-V_{0})+(S_{0}+\varepsilon V_{0})]-(\mu+p)E, \\ I' = pE-(\mu+\gamma)I, \\ B' = kI-\tau B. \end{array}\right. \end{equation} $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C32">定义Lyapunov函数</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label/><tex-math id="FE13"> $ L_{1}(S, V, E, I, B) = \int^{S}_{S_{0}}\frac{\theta-S_{0}}{\theta}{\rm d}\theta+ \int^{V}_{V_{0}}\frac{\theta-V_{0}}{\theta}{\rm d}\theta+E+\frac{\mu+p}{p}I+\frac{\alpha(S_{0}+\varepsilon V_{0})}{\tau}B. $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C33">函数<span class="formulaText"><inline-formula><tex-math id="M66">$ L_{1} $</tex-math></inline-formula></span>沿着模型(4.1)的全导数为</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label/><tex-math id="FE14"> $ \begin{eqnarray*} L'_{1}& = &(S-S_{0})[\lambda(1-\eta)(\frac{1}{S}-\frac{1}{S_{0}})-\beta I-\alpha B]+ (V-V_{0})[\lambda\eta(\frac{1}{V}-\frac{1}{V_{0}})-\varepsilon\beta I-\varepsilon\alpha B] \\ &\quad&+(\beta I+\alpha B)[(S-S_{0})+\varepsilon(V-V_{0})+(S_{0}+\varepsilon V_{0})]-(\mu+p)E \\ &\quad&+\frac{\mu+p}{p}[pE-(\mu+\gamma)I]+\frac{\alpha(S_{0}+\varepsilon V_{0})}{\tau}(kI-\tau B) \\ & = &\frac{(\mu+p)(\mu+\gamma)}{p}(R_{0}-1)I+\lambda(1-\eta)(2-\frac{S}{S_{0}}-\frac{S_{0}}{S}) +\lambda\eta(2-\frac{V}{V_{0}}-\frac{V_{0}}{V}). \end{eqnarray*} $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C34">当<span class="formulaText"><inline-formula><tex-math id="M67">$ R_{0} < 1 $</tex-math></inline-formula></span>时, <span class="formulaText"><inline-formula><tex-math id="M68">$ L'_{1}\leq0 $</tex-math></inline-formula></span><span class="formulaNumber">.</span>此外, <span class="formulaText"><inline-formula><tex-math id="M69">$ L'_{1} = 0 $</tex-math></inline-formula></span>当且仅当<span class="formulaText"><inline-formula><tex-math id="M70">$ S = S_{0}, V = V_{0}, I = 0 $</tex-math></inline-formula></span><span class="formulaNumber">.</span>所以单点集<span class="formulaText"><inline-formula><tex-math id="M71">$ \{P_{0}\} $</tex-math></inline-formula></span>是模型(2.2)在集合<span class="formulaText"><inline-formula><tex-math id="M72">$ \{(S, V, E, I, B)\in\Omega\mid L'_{1} = 0\} $</tex-math></inline-formula></span>上的最大不变集.由定理3.1和LaSalle不变集原理<sup>[<a class="demo-basic" href="javascript:;" onmouseover="jjaxxawwhah(this,'b11')">11</a>]</sup>,当<span class="formulaText"><inline-formula><tex-math id="M73">$ R_{0} < 1 $</tex-math></inline-formula></span>时,无病平衡点<span class="formulaText"><inline-formula><tex-math id="M74">$ P_{0} $</tex-math></inline-formula></span>在<span class="formulaText"><inline-formula><tex-math id="M75">$ \Omega $</tex-math></inline-formula></span>上是全局渐近稳定的.定理得证.</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C35"><strong>定理 4.2</strong>    当<span class="formulaText"><inline-formula><tex-math id="M76">$ R_{0} > 1 $</tex-math></inline-formula></span>时,模型<span class="formulaText"><inline-formula><tex-math id="M77">$ (2.2) $</tex-math></inline-formula></span>的地方病平衡点<span class="formulaText"><inline-formula><tex-math id="M78">$ P^{*}(S^{*}, V^{*}, E^{*}, I^{*}, B^{*}) $</tex-math></inline-formula></span>在<span class="formulaText"><inline-formula><tex-math id="M79">$ \Omega\setminus\{P_{0}\} $</tex-math></inline-formula></span>上是全局渐近稳定的.</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C36"><strong>证</strong>    对模型(2.2)的地方病平衡点<span class="formulaText"><inline-formula><tex-math id="M80">$ P^{*}(S^{*}, V^{*}, E^{*}, I^{*}, B^{*}) $</tex-math></inline-formula></span>,根据方程组(3.1),模型(2.2)可改写为</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label>((4.2))</label><tex-math id="E4.2"> $ \begin{equation} \left\{\begin{array}{ll} S' = S[\lambda(1-\eta)(\frac{1}{S}-\frac{1}{S^{*}})-\beta(I-I^{*})-\alpha(B-B^{*})], \\ V' = V[\lambda\eta(\frac{1}{V}-\frac{1}{V^{*}})-\varepsilon\beta(I-I^{*})-\varepsilon\alpha(B-B^{*})], \\ E' = E[\beta(\frac{SI}{E}-\frac{S^{*}I^{*}}{E^{*}})+\alpha(\frac{SB}{E}-\frac{S^{*}B^{*}}{E^{*}}) +\varepsilon\beta(\frac{VI}{E}-\frac{V^{*}I^{*}}{E^{*}}) +\varepsilon\alpha(\frac{VB}{E}-\frac{V^{*}B^{*}}{E^{*}})], \\ I' = pI(\frac{E}{I}-\frac{E^{*}}{I^{*}}), \quad B' = kB(\frac{I}{B}-\frac{I^{*}}{B^{*}}). \end{array}\right. \end{equation} $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C37">定义Lyapunov函数</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label/><tex-math id="FE15"> $ \begin{eqnarray*} L_{2}(S, V, E, I, B)& = &\int^{S}_{S^{*}}\frac{\theta-S^{*}}{\theta}{\rm d}\theta+ \int^{V}_{V^{*}}\frac{\theta-V^{*}}{\theta}{\rm d}\theta+\int^{E}_{E^{*}}\frac{\theta-E^{*}}{\theta}{\rm d}\theta \\ &\quad&+\frac{(S^{*}+\varepsilon V^{*})(\beta I^{*}+\alpha B^{*})}{pE^{*}}\int^{I}_{I^{*}}\frac{\theta-I^{*}}{\theta}{\rm d}\theta \\ &\quad&+\frac{\alpha B^{*}(S^{*}+\varepsilon V^{*})}{kI^{*}}\int^{B}_{B^{*}}\frac{\theta-B^{*}}{\theta}{\rm d}\theta. \end{eqnarray*} $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C38">容易验证<span class="formulaText"><inline-formula><tex-math id="M81">$ L_{2}(S, V, E, I, B) $</tex-math></inline-formula></span>是无穷大正定函数,则函数<span class="formulaText"><inline-formula><tex-math id="M82">$ L_{2} $</tex-math></inline-formula></span>沿着模型(4.2)的全导数为</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label/><tex-math id="FE16"> $ \begin{eqnarray*} L'_{2}& = &(S-S^{*})[\lambda(1-\eta)(\frac{1}{S}-\frac{1}{S^{*}})-\beta(I-I^{*})-\alpha(B-B^{*})] \\ &\quad&+(V-V^{*})[\lambda\eta(\frac{1}{V}-\frac{1}{V^{*}})-\varepsilon\beta(I-I^{*})-\varepsilon\alpha(B-B^{*})]+(E-E^{*}) \\ &\quad&\times[\beta(\frac{SI}{E}-\frac{S^{*}I^{*}}{E^{*}})+\alpha(\frac{SB}{E}-\frac{S^{*}B^{*}}{E^{*}}) +\varepsilon\beta(\frac{VI}{E}-\frac{V^{*}I^{*}}{E^{*}}) +\varepsilon\alpha(\frac{VB}{E}-\frac{V^{*}B^{*}}{E^{*}})] \\ &\quad&+\frac{(S^{*}+\varepsilon V^{*})(\beta I^{*}+\alpha B^{*})}{E^{*}}(I-I^{*})(\frac{E}{I}-\frac{E^{*}}{I^{*}})\\ &\quad&+\frac{\alpha B^{*}(S^{*}+\varepsilon V^{*})}{I^{*}}(B-B^{*})(\frac{I}{B}-\frac{I^{*}}{B^{*}}) \\ & = &2\lambda(1-\eta)+2\lambda\eta+\beta S^{*}I^{*}+2\alpha S^{*}B^{*}+\varepsilon\beta V^{*}I^{*}+2\varepsilon\alpha V^{*}B^{*} \\ &\quad&+[\lambda(1-\eta)-\beta S^{*}I^{*}-\alpha S^{*}B^{*}]\frac{S}{S^{*}}-(\lambda\eta-\varepsilon\beta V^{*}I^{*}-\varepsilon\alpha V^{*}B^{*})\frac{V}{V^{*}} \\ &\quad&-\lambda(1-\eta)\frac{S^{*}}{S}-\lambda\eta\frac{V^{*}}{V}-\beta\frac{SIE^{*}}{E}-\alpha\frac{SBE^{*}}{E} -\varepsilon\beta\frac{VIE^{*}}{E}-\varepsilon\alpha\frac{VBE^{*}}{E} \\ &\quad&-\frac{(S^{*}+\varepsilon V^{*})(\beta I^{*}+\alpha B^{*})}{E^{*}}\frac{EI^{*}}{I}-\frac{\alpha B^{*}(S^{*}+\varepsilon V^{*})}{I^{*}} \frac{IB^{*}}{B}. \end{eqnarray*} $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C39">因为</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label/><tex-math id="FE17"> $ \lambda(1-\eta) = \beta S^{*}I^{*}+\alpha S^{*}B^{*}+\mu S^{*}, \lambda\eta = \varepsilon\beta V^{*}I^{*}+\varepsilon\alpha V^{*}B^{*} +\mu V^{*}+\delta V^{*}, $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C40">所以</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label/><tex-math id="FE18"> $ \begin{eqnarray*} L'_{2}& = &\mu S^{*}(2-\frac{S}{S^{*}}-\frac{S^{*}}{S})+\beta S^{*}I^{*}(3-\frac{S^{*}}{S}-\frac{EI^{*}}{E^{*}I}-\frac{SIE^{*}}{S^{*}I^{*}E})\\ &\quad&+\alpha S^{*}B^{*}(4-\frac{S^{*}}{S}-\frac{EI^{*}}{E^{*}I}-\frac{IB^{*}}{I^{*}B}-\frac{SBE^{*}}{S^{*}B^{*}E})\\ &\quad&+(\mu V^{*}+\delta V^{*})(2-\frac{V}{V^{*}}-\frac{V^{*}}{V})\\ &\quad&+\varepsilon\beta V^{*}I^{*}(3-\frac{V^{*}}{V}-\frac{EI^{*}}{E^{*}I}-\frac{VIE^{*}}{V^{*}I^{*}E})\\ &\quad&+\varepsilon\alpha V^{*}B^{*}(4-\frac{V^{*}}{V}-\frac{EI^{*}}{E^{*}I}-\frac{IB^{*}}{I^{*}B}-\frac{VBE^{*}}{V^{*}B^{*}E}). \end{eqnarray*} $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C41">由均值不等式可知, <span class="formulaText"><inline-formula><tex-math id="M83">$ L'_{2}\leq0 $</tex-math></inline-formula></span><span class="formulaNumber">.</span>此外, <span class="formulaText"><inline-formula><tex-math id="M84">$ L'_{2} = 0 $</tex-math></inline-formula></span>当且仅当<span class="formulaText"><inline-formula><tex-math id="M85">$ S = S^{*}, V = V^{*}, \frac{E}{E^{*}} = \frac{I}{I^{*}} = \frac{B}{B^{*}} $</tex-math></inline-formula></span><span class="formulaNumber">.</span>令</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label/><tex-math id="FE19"> $ M = \{(S, V, E, I, B)\mid L'_{2} = 0\}. $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C42">显然<span class="formulaText"><inline-formula><tex-math id="M86">$ P^{*}\in M $</tex-math></inline-formula></span>,即<span class="formulaText"><inline-formula><tex-math id="M87">$ M $</tex-math></inline-formula></span>非空.任取<span class="formulaText"><inline-formula><tex-math id="M88">$ (S^{*}, V^{*}, E_{1}, I_{1}, B_{1})\in M $</tex-math></inline-formula></span>且<span class="formulaText"><inline-formula><tex-math id="M89">$ (S^{*}, V^{*}, E_{1}, I_{1}, B_{1}) $</tex-math></inline-formula></span>为模型(4.2)的解,则有</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label/><tex-math id="FE20"> $ E'_{1} = \beta I^{*}(S^{*}+\varepsilon V^{*})(\frac{I_{1}}{I^{*}}-\frac{E_{1}}{E^{*}})+\alpha B^{*}(S^{*}+\varepsilon V^{*})(\frac{B_{1}}{B^{*}}-\frac{E_{1}}{E^{*}}) = 0, $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label/><tex-math id="FE21"> $ I'_{1} = pE^{*}(\frac{E_{1}}{E^{*}}-\frac{I_{1}}{I^{*}}) = 0, \qquad B'_{1} = kB^{*}(\frac{I_{1}}{I^{*}}-\frac{B_{1}}{B^{*}}) = 0. $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C43">从而<span class="formulaText"><inline-formula><tex-math id="M90">$ (S^{*}, V^{*}, E_{1}, I_{1}, B_{1}) $</tex-math></inline-formula></span>为模型(4.2)的平衡点.由于<span class="formulaText"><inline-formula><tex-math id="M91">$ S^{*} < S_{0} $</tex-math></inline-formula></span>,从而<span class="formulaText"><inline-formula><tex-math id="M92">$ (S^{*}, V^{*}, E_{1}, I_{1}, B_{1}) $</tex-math></inline-formula></span>为平衡点<span class="formulaText"><inline-formula><tex-math id="M93">$ P^{*} $</tex-math></inline-formula></span><span class="formulaNumber">.</span>因此集合<span class="formulaText"><inline-formula><tex-math id="M94">$ M $</tex-math></inline-formula></span>内除<span class="formulaText"><inline-formula><tex-math id="M95">$ P^{*} $</tex-math></inline-formula></span>外不再包含模型(4.2)的其他轨线.由定理3.10 (文献[<a class="demo-basic" href="javascript:;" onmouseover="jjaxxawwhah(this,'b12')">12</a>])可知模型(4.2)的地方病平衡点是全局渐近稳定的.定理得证.</p> </div> </div> <h2 class="title-biaoti outline_anchor" level="1" id="outline_anchor_1"> 5数值仿真 </h2> <div class="paragraph"> <div class="content-zw-1"> <p id="C44">为了验证以上的理论分析,我们进行了数值模拟.