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>
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<div id="article_reference_meta_b1_title" class="title_">SV IR epidemic models with vaccination strategies</div>
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<div id="article_reference_meta_b1_nian">2008</div>
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<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>
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<div id="article_reference_meta_b2_title" class="title_">Modelling measles re-emergence as a result of waning of immunity in vaccinated populations</div>
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<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>
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<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>
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<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>
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<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} $</tex-math></inline-formula></span>为环境传播的基本再生数.从而,环境传播会促进疾病的传播. ...</div>
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<div id="article_reference_meta_b5">
<div id="article_reference_meta_b5_title" class="title_">考虑部分免疫和潜伏期的麻疹传染病模型的稳定性分析</div>
<div id="article_reference_meta_b5_citedNumber">2</div>
<div id="article_reference_meta_b5_nian">2017</div>
<div id="article_reference_meta_b5_jcr"></div>
<div id="article_reference_meta_b5_cjcr"></div>
<div id="article_reference_meta_b5_articleCitedText">
<div class="sentence">... 麻疹是已知最具有传染性的疾病之一.已经有许多学者对麻疹进行了研究.文献[<xref ref-type="bibr" rid="b1">1</xref>-<xref ref-type="bibr" rid="b3">3</xref>]研究了麻疹疫苗的有效性,其中文献[<xref ref-type="bibr" rid="b1">1</xref>]考虑了部分免疫对麻疹传播的影响.文献[<xref ref-type="bibr" rid="b4">4</xref>]研究了麻疹的潜伏期.文献[<xref ref-type="bibr" rid="b5">5</xref>]讨论了部分免疫和潜伏期共同作用下的麻疹传播动力学. Garba<sup>[<xref ref-type="bibr" rid="b6">6</xref>]</sup>等人在Trottier和Philippe<sup>[<xref ref-type="bibr" rid="b7">7</xref>-<xref ref-type="bibr" rid="b8">8</xref>]</sup>建立的标准传染率的麻疹传染病模型下,对感染麻疹的人群进行细化,把潜伏者分为接种过疫苗的潜伏者和未接种过疫苗的潜伏者,把染病者也分为接种过疫苗的染病者和未接种过疫苗的染病者,以及加入治疗者,研究了一个更为复杂的模型.文献[<xref ref-type="bibr" rid="b7">7</xref>, <xref ref-type="bibr" rid="b9">9</xref>]研究了麻疹的周期性.尽管我国提出2012年消除麻疹的目标,但全国麻疹疫情自2012年年底开始持续回升.文献[<xref ref-type="bibr" rid="b9">9</xref>]通过对我国麻疹月数据进行分析,得到麻疹随季节的周期性行为,进而进行数学建头治基于以上的各种研究,均没有考虑环境传播对麻疹的影响.基于此,本文考虑部分免疫和环境传播对麻疹流行性态的影响,构建数学模型,分析了模型的全局稳定性. ...</div>
<div class="boundary"><p class="ty-x"></p></div>
<div class="sentence">... <strong>注 3.1</strong> 由文献[<xref ref-type="bibr" rid="b5">5</xref>]知, <span class="formulaText"><inline-formula><tex-math id="M31">$ 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>
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</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>
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</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>
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<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>
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<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>
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<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>
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<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} $</tex-math></inline-formula></span><span class="formulaNumber">.</span> <xref ref-type="fig" rid="Fig1">图 1</xref>是拟合结果. ...</div>
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<div class="sentence">... <xref ref-type="fig" rid="Fig1">图 1</xref>模拟了中国2004-2016年的麻疹发病人数和模型(2.2)的染病者人数.虚线是<span class="formulaText"><inline-formula><tex-math id="M25">$ 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>
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