传染病垂直传染的传播动力学分析：以COVID⁃19为例

传染病垂直传染的传播动力学分析：以COVID⁃19为例 Transmission Dynamics of Vertical Infection of Infectious Diseases Based on COVID⁃19 1 first-author 陈姗姗，女，讲师，现从事复杂网络传播动力学、生物数学等方面研究。E-mail： sschen1214@163.com 陈姗姗，女，讲师，现从事复杂网络传播动力学、生物数学等方面研究。E-mail： sschen1214@163.com sschen1214@163.com 1 1 上海工程技术大学 电子电气工程学院，上海 201620 上海工程技术大学 电子电气工程学院，上海 201620 School of Electronic and Electrical Engineering， Shanghai University of Engineering Science， Shanghai 201620，China School of Electronic and Electrical Engineering， Shanghai University of Engineering Science， Shanghai 201620，China 为准确全面地分析新型冠状病毒肺炎（corona virus disease 2019，COVID-19）的传播特性和传播动力学行为，揭示病毒传播的内在规律，本文基于其传播特性，利用传染病动力学建模理论，提出了一类考虑潜在垂直传染风险及潜伏者感染性的SEIR传播动力学模型，给出了在垂直传染风险和潜伏期传染性的影响下COVID-19的基本再生数表达式，推导出垂直传染率和基本再生数线性正相关，潜伏者的转变率对基本再生数的影响比垂直传染率要大。通过构造Lyapunov函数和构造迭代序列的方法，分析了模型的全局稳定性，为COVID-19的防控以及类似传染病的传播研究与防控提供一定的理论依据。 In order to analyze propagation characteristics and propagation dynamics of COVID⁃19 (corona virus disease 2019) accurately and comprehensively, and reveal the internal law of virus transmission， a SEIR propagation dynamics model considering vertical infection risk and latent infection is proposed. Based on its propagation characteristics and the theory of dynamics modeling of infectious diseases， we derived the basic reproduction number of disease related to the vertical infection rate and Latent infectivity. The basic reproduction number is positively linear with the vertical infection rate. The conversion rate of incubation has greater influence on the basic regeneration number than the vertical infection rate. By constructing Lyapunov function and iterative sequence, the global stability of the model is analyzed, which can provide a theoretical basis for the prevention and control of new coronavirus pneumonia and the research and prevention of similar infectious diseases. 传播动力学 新型冠状病毒肺炎 基本再生数 稳定性分析 epidemic dynamics COVID⁃19 (corona virus disease 2019) basic reproduction number stability analysis 陈姗姗,黄勃,方志军.传染病垂直传染的传播动力学分析：以COVID-19为例[J].武汉大学学报（理学版）,2020,66(5):433-441. DOI:10.14188/j.1671-8836.2020.0129. 0　 引 言 1 传染病的防控关系到国计民生，对传染病的流行规律进行定性分析是疫情防控的重要依据 ［ 1 1 ~ 4 4 1~4 ］ 。传染病动力学就是根据疾病发展的情况，建立反映其变化规律的数学模型，并通过传播动力学行为分析疾病的发展过程、产生的原因，为疾病的防控提供重要理论依据的方法。SIR模型 ［ 4 4 ］ 、SEIR模型 ［ 5 5 ］ 是研究可以获得自然免疫或者治愈后不再复发的传染病动力学模型的主要模型，其主要思想为将人群按照传染病传播过程中个体的患病状态进行划分，根据传染病传播机制建模分析单位时间内各状态的变化规律和动力学行为。 新型冠状病毒肺炎（corona virus disease 2019，COVID-19）传播以来，很多流行病学专家针对其传播特性和确诊病例数对其进行预测，为有效防控COVID-19传播和制定相应政策提供了有价值的参考 ［ 6 6 ~ 9 9 6~9 ］ 。对于COVID-19，各个年龄段的人都可能被感染，目前确诊的案例中被感染的主要是成年人、老年人和体弱多病的人，他们由于自身免疫力较弱为易感人群。文献［ 10 10 ］揭示了COVID-19可感染婴儿。2020年2月2日，《楚天都市报》报道，出生仅30小时的男婴确诊COVID-19，男婴的母亲为确诊的新型冠状病毒携带者。现有资料显示新生儿并不能豁免感染，防控母婴垂直传播相当重要。新型冠状病毒是一种全新的病毒，对其认识还在不断地深化。因此，对垂直传染风险的COVID-19模型的传播特性和传播动力学行为进行分析，有利于早发现、早预防的防控措施的执行。现有的关于COVID-19模型主要基于SEIR、SIR传染病模型及其变形，在大多建模研究中没有考虑可能潜在的母婴垂直传染的风险以及潜伏期无症状患者的传染性。对于有症状的染病者，可以通过染病者的症状，如咳嗽、发烧和呼吸困难等进行排查。但是，处于潜伏期的无症状患者由于他们的无症状性，进行疫情防控比较困难。 本文基于COVID-19可能潜在的母婴垂直传染风险和潜伏期个体的传染性，从流行病学理论分析的角度，建立SEIR的传染病动力学模型，通过对基本再生数和模型稳定性的分析，分析COVID-19在垂直传染影响下的传播动力学行为和规律，为COVID-19的防控提供一定的参考。 1　 基于垂直传染的SEIR传染病模型 1 1.1　模型建立 2 根据流行病学理论分析的建模思想及COVID-19的传播特性，单位时间内将人群分为以下几类：易感者（ S （ t ））、潜伏者（ E （ t ））、感染者（ I （ t ））和免疫者（ R （ t ））。潜伏者虽然没有表现出症状，但是与易感者接触后仍然具有传染性。因此，传统的SEIR模型在描述COVID-19的传播特性时并不合理。因此，本文考虑单位时间内，易感个体与潜伏或感染的个体接触后均以一定的概率被感染，被感染的个体由于传染病的潜伏期转变为潜伏者，潜伏期过后，潜伏者转化为感染者，易感者经过治疗，被治愈后获得免疫。设人群总人口密度为 N （ t ），满足 N （ t ） =S （ t ） +E （ t ） +I （ t ） +R （ t ） 。 本文主要参数的意义如下： • b，d 分别表示个体的自然出生率和死亡率的正常数，这里不考虑因病死亡。 • 参数 β 为病毒的传播率， γ 为潜伏者转化为染病者的转变率， μ 为染病者转化成免疫者的恢复率。 • 参数 p 为病毒垂直传染的可能性（概率）。 单位时间内模型的主要传播要素和过程包括： 1）　出生和死亡 3 本文模型考虑个体的自然出生和死亡。出生过程主要包括：健康个体（易感者和免疫者）以概率 b 产生新的健康个体（易感状态的新生儿）；假设COVID-19的垂直传染率为 p 。那么，潜伏者和染病者以（ 1-p ） b 的概率生成健康的新个体（易感状态的新生儿），以 pb 概率产生垂直传染的可能，产生带病毒的处于潜伏状态的新生儿。死亡的过程，本文仅考虑个体的自然死亡率，个体以概率 d 死亡。本文不考虑因病死亡，由于COVID-19是可治愈的，相对于人群的自然死亡率来说，做忽略因病死亡的假设。 2） S <math id="M1"><mo>→</mo><mi mathvariant="normal">E</mi><mo>→</mo><mi mathvariant="normal">I</mi><mo>→</mo><mi mathvariant="normal">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558458&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558454&type= 20.15066719 2.79399991 过程 3 单位时间内，易感个体与潜伏或感染的个体接触后均可能被感染，以感染率 β 被感染，变成潜伏者，潜伏者经过潜伏期以转变率 γ 转变为染病者（有症状），有症状的染病者经过治疗，以恢复率 μ 被治愈获得免疫。 根据上述传播过程和机制，建立带有垂直传染的SEIR传染病模型如下 <math display="block" id="M2"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>b</mi><mi>N</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>γ</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>γ</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>μ</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>μ</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558467&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558462&type= 75.43800354 32.59666443 由 N（t）= <math id="M3"><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558474&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558472&type= 41.74066544 3.55599999 及模型（1）可得： <math display="block" id="M4"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>N</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>b</mi><mi>N</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>N</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mtext> </mtext></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558483&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558479&type= 41.23266602 7.70466709 很显然，当 b<d 时 ， <math id="M5"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>t</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mtext> </mtext><mi>N</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558490&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558485&type= 23.11400032 4.65666676 总人口趋于消亡，没有实际意义。