</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C45"><a class="table-icon" style="color:#2150f9" href="#Fig2"; id="inline_content图 2">图 2</a>参数取值见<a class="table-icon" style="color:#2150f9" href="#T1"; id="inline_content表 1">表 1</a>,经过计算<span class="formulaText"><inline-formula><tex-math id="M96">$ R_{0} < 1 $</tex-math></inline-formula></span><span class="formulaNumber">.</span>模型(2.2)的无病平衡点是全局渐近稳定的.</p> </div> </div> <h3 style="position: absolute; opacity: 0; filter:Alpha(opacity=0);">图图 2</h3> <div class="content-zw-img" id=""> <div class="content-zw-img-img figure outline_anchor" onmouseleave="likai(this);"> <img src="sxwlxb-39-4-909-2.jpg" onclick="clickss(this)" onmouseover="huoqukuanduimg(this);" class="tupian"> <p class="tishi"> <a href="sxwlxb-39-4-909-2.jpg.html" target="_blank">新窗口打开</a>| <a href="sxwlxb-39-4-909-2.jpg.zip">下载原图ZIP</a>| <a href="sxwlxb-39-4-909-2.jpg.ppt">生成PPT</a> </p> </div> <div class="content-zw-img-shuoming"> <p class="content-zw-img-shuoming-title-cn"><b>图图 2   <title/> </b></p> </div> </div> <br> <div class="paragraph"> <div class="content-zw-1"> <p id="C46"><a class="table-icon" style="color:#2150f9" href="#Fig3"; id="inline_content图 3">图 3</a>取参数<span class="formulaText"><inline-formula><tex-math id="M99">$ \lambda = 1.728\times 10^{10} $</tex-math></inline-formula></span>,其余参数取值见<a class="table-icon" style="color:#2150f9" href="#T1"; id="inline_content表 1">表 1</a>,经过计算<span class="formulaText"><inline-formula><tex-math id="M100">$ R_{0} > 1 $</tex-math></inline-formula></span><span class="formulaNumber">.</span>模型(2.2)的地方病平衡点是全局渐近稳定的.</p> </div> </div> <h3 style="position: absolute; opacity: 0; filter:Alpha(opacity=0);">图图 3</h3> <div class="content-zw-img" id=""> <div class="content-zw-img-img figure outline_anchor" onmouseleave="likai(this);"> <img src="sxwlxb-39-4-909-3.jpg" onclick="clickss(this)" onmouseover="huoqukuanduimg(this);" class="tupian"> <p class="tishi"> <a href="sxwlxb-39-4-909-3.jpg.html" target="_blank">新窗口打开</a>| <a href="sxwlxb-39-4-909-3.jpg.zip">下载原图ZIP</a>| <a href="sxwlxb-39-4-909-3.jpg.ppt">生成PPT</a> </p> </div> <div class="content-zw-img-shuoming"> <p class="content-zw-img-shuoming-title-cn"><b>图图 3   <title/> </b></p> </div> </div> <br> <h2 class="title-biaoti outline_anchor" level="1" id="outline_anchor_1"> 6结束语 </h2> <div class="paragraph"> <div class="content-zw-1"> <p id="C47">本文建立了一个考虑部分免疫的麻疹传染病模型,并引入了环境传播的影响,得到了模型的基本再生数<span class="formulaText"><inline-formula><tex-math id="M103">$ R_{0} $</tex-math></inline-formula></span><span class="formulaNumber">.</span>当<span class="formulaText"><inline-formula><tex-math id="M104">$ R_{0} < 1 $</tex-math></inline-formula></span>时,模型仅存在可行域上全局渐近稳定的无病平衡点,即麻疹病毒最终灭绝;当<span class="formulaText"><inline-formula><tex-math id="M105">$ R_{0} > 1 $</tex-math></inline-formula></span>时,模型除无病平衡点外还存在唯一的地方病平衡点,此时地方病平衡点在可行域内是全局渐近稳定的,即麻疹病毒的传播保持在一个稳定的状态.</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C48">当模型(2.2)中不考虑环境传播的影响,我们可得到基本再生数</p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p><disp-formula><label/><tex-math id="FE22"> $ R_{1} = \frac{p\beta(S_{0}+\varepsilon V_{0})}{(\mu+p)(\mu+\gamma)} = R^{I}_{0}. $ </tex-math></disp-formula></p> </div> </div> <div class="paragraph"> <div class="content-zw-1"> <p id="C49">显然有<span class="formulaText"><inline-formula><tex-math id="M106">$ R_{1} < R_{0} $</tex-math></inline-formula></span>,所以忽略环境传播的影响会低估基本再生数.由此可见,对病人的生活环境加强管理,及时对空气以及病人使用过的物品消毒,避免直接接触染病者鼻腔或口腔的分泌物,能够有效地控制疾病的传播.</p> </div> </div> <div class="cankaowenxian1"></div> <h2 class="title-biaoti"> <span class="outline_anchor" level="1">参考文献 </span> <div class="btn-group"> <button style="font-size:11px;padding:3px;" type="button" onclick="sddas();" class="btn btn-info dropdown-toggle" 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<trans-source xml:lang="en">The Mathematical Modeling and Research of Infectious Diseases</trans-source>. <publisher-loc>Beijing</publisher-loc>: <publisher-name>Science Press</publisher-name>, <year>2004</year> </mixed-citation> </div> <p class="cankaowenxian-xx-x" id="linked_"> <span class="jcr_"></span> <span class="cjcr_"></span> <span class="cited" id="bd_cited_count_"></span> </p> <div class="xiangxicankao"> <p> </p> </div> <div> <div id="article_reference_meta" style="display: none;"> <div id="article_reference_meta_b1"> <div id="article_reference_meta_b1_title" class="title_">SV IR epidemic models with vaccination strategies</div> <div id="article_reference_meta_b1_citedNumber">3</div> <div id="article_reference_meta_b1_nian">2008</div> <div id="article_reference_meta_b1_jcr"></div> <div id="article_reference_meta_b1_cjcr"></div> <div id="article_reference_meta_b1_articleCitedText"> <div class="sentence">... 麻疹是儿童最常见的急性传染病,是一种传染性很强的呼吸道疾病,因此通过数学建模来了解它的动力学性态是非常有必要的<sup>[<xref ref-type="bibr" rid="b1">1</xref>-<xref ref-type="bibr" rid="b9">9</xref>]</sup>.人群普遍为易感者,接种疫苗是预防麻疹的最有效措施.由于麻疹疫苗需接种两次且间隔时间较长,导致漏种的情况及接种疫苗后免疫失败,所以接种疫苗者是部分免疫.此外,麻疹病毒有9-14天的潜伏期,接触过麻疹患者并在潜伏期接受被动免疫者可延至3-4周.未接种过疫苗的人,尤其是幼儿,具有罹患麻疹及其并发症(包括死亡)的最高风险.高传染性的麻疹病毒是通过空气中的飞沫传播(由于咳嗽和打喷嚏),或者是与染病者密切接触,或者是直接接触到他们的鼻腔和口腔分泌物进行传播.病毒在空气中或受染物表体的活力和传染性可维持两个小时.因此环境因素对于麻疹的传播有着重要的影响. ...</div> <div class="boundary"><p class="ty-x"></p></div> <div class="sentence">... 麻疹是已知最具有传染性的疾病之一.已经有许多学者对麻疹进行了研究.文献[<xref ref-type="bibr" rid="b1">1</xref>-<xref ref-type="bibr" rid="b3">3</xref>]研究了麻疹疫苗的有效性,其中文献[<xref ref-type="bibr" rid="b1">1</xref>]考虑了部分免疫对麻疹传播的影响.文献[<xref ref-type="bibr" rid="b4">4</xref>]研究了麻疹的潜伏期.文献[<xref ref-type="bibr" rid="b5">5</xref>]讨论了部分免疫和潜伏期共同作用下的麻疹传播动力学. Garba<sup>[<xref ref-type="bibr" rid="b6">6</xref>]</sup>等人在Trottier和Philippe<sup>[<xref ref-type="bibr" rid="b7">7</xref>-<xref ref-type="bibr" rid="b8">8</xref>]</sup>建立的标准传染率的麻疹传染病模型下,对感染麻疹的人群进行细化,把潜伏者分为接种过疫苗的潜伏者和未接种过疫苗的潜伏者,把染病者也分为接种过疫苗的染病者和未接种过疫苗的染病者,以及加入治疗者,研究了一个更为复杂的模型.文献[<xref ref-type="bibr" rid="b7">7</xref>, <xref ref-type="bibr" rid="b9">9</xref>]研究了麻疹的周期性.尽管我国提出2012年消除麻疹的目标,但全国麻疹疫情自2012年年底开始持续回升.文献[<xref ref-type="bibr" rid="b9">9</xref>]通过对我国麻疹月数据进行分析,得到麻疹随季节的周期性行为,进而进行数学建头治基于以上的各种研究,均没有考虑环境传播对麻疹的影响.基于此,本文考虑部分免疫和环境传播对麻疹流行性态的影响,构建数学模型,分析了模型的全局稳定性. ...</div> <div class="boundary"><p class="ty-x"></p></div> <div class="sentence">... ]研究了麻疹疫苗的有效性,其中文献[<xref ref-type="bibr" rid="b1">1</xref>]考虑了部分免疫对麻疹传播的影响.文献[<xref ref-type="bibr" rid="b4">4</xref>]研究了麻疹的潜伏期.文献[<xref ref-type="bibr" rid="b5">5</xref>]讨论了部分免疫和潜伏期共同作用下的麻疹传播动力学. Garba<sup>[<xref ref-type="bibr" rid="b6">6</xref>]</sup>等人在Trottier和Philippe<sup>[<xref ref-type="bibr" rid="b7">7</xref>-<xref ref-type="bibr" rid="b8">8</xref>]</sup>建立的标准传染率的麻疹传染病模型下,对感染麻疹的人群进行细化,把潜伏者分为接种过疫苗的潜伏者和未接种过疫苗的潜伏者,把染病者也分为接种过疫苗的染病者和未接种过疫苗的染病者,以及加入治疗者,研究了一个更为复杂的模型.文献[<xref ref-type="bibr" rid="b7">7</xref>, <xref ref-type="bibr" rid="b9">9</xref>]研究了麻疹的周期性.尽管我国提出2012年消除麻疹的目标,但全国麻疹疫情自2012年年底开始持续回升.文献[<xref ref-type="bibr" rid="b9">9</xref>]通过对我国麻疹月数据进行分析,得到麻疹随季节的周期性行为,进而进行数学建头治基于以上的各种研究,均没有考虑环境传播对麻疹的影响.基于此,本文考虑部分免疫和环境传播对麻疹流行性态的影响,构建数学模型,分析了模型的全局稳定性. ...</div> </div> </div> <div id="article_reference_meta_b2"> <div id="article_reference_meta_b2_title" class="title_">Modelling measles re-emergence as a result of waning of immunity in vaccinated populations</div> <div id="article_reference_meta_b2_citedNumber">0</div> <div id="article_reference_meta_b2_nian">2003</div> <div id="article_reference_meta_b2_jcr"></div> <div id="article_reference_meta_b2_cjcr"></div> </div> <div id="article_reference_meta_b3"> <div id="article_reference_meta_b3_title" class="title_">Real-time investigation of measles of vaccine efficacy</div> <div id="article_reference_meta_b3_citedNumber">1</div> <div id="article_reference_meta_b3_nian">2012</div> <div id="article_reference_meta_b3_jcr"></div> <div id="article_reference_meta_b3_cjcr"></div> <div id="article_reference_meta_b3_articleCitedText"> <div class="sentence">... 麻疹是已知最具有传染性的疾病之一.已经有许多学者对麻疹进行了研究.文献[<xref ref-type="bibr" rid="b1">1</xref>-<xref ref-type="bibr" rid="b3">3</xref>]研究了麻疹疫苗的有效性,其中文献[<xref ref-type="bibr" rid="b1">1</xref>]考虑了部分免疫对麻疹传播的影响.文献[<xref ref-type="bibr" rid="b4">4</xref>]研究了麻疹的潜伏期.文献[<xref ref-type="bibr" rid="b5">5</xref>]讨论了部分免疫和潜伏期共同作用下的麻疹传播动力学. Garba<sup>[<xref ref-type="bibr" rid="b6">6</xref>]</sup>等人在Trottier和Philippe<sup>[<xref ref-type="bibr" rid="b7">7</xref>-<xref ref-type="bibr" rid="b8">8</xref>]</sup>建立的标准传染率的麻疹传染病模型下,对感染麻疹的人群进行细化,把潜伏者分为接种过疫苗的潜伏者和未接种过疫苗的潜伏者,把染病者也分为接种过疫苗的染病者和未接种过疫苗的染病者,以及加入治疗者,研究了一个更为复杂的模型.文献[<xref ref-type="bibr" rid="b7">7</xref>, <xref ref-type="bibr" rid="b9">9</xref>]研究了麻疹的周期性.尽管我国提出2012年消除麻疹的目标,但全国麻疹疫情自2012年年底开始持续回升.文献[<xref ref-type="bibr" rid="b9">9</xref>]通过对我国麻疹月数据进行分析,得到麻疹随季节的周期性行为,进而进行数学建头治基于以上的各种研究,均没有考虑环境传播对麻疹的影响.基于此,本文考虑部分免疫和环境传播对麻疹流行性态的影响,构建数学模型,分析了模型的全局稳定性. ...</div> </div> </div> <div id="article_reference_meta_b4"> <div id="article_reference_meta_b4_title" class="title_">On the dynamical analysis of a new model for measles infection</div> <div id="article_reference_meta_b4_citedNumber">1</div> <div id="article_reference_meta_b4_nian">2014</div> <div id="article_reference_meta_b4_jcr"></div> <div id="article_reference_meta_b4_cjcr"></div> <div id="article_reference_meta_b4_articleCitedText"> <div class="sentence">... 麻疹是已知最具有传染性的疾病之一.已经有许多学者对麻疹进行了研究.文献[<xref ref-type="bibr" rid="b1">1</xref>-<xref ref-type="bibr" rid="b3">3</xref>]研究了麻疹疫苗的有效性,其中文献[<xref ref-type="bibr" rid="b1">1</xref>]考虑了部分免疫对麻疹传播的影响.文献[<xref ref-type="bibr" rid="b4">4</xref>]研究了麻疹的潜伏期.文献[<xref ref-type="bibr" rid="b5">5</xref>]讨论了部分免疫和潜伏期共同作用下的麻疹传播动力学. Garba<sup>[<xref ref-type="bibr" rid="b6">6</xref>]</sup>等人在Trottier和Philippe<sup>[<xref ref-type="bibr" rid="b7">7</xref>-<xref ref-type="bibr" rid="b8">8</xref>]</sup>建立的标准传染率的麻疹传染病模型下,对感染麻疹的人群进行细化,把潜伏者分为接种过疫苗的潜伏者和未接种过疫苗的潜伏者,把染病者也分为接种过疫苗的染病者和未接种过疫苗的染病者,以及加入治疗者,研究了一个更为复杂的模型.文献[<xref ref-type="bibr" rid="b7">7</xref>, <xref ref-type="bibr" rid="b9">9</xref>]研究了麻疹的周期性.尽管我国提出2012年消除麻疹的目标,但全国麻疹疫情自2012年年底开始持续回升.文献[<xref ref-type="bibr" rid="b9">9</xref>]通过对我国麻疹月数据进行分析,得到麻疹随季节的周期性行为,进而进行数学建头治基于以上的各种研究,均没有考虑环境传播对麻疹的影响.基于此,本文考虑部分免疫和环境传播对麻疹流行性态的影响,构建数学模型,分析了模型的全局稳定性. ...</div> </div> </div> <div id="article_reference_meta_b5"> <div id="article_reference_meta_b5_title" class="title_">考虑部分免疫和潜伏期的麻疹传染病模型的稳定性分析</div> <div id="article_reference_meta_b5_citedNumber">2</div> <div id="article_reference_meta_b5_nian">2017</div> <div id="article_reference_meta_b5_jcr"></div> <div id="article_reference_meta_b5_cjcr"></div> <div id="article_reference_meta_b5_articleCitedText"> <div class="sentence">... 麻疹是已知最具有传染性的疾病之一.已经有许多学者对麻疹进行了研究.文献[<xref ref-type="bibr" rid="b1">1</xref>-<xref ref-type="bibr" rid="b3">3</xref>]研究了麻疹疫苗的有效性,其中文献[<xref ref-type="bibr" rid="b1">1</xref>]考虑了部分免疫对麻疹传播的影响.