因此，考虑 b≥d 的情况，不妨设 b=d ，满足 <math id="M6"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>N</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558498&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558495&type= 17.86466599 7.70466709 ，则在一定的时间内，满足 <math id="M7"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>t</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>N</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558503&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558501&type= 23.02933502 4.65666676 ，其中 <math id="M8"><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558510&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558507&type= 4.31799984 3.13266659 为常数，此时 <math id="M9"><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>=</mo><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558519&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558514&type= 32.25799942 3.81000018 <math id="M10"><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558526&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558522&type= 18.28800011 3.55599999 。 为了讨论 b=d 时模型（1）的动力学行为，根据上面的分析及 <math id="M11"><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>=</mo><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558535&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558531&type= 51.13866806 3.81000018 得到如下动力学模型 <math display="block" id="M12"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mfrac linethickness="0pt"><mrow><mfrac><mrow><mi mathvariant="normal">d</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mfrac><mrow><mi mathvariant="normal">d</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>γ</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>μ</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mtext> </mtext></mtd></mtr><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>μ</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558542&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558539&type= 72.38999939 31.66533279 其中，模型的初始条件满足（ <math id="M13"><mi>S</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558550&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558546&type= 7.53533316 3.55599999 ， <math id="M14"><mi>E</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558557&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558552&type= 8.04333305 3.55599999 ， <math id="M15"><mi>I</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558564&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558559&type= 6.94266701 3.55599999 ， <math id="M16"><mi>R</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∈</mo><msubsup><mrow><mi mathvariant="normal">ℝ</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558571&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558568&type= 17.86466599 4.65666676 且 <math id="M17"><mi>S</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>+</mo><mi>E</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>+</mo><mi>R</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558579&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558575&type= 52.83199692 3.81000018 。 1.2　准备工作 2 主要介绍本文主要应用的引理和模型的正定性与有界性的证明。 引理 1 (第二比较定理) 对于一阶微分方程问题 <math display="block" id="M18"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><mi>y</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>y</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558587&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558583&type= 22.60599899 14.30866623 (E <math display="block" id="M19"><msub><mrow/><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558593&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558590&type= 1.18533337 4.06400013 ） <math display="block" id="M20"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><mi>y</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>y</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558602&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558597&type= 23.36800003 14.30866623 (E <math display="block" id="M21"><msub><mrow/><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558608&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558604&type= 1.18533337 4.06400013 ） 其中 f 和 F 均在 G ⊆ <math id="M22"><mi mathvariant="normal">ℝ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558614&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558611&type= 2.45533323 2.79399991 × <math id="M23"><mi mathvariant="normal">ℝ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558622&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558618&type= 2.45533323 2.79399991 内连续，对 y 满足局部Lipschitz条件，（ <math id="M24"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558629&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558626&type= 8.12800026 4.57200003 ）∈ G ，并设 <math id="M25"><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558636&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558633&type= 3.64066648 3.64066648 ， <math id="M26"><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558643&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558640&type= 3.64066648 3.64066648 的解均在闭区间［ a ， b ］上存在，分别记作 <math id="M27"><mi>y</mi><mo>=</mo><mi>y</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558650&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558647&type= 14.47799969 3.81000018 与 y=Y （ x ）， x ∈［ a ， b ］。若 <math id="M28"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558657&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558654&type= 28.36333275 3.89466667 成立，∀ x ∈ G ，则 y（x ）≤ Y （ x ）， x ∈［ <math id="M29"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558664&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558662&type= 7.02733326 4.14866638 ］； y （ x ）≥ Y （ x ）， x ∈［ <math id="M30"><mi>a</mi><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558673&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558668&type= 6.85800028 4.14866638 ］成立。 引理 2 ［ 11 11 ］ 如果 w （ t ）是系统 <math id="M31"><mover accent="true"><mrow><mi>X</mi></mrow><mo stretchy="true">˙</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558679&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558675&type= 2.62466669 3.55599999 =TX ， X （0） = <math id="M32"><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558688&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558683&type= 3.21733332 3.72533321 的一个Lyapunov函数，满足 x （ n ）是系统的解，对任意的 <math id="M33"><mi>n</mi><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558694&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558690&type= 8.55133343 3.30199981 存在一个常数 c 使得 <math id="M34"><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558701&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558696&type= 10.24466705 2.79399991 时， <math id="M35"><mi>x</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>M</mi><mo>⋂</mo><msup><mrow><mi>W</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558706&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558704&type= 32.34266663 3.81000018 ，其中， M 是 <math id="M36"><mi>G</mi><mo>=</mo><mo stretchy="false">{</mo><mfenced open="" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced><msup><mrow><mi>W</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558715&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558711&type= 31.15733337 3.97933340 的最大不变集， <math id="M37"><msup><mrow><mi>W</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mfenced open="" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced><mi>W</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558723&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558719&type= 40.63999939 3.97933340 。 引理3 ［ 12 12 ］ 当 <math id="M38"><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558730&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558726&type= 7.95866680 2.79399991 ， <math id="M39"><mi>x</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558737&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558734&type= 13.97000027 3.55599999 时，如果 <math id="M40"><mi>f</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558743&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558740&type= 8.55133343 3.64066648 ， <math id="M41"><mi>g</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558754&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558748&type= 8.80533314 3.64066648 且满足 <math id="M42"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">t</mi></mrow></mfrac><mo>≥</mo><mi>f</mi><mo>-</mo><mi>g</mi><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558761&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558757&type= 30.22600174 7.70466709 ，则有 <math id="M43"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>t</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">f</mi><mtext> </mtext><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558768&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558766&type= 17.52599907 4.65666676 ≥ <math id="M44"><mfrac><mrow><mi>f</mi></mrow><mrow><mi>g</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558776&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558772&type= 2.62466669 8.38199997 。如果 <math id="M45"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>≤</mo><mi>f</mi><mo>-</mo><mi>g</mi><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558785&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558780&type= 30.39533424 7.70466709 ，则有 <math id="M46"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>t</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">p</mi><mtext> </mtext><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo><mfrac><mrow><mi>f</mi></mrow><mrow><mi>g</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558791&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558787&type= 26.58533287 8.38199997 。 引理4 ［ 13 13 ］ 设 X （ t ）是［ m， <math id="M47"><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558795&type= 3.64066648 2.79399991 ）上的有界可微函数，则存在序列 <math id="M48"><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>→</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558806&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558803&type= 11.09133339 4.57200003 ， <math id="M49"><msub><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>→</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558814&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558810&type= 11.85333347 4.57200003 ，使得对所有的 <math id="M50"><mtext> </mtext><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558821&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558818&type= 11.09133339 2.79399991 满足：1） <math id="M51"><msup><mrow><mi>X</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>→</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558828&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558825&type= 16.59466553 4.91066647 ， <math id="M52"><mi>X</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>→</mo><msup><mrow><mi>X</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558836&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558832&type= 18.88066673 4.91066647 其中 <math id="M53"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">p</mi><mi>X</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>X</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558843&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558841&type= 30.56466866 4.65666676 ；2） <math id="M54"><msup><mrow><mi>X</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>→</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558852&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558847&type= 16.84866714 4.91066647 ， <math id="M55"><msup><mrow><mi>X</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>→</mo><msub><mrow><mi>X</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558858&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558854&type= 19.64266586 4.91066647 ，其中 <math id="M56"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">f</mi><mtext> </mtext><mi>X</mi><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558867&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558863&type= 29.54866600 4.65666676 。 