文献[<xref ref-type="bibr" rid="b4">4</xref>]研究了麻疹的潜伏期.文献[<xref ref-type="bibr" rid="b5">5</xref>]讨论了部分免疫和潜伏期共同作用下的麻疹传播动力学. Garba<sup>[<xref ref-type="bibr" rid="b6">6</xref>]</sup>等人在Trottier和Philippe<sup>[<xref ref-type="bibr" rid="b7">7</xref>-<xref ref-type="bibr" rid="b8">8</xref>]</sup>建立的标准传染率的麻疹传染病模型下,对感染麻疹的人群进行细化,把潜伏者分为接种过疫苗的潜伏者和未接种过疫苗的潜伏者,把染病者也分为接种过疫苗的染病者和未接种过疫苗的染病者,以及加入治疗者,研究了一个更为复杂的模型.文献[<xref ref-type="bibr" rid="b7">7</xref>, <xref ref-type="bibr" rid="b9">9</xref>]研究了麻疹的周期性.尽管我国提出2012年消除麻疹的目标,但全国麻疹疫情自2012年年底开始持续回升.文献[<xref ref-type="bibr" rid="b9">9</xref>]通过对我国麻疹月数据进行分析,得到麻疹随季节的周期性行为,进而进行数学建头治基于以上的各种研究,均没有考虑环境传播对麻疹的影响.基于此,本文考虑部分免疫和环境传播对麻疹流行性态的影响,构建数学模型,分析了模型的全局稳定性. ...</div> <div class="boundary"><p class="ty-x"></p></div> <div class="sentence">... <strong>注 3.1</strong>    由文献[<xref ref-type="bibr" rid="b5">5</xref>]知, <span class="formulaText"><inline-formula><tex-math id="M31">$ R^{I}_{0} $</tex-math></inline-formula></span>为不考虑环境传播(即<span class="formulaText"><inline-formula><tex-math id="M32">$ \alpha = 0 $</tex-math></inline-formula></span><span class="formulaNumber">)</span>时模型<span class="formulaText"><inline-formula><tex-math id="M33">$ (2.2) $</tex-math></inline-formula></span>的基本再生数.此外, <span class="formulaText"><inline-formula><tex-math id="M34">$ R^{B}_{0} $</tex-math></inline-formula></span>为环境传播的基本再生数.从而,环境传播会促进疾病的传播. ...</div> </div> </div> <div id="article_reference_meta_b5"> <div id="article_reference_meta_b5_title" class="title_">考虑部分免疫和潜伏期的麻疹传染病模型的稳定性分析</div> <div id="article_reference_meta_b5_citedNumber">2</div> <div id="article_reference_meta_b5_nian">2017</div> <div id="article_reference_meta_b5_jcr"></div> <div id="article_reference_meta_b5_cjcr"></div> <div id="article_reference_meta_b5_articleCitedText"> <div class="sentence">... 麻疹是已知最具有传染性的疾病之一.已经有许多学者对麻疹进行了研究.文献[<xref ref-type="bibr" rid="b1">1</xref>-<xref ref-type="bibr" rid="b3">3</xref>]研究了麻疹疫苗的有效性,其中文献[<xref ref-type="bibr" rid="b1">1</xref>]考虑了部分免疫对麻疹传播的影响.文献[<xref ref-type="bibr" rid="b4">4</xref>]研究了麻疹的潜伏期.文献[<xref ref-type="bibr" rid="b5">5</xref>]讨论了部分免疫和潜伏期共同作用下的麻疹传播动力学. Garba<sup>[<xref ref-type="bibr" rid="b6">6</xref>]</sup>等人在Trottier和Philippe<sup>[<xref ref-type="bibr" rid="b7">7</xref>-<xref ref-type="bibr" rid="b8">8</xref>]</sup>建立的标准传染率的麻疹传染病模型下,对感染麻疹的人群进行细化,把潜伏者分为接种过疫苗的潜伏者和未接种过疫苗的潜伏者,把染病者也分为接种过疫苗的染病者和未接种过疫苗的染病者,以及加入治疗者,研究了一个更为复杂的模型.文献[<xref ref-type="bibr" rid="b7">7</xref>, <xref ref-type="bibr" rid="b9">9</xref>]研究了麻疹的周期性.尽管我国提出2012年消除麻疹的目标,但全国麻疹疫情自2012年年底开始持续回升.文献[<xref ref-type="bibr" rid="b9">9</xref>]通过对我国麻疹月数据进行分析,得到麻疹随季节的周期性行为,进而进行数学建头治基于以上的各种研究,均没有考虑环境传播对麻疹的影响.基于此,本文考虑部分免疫和环境传播对麻疹流行性态的影响,构建数学模型,分析了模型的全局稳定性. ...</div> <div class="boundary"><p class="ty-x"></p></div> <div class="sentence">... <strong>注 3.1</strong>    由文献[<xref ref-type="bibr" rid="b5">5</xref>]知, <span class="formulaText"><inline-formula><tex-math id="M31">$ R^{I}_{0} $</tex-math></inline-formula></span>为不考虑环境传播(即<span class="formulaText"><inline-formula><tex-math id="M32">$ \alpha = 0 $</tex-math></inline-formula></span><span class="formulaNumber">)</span>时模型<span class="formulaText"><inline-formula><tex-math id="M33">$ (2.2) $</tex-math></inline-formula></span>的基本再生数.此外, <span class="formulaText"><inline-formula><tex-math id="M34">$ R^{B}_{0} $</tex-math></inline-formula></span>为环境传播的基本再生数.从而,环境传播会促进疾病的传播. ...</div> </div> </div> <div id="article_reference_meta_b6"> <div id="article_reference_meta_b6_title" class="title_">Mathematical model for assessing the impact of vaccination and treatment on measles transmission dynamics</div> <div id="article_reference_meta_b6_citedNumber">1</div> <div id="article_reference_meta_b6_nian">2017</div> <div id="article_reference_meta_b6_jcr"></div> <div id="article_reference_meta_b6_cjcr"></div> <div id="article_reference_meta_b6_articleCitedText"> <div class="sentence">... 麻疹是已知最具有传染性的疾病之一.已经有许多学者对麻疹进行了研究.文献[<xref ref-type="bibr" rid="b1">1</xref>-<xref ref-type="bibr" rid="b3">3</xref>]研究了麻疹疫苗的有效性,其中文献[<xref ref-type="bibr" rid="b1">1</xref>]考虑了部分免疫对麻疹传播的影响.文献[<xref ref-type="bibr" rid="b4">4</xref>]研究了麻疹的潜伏期.文献[<xref ref-type="bibr" rid="b5">5</xref>]讨论了部分免疫和潜伏期共同作用下的麻疹传播动力学. Garba<sup>[<xref ref-type="bibr" rid="b6">6</xref>]</sup>等人在Trottier和Philippe<sup>[<xref ref-type="bibr" rid="b7">7</xref>-<xref ref-type="bibr" rid="b8">8</xref>]</sup>建立的标准传染率的麻疹传染病模型下,对感染麻疹的人群进行细化,把潜伏者分为接种过疫苗的潜伏者和未接种过疫苗的潜伏者,把染病者也分为接种过疫苗的染病者和未接种过疫苗的染病者,以及加入治疗者,研究了一个更为复杂的模型.文献[<xref ref-type="bibr" rid="b7">7</xref>, <xref ref-type="bibr" rid="b9">9</xref>]研究了麻疹的周期性.尽管我国提出2012年消除麻疹的目标,但全国麻疹疫情自2012年年底开始持续回升.文献[<xref ref-type="bibr" rid="b9">9</xref>]通过对我国麻疹月数据进行分析,得到麻疹随季节的周期性行为,进而进行数学建头治基于以上的各种研究,均没有考虑环境传播对麻疹的影响.基于此,本文考虑部分免疫和环境传播对麻疹流行性态的影响,构建数学模型,分析了模型的全局稳定性. ...</div> </div> </div> <div id="article_reference_meta_b7"> <div id="article_reference_meta_b7_title" class="title_">Deterministic modelling of infectious diseases:measles cycles and the role of births and vaccination</div> <div id="article_reference_meta_b7_citedNumber">2</div> <div id="article_reference_meta_b7_nian">2002</div> <div id="article_reference_meta_b7_jcr"></div> <div id="article_reference_meta_b7_cjcr"></div> <div id="article_reference_meta_b7_articleCitedText"> <div class="sentence">... 麻疹是已知最具有传染性的疾病之一.