模型（1）的解存在唯一性及连续性显然满足。模型的生物意义要求模型（1）的解必须是正定及有界的。根据引理1可证明模型（1）解的正定性及有界性，即命题1成立。 命题1 对所有 <math id="M57"><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558874&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558869&type= 7.95866680 2.79399991 ，模型（3）所有满足初始条件的解（ <math id="M58"><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558877&type= 34.96733475 3.55599999 ）都满足 <math id="M59"><mn mathvariant="normal">0</mn><mo>≤</mo><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558889&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558884&type= 13.63133430 3.64066648 ， <math id="M60"><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo><msup><mrow><mi>N</mi></mrow><mrow><mi mathvariant="bold-italic">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558896&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558891&type= 34.88266754 3.89466667 。 证 模型（3）的初始解满足（ <math id="M61"><mi>S</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558902&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558900&type= 7.53533316 3.55599999 ， <math id="M62"><mi>E</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558911&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558907&type= 8.04333305 3.55599999 ， <math id="M63"><mi>I</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>R</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558918&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558914&type= 17.10266685 3.55599999 ） <math id="M64"><mo>∈</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558926&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558924&type= 8.12800026 4.23333359 ， <math id="M65"><mi>S</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>+</mo><mi>E</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558936&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558931&type= 19.38866615 3.55599999 <math id="M66"><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558946&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558939&type= 3.72533321 2.96333337 <math id="M67"><mi>I</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558952&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558949&type= 6.94266701 3.55599999 <math id="M68"><mo>+</mo><mi>R</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558960&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558957&type= 15.91733360 3.55599999 <math id="M69"><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558968&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558963&type= 4.31799984 3.13266659 。首先，证明对任意的 t > 0， <math id="M70"><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558977&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558972&type= 14.05466652 3.64066648 成立。如果不成立，因为 E （0）≥0，则至少存在一点 t 使得 E （ t ）=0。那么，必存在 <math id="M71"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn><mtext> </mtext></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558983&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558979&type= 9.56733322 4.14866638 使得 <math display="block" id="M72"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn><mtext> </mtext></mrow></msub><mo>=</mo><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">f</mi><mo stretchy="false">{</mo><mi>t</mi><mfenced open="|" close="" separators="|"><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></mrow></mfenced><mo>,</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">t</mi></mrow></mfrac><mo><</mo><mn mathvariant="normal">0</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558990&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558988&type= 52.74732971 7.70466709 根据 <math id="M73"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn><mtext> </mtext></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558998&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4558994&type= 2.62466669 3.72533321 的定义得， <math id="M74"><mi>E</mi><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn><mtext> </mtext></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559004&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559001&type= 14.98599911 4.48733330 ， <math id="M75"><msup><mrow><mi>E</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn><mtext> </mtext></mrow></msub><mo stretchy="false">)</mo><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559013&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559008&type= 15.83266640 4.65666676 ，且当 <math id="M76"><mn mathvariant="normal">0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn><mtext> </mtext></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559023&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559018&type= 15.32466698 4.23333359 时， <math id="M77"><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559029&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559026&type= 14.05466652 3.64066648 成立。