已经有许多学者对麻疹进行了研究.文献[<xref ref-type="bibr" rid="b1">1</xref>-<xref ref-type="bibr" rid="b3">3</xref>]研究了麻疹疫苗的有效性,其中文献[<xref ref-type="bibr" rid="b1">1</xref>]考虑了部分免疫对麻疹传播的影响.文献[<xref ref-type="bibr" rid="b4">4</xref>]研究了麻疹的潜伏期.文献[<xref ref-type="bibr" rid="b5">5</xref>]讨论了部分免疫和潜伏期共同作用下的麻疹传播动力学. Garba<sup>[<xref ref-type="bibr" rid="b6">6</xref>]</sup>等人在Trottier和Philippe<sup>[<xref ref-type="bibr" rid="b7">7</xref>-<xref ref-type="bibr" rid="b8">8</xref>]</sup>建立的标准传染率的麻疹传染病模型下,对感染麻疹的人群进行细化,把潜伏者分为接种过疫苗的潜伏者和未接种过疫苗的潜伏者,把染病者也分为接种过疫苗的染病者和未接种过疫苗的染病者,以及加入治疗者,研究了一个更为复杂的模型.文献[<xref ref-type="bibr" rid="b7">7</xref>, <xref ref-type="bibr" rid="b9">9</xref>]研究了麻疹的周期性.尽管我国提出2012年消除麻疹的目标,但全国麻疹疫情自2012年年底开始持续回升.文献[<xref ref-type="bibr" rid="b9">9</xref>]通过对我国麻疹月数据进行分析,得到麻疹随季节的周期性行为,进而进行数学建头治基于以上的各种研究,均没有考虑环境传播对麻疹的影响.基于此,本文考虑部分免疫和环境传播对麻疹流行性态的影响,构建数学模型,分析了模型的全局稳定性. ...</div> <div class="boundary"><p class="ty-x"></p></div> <div class="sentence">... 建立的标准传染率的麻疹传染病模型下,对感染麻疹的人群进行细化,把潜伏者分为接种过疫苗的潜伏者和未接种过疫苗的潜伏者,把染病者也分为接种过疫苗的染病者和未接种过疫苗的染病者,以及加入治疗者,研究了一个更为复杂的模型.文献[<xref ref-type="bibr" rid="b7">7</xref>, <xref ref-type="bibr" rid="b9">9</xref>]研究了麻疹的周期性.尽管我国提出2012年消除麻疹的目标,但全国麻疹疫情自2012年年底开始持续回升.文献[<xref ref-type="bibr" rid="b9">9</xref>]通过对我国麻疹月数据进行分析,得到麻疹随季节的周期性行为,进而进行数学建头治基于以上的各种研究,均没有考虑环境传播对麻疹的影响.基于此,本文考虑部分免疫和环境传播对麻疹流行性态的影响,构建数学模型,分析了模型的全局稳定性. ...</div> </div> </div> <div id="article_reference_meta_b8"> <div id="article_reference_meta_b8_title" class="title_">Deterministic modelling of infectious diseases:applications to measles and other similar infections</div> <div id="article_reference_meta_b8_citedNumber">1</div> <div id="article_reference_meta_b8_nian">2001</div> <div id="article_reference_meta_b8_jcr"></div> <div id="article_reference_meta_b8_cjcr"></div> <div id="article_reference_meta_b8_articleCitedText"> <div class="sentence">... 麻疹是已知最具有传染性的疾病之一.已经有许多学者对麻疹进行了研究.文献[<xref ref-type="bibr" rid="b1">1</xref>-<xref ref-type="bibr" rid="b3">3</xref>]研究了麻疹疫苗的有效性,其中文献[<xref ref-type="bibr" rid="b1">1</xref>]考虑了部分免疫对麻疹传播的影响.文献[<xref ref-type="bibr" rid="b4">4</xref>]研究了麻疹的潜伏期.文献[<xref ref-type="bibr" rid="b5">5</xref>]讨论了部分免疫和潜伏期共同作用下的麻疹传播动力学. Garba<sup>[<xref ref-type="bibr" rid="b6">6</xref>]</sup>等人在Trottier和Philippe<sup>[<xref ref-type="bibr" rid="b7">7</xref>-<xref ref-type="bibr" rid="b8">8</xref>]</sup>建立的标准传染率的麻疹传染病模型下,对感染麻疹的人群进行细化,把潜伏者分为接种过疫苗的潜伏者和未接种过疫苗的潜伏者,把染病者也分为接种过疫苗的染病者和未接种过疫苗的染病者,以及加入治疗者,研究了一个更为复杂的模型.文献[<xref ref-type="bibr" rid="b7">7</xref>, <xref ref-type="bibr" rid="b9">9</xref>]研究了麻疹的周期性.尽管我国提出2012年消除麻疹的目标,但全国麻疹疫情自2012年年底开始持续回升.文献[<xref ref-type="bibr" rid="b9">9</xref>]通过对我国麻疹月数据进行分析,得到麻疹随季节的周期性行为,进而进行数学建头治基于以上的各种研究,均没有考虑环境传播对麻疹的影响.基于此,本文考虑部分免疫和环境传播对麻疹流行性态的影响,构建数学模型,分析了模型的全局稳定性. ...</div> </div> </div> <div id="article_reference_meta_b9"> <div id="article_reference_meta_b9_title" class="title_">Seasonal transmission dynamics of measles in China</div> <div id="article_reference_meta_b9_citedNumber">5</div> <div id="article_reference_meta_b9_nian">2018</div> <div id="article_reference_meta_b9_jcr"></div> <div id="article_reference_meta_b9_cjcr"></div> <div id="article_reference_meta_b9_articleCitedText"> <div class="sentence">... 麻疹是儿童最常见的急性传染病,是一种传染性很强的呼吸道疾病,因此通过数学建模来了解它的动力学性态是非常有必要的<sup>[<xref ref-type="bibr" rid="b1">1</xref>-<xref ref-type="bibr" rid="b9">9</xref>]</sup>.人群普遍为易感者,接种疫苗是预防麻疹的最有效措施.由于麻疹疫苗需接种两次且间隔时间较长,导致漏种的情况及接种疫苗后免疫失败,所以接种疫苗者是部分免疫.此外,麻疹病毒有9-14天的潜伏期,接触过麻疹患者并在潜伏期接受被动免疫者可延至3-4周.未接种过疫苗的人,尤其是幼儿,具有罹患麻疹及其并发症(包括死亡)的最高风险.高传染性的麻疹病毒是通过空气中的飞沫传播(由于咳嗽和打喷嚏),或者是与染病者密切接触,或者是直接接触到他们的鼻腔和口腔分泌物进行传播.病毒在空气中或受染物表体的活力和传染性可维持两个小时.因此环境因素对于麻疹的传播有着重要的影响. ...</div> <div class="boundary"><p class="ty-x"></p></div> <div class="sentence">... 麻疹是已知最具有传染性的疾病之一.已经有许多学者对麻疹进行了研究.文献[<xref ref-type="bibr" rid="b1">1</xref>-<xref ref-type="bibr" rid="b3">3</xref>]研究了麻疹疫苗的有效性,其中文献[<xref ref-type="bibr" rid="b1">1</xref>]考虑了部分免疫对麻疹传播的影响.文献[<xref ref-type="bibr" rid="b4">4</xref>]研究了麻疹的潜伏期.文献[<xref ref-type="bibr" rid="b5">5</xref>]讨论了部分免疫和潜伏期共同作用下的麻疹传播动力学. Garba<sup>[<xref ref-type="bibr" rid="b6">6</xref>]</sup>等人在Trottier和Philippe<sup>[<xref ref-type="bibr" rid="b7">7</xref>-<xref ref-type="bibr" rid="b8">8</xref>]</sup>建立的标准传染率的麻疹传染病模型下,对感染麻疹的人群进行细化,把潜伏者分为接种过疫苗的潜伏者和未接种过疫苗的潜伏者,把染病者也分为接种过疫苗的染病者和未接种过疫苗的染病者,以及加入治疗者,研究了一个更为复杂的模型.文献[<xref ref-type="bibr" rid="b7">7</xref>, <xref ref-type="bibr" rid="b9">9</xref>]研究了麻疹的周期性.尽管我国提出2012年消除麻疹的目标,但全国麻疹疫情自2012年年底开始持续回升.文献[<xref ref-type="bibr" rid="b9">9</xref>]通过对我国麻疹月数据进行分析,得到麻疹随季节的周期性行为,进而进行数学建头治基于以上的各种研究,均没有考虑环境传播对麻疹的影响.基于此,本文考虑部分免疫和环境传播对麻疹流行性态的影响,构建数学模型,分析了模型的全局稳定性. ...</div> <div class="boundary"><p class="ty-x"></p></div> <div class="sentence">... ]研究了麻疹的周期性.尽管我国提出2012年消除麻疹的目标,但全国麻疹疫情自2012年年底开始持续回升.文献[<xref ref-type="bibr" rid="b9">9</xref>]通过对我国麻疹月数据进行分析,得到麻疹随季节的周期性行为,进而进行数学建头治基于以上的各种研究,均没有考虑环境传播对麻疹的影响.基于此,本文考虑部分免疫和环境传播对麻疹流行性态的影响,构建数学模型,分析了模型的全局稳定性. ...</div> <div class="boundary"><p class="ty-x"></p></div> <div class="sentence">... 