因此，根据模型（3）的第三个方程可得，当 <math id="M78"><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn><mtext> </mtext></mrow></msub><mo stretchy="false">]</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559035&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559034&type= 16.76399994 4.48733330 以下方程成立 <math display="block" id="M79"><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>γ</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>μ</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo></mtd></mtr><mtr><mtd><mo>-</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn><mtext> </mtext></mrow></msub><mo stretchy="false">]</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559044&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559040&type= 53.59399796 14.13933372 由比较法则可知， <math id="M80"><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>I</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mfenced separators="|"><mrow><mi>d</mi><mo>+</mo><mi>μ</mi></mrow></mfenced><mi>t</mi></mrow></msup><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559050&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559047&type= 36.32199860 4.23333359 ， <math id="M81"><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn><mtext> </mtext></mrow></msub><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559057&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559053&type= 14.22399998 4.48733330 。根据模型（3）的第四个方程得 <math id="M82"><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn><mtext> </mtext></mrow></msub><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559065&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559061&type= 14.22399998 4.48733330 <math display="block" id="M83"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>μ</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><mo>-</mo><mi>d</mi><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn><mtext> </mtext></mrow></msub><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559072&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559070&type= 71.71266174 7.70466709 由比较法则可知， <math id="M84"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>R</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>d</mi><mi>t</mi></mrow></msup><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559082&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559077&type= 33.02000046 3.89466667 ， <math id="M85"><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn><mtext> </mtext></mrow></msub><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559089&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559085&type= 14.22399998 4.48733330 成立。 根据模型（3）的第一个方程及 <math id="M86"><mi>b</mi><mo>=</mo><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559097&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559093&type= 8.72066689 2.79399991 得 <math id="M87"><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn><mtext> </mtext></mrow></msub><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559103&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559099&type= 14.22399998 4.48733330 且 <math display="block" id="M88"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mi>p</mi><mi>b</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>β</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559110&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559107&type= 157.22599792 13.03866673 <math display="block" id="M89"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>≥</mo><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mi>b</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>β</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>b</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>β</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559118&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559113&type= 158.49598694 13.03866673 <math display="block" id="M90"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>≥</mo><mo>-</mo><mi>β</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn><mtext> </mtext></mrow></msub><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559124&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559121&type= 66.37866211 7.70466709 由比较法则可知， <math id="M91"><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>S</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>β</mi><mfenced separators="|"><mrow><mi>E</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mi>I</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mi>t</mi></mrow></msup><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn><mtext> </mtext></mrow></msub><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559132&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559128&type= 59.77466583 5.16466665 成立。 故当 <math id="M92"><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn><mtext> </mtext></mrow></msub><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559140&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559136&type= 14.22399998 4.48733330 ，有 <math display="block" id="M93"><msub><mrow><mfenced open="" close="|" separators="|"><mrow><mfrac><mrow><mi mathvariant="normal">d</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>t</mi><mo>=</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>S</mi><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mi>E</mi><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559147&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559142&type= 91.18599701 10.49866676 由于 <math id="M94"><mi>E</mi><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559153&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559150&type= 14.64733315 4.48733330 ， <math id="M95"><msub><mrow><mfenced open="" close="|" separators="|"><mrow><mfrac><mrow><mi mathvariant="normal">d</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>t</mi><mo>=</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559160&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559157&type= 16.42533302 10.49866676 = <math id="M96"><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559166&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559165&type= 5.75733376 3.72533321 + <math id="M97"><mi>β</mi><mi>S</mi><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559173&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559170&type= 18.88066673 4.82599974 <math id="M98"><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559182&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559177&type= 6.18066692 3.30199981 ，即 <math id="M99"><msup><mrow><mi>E</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">0</mn><mtext> </mtext></mrow></msub><mo stretchy="false">)</mo><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559190&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559185&type= 15.83266640 4.74133348 。显然，这与假设矛盾，因此假设不成立，对任意的 t > 0， <math id="M100"><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559198&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559194&type= 14.05466652 3.64066648 成立。 根据上述思路，有 <math id="M101"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>γ</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559204&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559201&type= 36.74533463 7.70466709 - <math id="M102"><mi>μ</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><mo>-</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559212&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559208&type= 35.47533417 3.81000018 ，得 <math id="M103"><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>I</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mi>t</mi></mrow></msup><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559217&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559215&type= 36.15266418 3.89466667 ， t ≥0。由 <math id="M104"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>μ</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><mo>-</mo><mi>d</mi><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559224&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559221&type= 55.71066666 7.70466709 ，得 <math id="M105"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>d</mi><mi>t</mi></mrow></msup><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559231&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559228&type= 32.51200104 3.89466667 ， <math id="M106"><mi>t</mi><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559239&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559235&type= 7.95866680 3.30199981 。进而得到 <math id="M107"><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>t</mi><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559246&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559244&type= 23.28333473 3.64066648 。 由 <math id="M108"><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>=</mo><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559257&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559253&type= 51.13866806 3.81000018 可得，对模型（3）的任意解满足 <math id="M109"><mn mathvariant="normal">0</mn><mo>≤</mo><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559263&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559258&type= 31.91933250 3.64066648 ， <math id="M110"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559271&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559267&type= 16.59466553 3.89466667 。因此，模型（3）的解具有正定性和有界性，满足生物学意义。 证毕。 2　 模型的基本再生数和稳定性分析 1 本小节主要分析系统的基本再生数、平衡点和全局稳定性。由于模型（3）中前3个方程不涉及变量 R （ t ），因此，分析前3个方程组成的动力系统可等价分析模型（3）的动力学行为，即模型（4）的动力学行为等价于模型（3）。 <math display="block" id="M111"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>γ</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>μ</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559283&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559279&type= 72.38999939 24.12999916 由命题1，解的正定性和有界性可知，模型（4）的解状态空间 <math id="M112"><mi>D</mi><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∈</mo><msubsup><mrow><mi mathvariant="normal">ℝ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mo>+</mo></mrow></msubsup><mfenced open="|" close="" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>≤</mo><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559290&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559285&type= 60.02866364 3.97933340 <math id="M113"><mo>+</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo><msup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">*</mi></mrow></msup><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559297&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559293&type= 32.68133163 3.97933340 为正向不变集。下面在解状态空间 D 上分析模型（4）的动力学行为。模型（4）有生物意义的稳定解满足下列方程 <math display="block" id="M114"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>γ</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mi>γ</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>μ</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559305&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559301&type= 79.24799347 16.67933464 在解状态空间 D 中，模型始终具有系统的无病平衡点 <math id="M115"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mn mathvariant="normal">0,0</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559312&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559309&type= 22.52133369 4.48733330 ， <math id="M116"><mi mathvariant="normal">其</mi><mi mathvariant="normal">中</mi><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mi>d</mi></mrow></mfrac><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559320&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559317&type= 26.16199875 7.53533316 。 