模型中参数的意义及取值(单位:年)</p></caption><table> <thead> <tr> <td style="border-top:1px solid #000;" align="center">参数</td> <td style="border-top:1px solid #000;" align="center">取值</td> <td style="border-top:1px solid #000;" align="center">意义</td> <td style="border-top:1px solid #000;" align="center">来源</td> </tr></thead> <tbody> <tr> <td style="border-top:1px solid #000;" align="center"><span class="formulaText"><inline-formula><tex-math id="M9">$\lambda$</tex-math></inline-formula></span></td> <td style="border-top:1px solid #000;" align="center"><span class="formulaText"><inline-formula><tex-math id="M10">$1.728\times 10^{7}$</tex-math></inline-formula></span></td> <td style="border-top:1px solid #000;" align="center">新生儿的出生率</td> <td style="border-top:1px solid #000;" align="center">国家统计局(2017)</td> </tr> <tr> <td align="center"><span class="formulaText"><inline-formula><tex-math id="M11">$\mu$</tex-math></inline-formula></span></td> <td align="center">0.0131</td> <td align="center">自然死亡率</td> <td align="center">国家统计局(2017)</td> </tr> <tr> <td align="center"><span class="formulaText"><inline-formula><tex-math id="M12">$\eta$</tex-math></inline-formula></span></td> <td align="center">0.8216</td> <td align="center">新生儿接种疫苗的比例</td> <td align="center">文献[<xref ref-type="bibr" rid="b9">9</xref>]</td> </tr> <tr> <td align="center"><span class="formulaText"><inline-formula><tex-math id="M13">$\beta$</tex-math></inline-formula></span></td> <td align="center"><span class="formulaText"><inline-formula><tex-math id="M14">$9.6696\times 10^{-9}$</tex-math></inline-formula></span></td> <td align="center">人与人的感染率</td> <td align="center">估计</td> </tr> <tr> <td align="center"><span class="formulaText"><inline-formula><tex-math id="M15">$\alpha$</tex-math></inline-formula></span></td> <td align="center"><span class="formulaText"><inline-formula><tex-math id="M16">$1.1352\times 10^{-9}$</tex-math></inline-formula></span></td> <td align="center">环境对人的感染率</td> <td align="center">估计</td> </tr> <tr> <td align="center"><span class="formulaText"><inline-formula><tex-math id="M17">$\varepsilon$</tex-math></inline-formula></span></td> <td align="center">0.15</td> <td align="center">无效的接种率</td> <td align="center">中国疾病预防控制中心(2017)</td> </tr> <tr> <td align="center"><span class="formulaText"><inline-formula><tex-math id="M18">$\delta$</tex-math></inline-formula></span></td> <td align="center">0.95</td> <td align="center">接种疫苗后获得免疫并成为恢复者的转移率</td> <td align="center">中国疾病预防控制中心(2017)</td> </tr> <tr> <td align="center"><span class="formulaText"><inline-formula><tex-math id="M19">$p$</tex-math></inline-formula></span></td> <td align="center">26</td> <td align="center">潜伏者变为染病者的比例</td> <td align="center">中国疾病预防控制中心(2017)</td> </tr> <tr> <td align="center"><span class="formulaText"><inline-formula><tex-math id="M20">$\gamma$</tex-math></inline-formula></span></td><td align="center">20</td> <td align="center">染病者的恢复率</td> <td align="center">中国疾病预防控制中心(2017)</td> </tr> <tr> <td align="center"><span class="formulaText"><inline-formula><tex-math id="M21">$k$</tex-math></inline-formula></span></td> <td align="center">8</td> <td align="center">染病者排放到环境中的病毒的速率</td> <td align="center">假设</td> </tr> <tr> <td style="border-bottom:1px solid #000;" align="center"><span class="formulaText"><inline-formula><tex-math id="M22">$\tau$</tex-math></inline-formula></span></td> <td style="border-bottom:1px solid #000;" align="center">1.6</td> <td style="border-bottom:1px solid #000;" align="center">环境中病毒的失效率</td> <td style="border-bottom:1px solid #000;" align="center">假设</td> </tr></tbody> </table></table-wrap><p id="C9">我们假设染病者把病毒排放到环境中的速率为8,麻疹病毒的失效率为1.6,然后使用2004-2016年的麻疹发病人数和模型(2.2)进行数值拟合,得到人与人的感染率<span class="formulaText"><inline-formula><tex-math id="M23">$ \beta = 9.6696\times 10^{-9} $</tex-math></inline-formula></span>,环境对人的感染率<span class="formulaText"><inline-formula><tex-math id="M24">$ \alpha = 1.1352\times 10^{-9} $</tex-math></inline-formula></span><span class="formulaNumber">.</span> <xref ref-type="fig" rid="Fig1">图 1</xref>是拟合结果. ...</div> <div class="boundary"><p class="ty-x"></p></div> <div class="sentence">... <xref ref-type="fig" rid="Fig1">图 1</xref>模拟了中国2004-2016年的麻疹发病人数和模型(2.2)的染病者人数.虚线是<span class="formulaText"><inline-formula><tex-math id="M25">$ 2004-2016 $</tex-math></inline-formula></span>年的麻疹年数据,实线是模型(2.2)的<span class="formulaText"><inline-formula><tex-math id="M26">$ I(t) $</tex-math></inline-formula></span><span class="formulaNumber">.</span>根据文献[<xref ref-type="bibr" rid="b9">9</xref>]和中国疾病预防控制中心(2017)中的数据,取 ...</div> </div> </div> <div id="article_reference_meta_b10"> <div id="article_reference_meta_b10_title" class="title_">Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission</div> <div id="article_reference_meta_b10_citedNumber">2</div> <div id="article_reference_meta_b10_nian">2002</div> <div id="article_reference_meta_b10_jcr"></div> <div id="article_reference_meta_b10_cjcr"></div> <div id="article_reference_meta_b10_articleCitedText"> <div class="sentence">... 模型(2.2)存在无病平衡点<span class="formulaText"><inline-formula><tex-math id="M28">$ P_{0}(S_{0}, V_{0}, 0, 0, 0) $</tex-math></inline-formula></span>,其中<span class="formulaText"><inline-formula><tex-math id="M29">$ S_{0} = \frac{\lambda(1-\eta)}{\mu} $</tex-math></inline-formula></span>, <span class="formulaText"><inline-formula><tex-math id="M30">$ V_{0} = \frac{\lambda\eta}{\mu+\delta} $</tex-math></inline-formula></span><span class="formulaNumber">.</span>通过下一代矩阵方法<sup>[<xref ref-type="bibr" rid="b10">10</xref>]</sup>可得模型(2.2)的基本再生数为 ...</div> <div class="boundary"><p class="ty-x"></p></div> <div class="sentence">... 由文献[<xref ref-type="bibr" rid="b10">10</xref>]知,下列结论成立. ...</div> </div> </div> <div id="article_reference_meta_b11"> <div id="article_reference_meta_b11_title" class="title_"></div> <div id="article_reference_meta_b11_citedNumber">1</div> <div id="article_reference_meta_b11_nian">1976</div> <div id="article_reference_meta_b11_jcr"></div> <div id="article_reference_meta_b11_cjcr"></div> <div id="article_reference_meta_b11_articleCitedText"> <div class="sentence">... 当<span class="formulaText"><inline-formula><tex-math id="M67">$ R_{0} < 1 $</tex-math></inline-formula></span>时, <span class="formulaText"><inline-formula><tex-math id="M68">$ L'_{1}\leq0 $</tex-math></inline-formula></span><span class="formulaNumber">.</span>此外, <span class="formulaText"><inline-formula><tex-math id="M69">$ L'_{1} = 0 $</tex-math></inline-formula></span>当且仅当<span class="formulaText"><inline-formula><tex-math id="M70">$ S = S_{0}, V = V_{0}, I = 0 $</tex-math></inline-formula></span><span class="formulaNumber">.</span>所以单点集<span class="formulaText"><inline-formula><tex-math id="M71">$ \{P_{0}\} $</tex-math></inline-formula></span>是模型(2.2)在集合<span class="formulaText"><inline-formula><tex-math id="M72">$ \{(S, V, E, I, B)\in\Omega\mid L'_{1} = 0\} $</tex-math></inline-formula></span>上的最大不变集.由定理3.1和LaSalle不变集原理<sup>[<xref ref-type="bibr" rid="b11">11</xref>]</sup>,当<span class="formulaText"><inline-formula><tex-math id="M73">$ R_{0} < 1 $</tex-math></inline-formula></span>时,无病平衡点<span class="formulaText"><inline-formula><tex-math id="M74">$ P_{0} $</tex-math></inline-formula></span>在<span class="formulaText"><inline-formula><tex-math id="M75">$ \Omega $</tex-math></inline-formula></span>上是全局渐近稳定的.定理得证. ...</div> </div> </div> <div id="article_reference_meta_b12"> <div id="article_reference_meta_b12_title" class="title_"></div> <div id="article_reference_meta_b12_citedNumber">1</div> <div id="article_reference_meta_b12_nian">2015</div> <div id="article_reference_meta_b12_jcr"></div> <div id="article_reference_meta_b12_cjcr"></div> <div id="article_reference_meta_b12_articleCitedText"> <div class="sentence">... 从而<span class="formulaText"><inline-formula><tex-math id="M90">$ (S^{*}, V^{*}, E_{1}, I_{1}, B_{1}) $</tex-math></inline-formula></span>为模型(4.2)的平衡点.由于<span class="formulaText"><inline-formula><tex-math id="M91">$ S^{*} < S_{0} $</tex-math></inline-formula></span>,从而<span class="formulaText"><inline-formula><tex-math id="M92">$ (S^{*}, V^{*}, E_{1}, I_{1}, B_{1}) $</tex-math></inline-formula></span>为平衡点<span class="formulaText"><inline-formula><tex-math id="M93">$ P^{*} $</tex-math></inline-formula></span><span class="formulaNumber">.</span>因此集合<span class="formulaText"><inline-formula><tex-math id="M94">$ M $</tex-math></inline-formula></span>内除<span class="formulaText"><inline-formula><tex-math id="M95">$ P^{*} $</tex-math></inline-formula></span>外不再包含模型(4.2)的其他轨线.由定理3.10 (文献[<xref ref-type="bibr" rid="b12">12</xref>])可知模型(4.2)的地方病平衡点是全局渐近稳定的.定理得证. ...</div> </div> </div> <div id="article_reference_meta_b12"> <div id="article_reference_meta_b12_title" class="title_"></div> <div id="article_reference_meta_b12_citedNumber">1</div> <div id="article_reference_meta_b12_nian">2015</div> <div id="article_reference_meta_b12_jcr"></div> <div id="article_reference_meta_b12_cjcr"></div> <div id="article_reference_meta_b12_articleCitedText"> <div class="sentence">... 从而<span class="formulaText"><inline-formula><tex-math id="M90">$ (S^{*}, V^{*}, E_{1}, I_{1}, B_{1}) $</tex-math></inline-formula></span>为模型(4.2)的平衡点.由于<span class="formulaText"><inline-formula><tex-math id="M91">$ S^{*} < S_{0} $</tex-math></inline-formula></span>,从而<span class="formulaText"><inline-formula><tex-math id="M92">$ (S^{*}, V^{*}, E_{1}, I_{1}, B_{1}) $</tex-math></inline-formula></span>为平衡点<span class="formulaText"><inline-formula><tex-math id="M93">$ P^{*} $</tex-math></inline-formula></span><span class="formulaNumber">.</span>因此集合<span class="formulaText"><inline-formula><tex-math id="M94">$ M $</tex-math></inline-formula></span>内除<span class="formulaText"><inline-formula><tex-math id="M95">$ P^{*} $</tex-math></inline-formula></span>外不再包含模型(4.2)的其他轨线.由定理3.10 (文献[<xref ref-type="bibr" rid="b12">12</xref>])可知模型(4.2)的地方病平衡点是全局渐近稳定的.定理得证. ...</div> </div> </div> <div id="article_reference_meta_b13"> <div id="article_reference_meta_b13_title" class="title_"></div> <div id="article_reference_meta_b13_citedNumber">0</div> <div id="article_reference_meta_b13_nian">2004</div> <div id="article_reference_meta_b13_jcr"></div> <div id="article_reference_meta_b13_cjcr"></div> </div> <div id="article_reference_meta_b13"> <div id="article_reference_meta_b13_title" class="title_"></div> <div id="article_reference_meta_b13_citedNumber">0</div> <div id="article_reference_meta_b13_nian">2004</div> <div id="article_reference_meta_b13_jcr"></div> <div id="article_reference_meta_b13_cjcr"></div> </div> </div> </div> </div> </div> <div class="cankaowenxian1"></div> <div class="col-xs-3"> <div class="slide" style="top: 367px; display: block;"> <div id="sideToolbar"> <div id="sideCatalog" class="sideCatalogBg"> <div id="sideCatalog-sidebar"> <div class="sideCatalog-sidebar-top"></div> <div 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