令 <math id="M117"><mi>ρ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559327&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559322&type= 29.63333321 3.72533321 ，模型的正平衡点满足： <math display="block" id="M118"><mi>S</mi><mo>=</mo><mfrac><mrow><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mi>p</mi><mi>b</mi><mi>ρ</mi></mrow><mrow><mi>β</mi><mi>ρ</mi><mo>+</mo><mi>d</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559335&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559329&type= 24.72266769 8.88999939 ， <math display="block" id="M119"><mi>I</mi><mo>=</mo><mfrac><mrow><mi>γ</mi><mi>E</mi></mrow><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559346&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559338&type= 19.72733307 8.55133343 <math display="block" id="M120"><mi>E</mi><mo>=</mo><mi>ρ</mi><mfrac><mrow><mi>β</mi><mo stretchy="false">(</mo><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mi>p</mi><mi>b</mi><mi>ρ</mi><mo stretchy="false">)</mo><mo>+</mo><mi>p</mi><mi>b</mi><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>β</mi><mi>ρ</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>β</mi><mi>ρ</mi><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559355&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559351&type= 54.94866943 8.97466660 将 <math id="M121"><mi>E</mi><mo>,</mo><mi>I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559363&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559358&type= 5.58799982 3.21733332 代入 <math id="M122"><mi>ρ</mi><mo>=</mo><mi>E</mi><mo>+</mo><mi>I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559370&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559366&type= 15.24000072 3.72533321 得 <math display="block" id="M123"><mi>ρ</mi><mi>f</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559378&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559375&type= 15.66333389 3.81000018 其中， <math display="block" id="M124"><mi>f</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>γ</mi><mo>+</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mi>d</mi><mo>+</mo><mi>β</mi><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>β</mi><mi>ρ</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559385&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559381&type= 60.87533569 8.97466660 显然， <math id="M125"><msup><mrow><mi>f</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><mtext> </mtext><mi>f</mi><mo>″</mo><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559393&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559389&type= 31.58066750 3.97933340 ， <math id="M126"><mtext> </mtext><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>ρ</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559402&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559398&type= 28.70199966 5.07999992 。因此，方程 <math id="M127"><mi>ρ</mi><mi>f</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559409&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559406&type= 15.66333389 3.81000018 具有唯一的正平衡点，当且仅当 <math id="M128"><mi>f</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559416&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559414&type= 13.80066681 3.81000018 成立，即当且仅当满足 <math display="block" id="M129"><mfrac><mrow><mo stretchy="false">(</mo><mi>γ</mi><mo>+</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mi>d</mi><mo>+</mo><mi>β</mi><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559425&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559420&type= 49.44533539 8.72066689 由上面的分析，我们得到以下定理： 定理1 定义系统（4）的基本再生数为 <math display="block" id="M130"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>γ</mi><mo>+</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mi>d</mi><mo>+</mo><mi>β</mi><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559432&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559427&type= 51.56200027 8.72066689 模型（4）存在无病平衡点 <math id="M131"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mn mathvariant="normal">0</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mi>d</mi></mrow></mfrac><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559439&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559435&type= 41.40200043 7.53533316 。当 <math id="M132"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559447&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559443&type= 10.75266743 3.72533321 时，模型（4）有唯一的正平衡点 <math id="M133"><msup><mrow><mi>E</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>=</mo><mo stretchy="false">(</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>,</mo><msup><mrow><mi>E</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>,</mo><msup><mrow><mi>I</mi></mrow><mrow><mi>*</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559455&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559450&type= 25.06133270 3.81000018 ，满足 <math display="block" id="M134"><msup><mrow><mi>S</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>=</mo><mfrac><mrow><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mi>p</mi><mi>b</mi><msup><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></mrow><mrow><mi>β</mi><msup><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>+</mo><mi>d</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559464&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559460&type= 27.09333420 9.05933285 , <math display="block" id="M135"><msup><mrow><mi>I</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>=</mo><mfrac><mrow><mi>γ</mi><msup><mrow><mi>E</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559471&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559468&type= 21.08199883 8.72066689 <math display="block" id="M136"><msup><mrow><mi>E</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>=</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mfrac><mrow><mi>β</mi><mo stretchy="false">(</mo><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mi>p</mi><mi>b</mi><msup><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><mi>p</mi><mi>b</mi><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>β</mi><msup><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>β</mi><msup><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559475&type= 58.84333038 9.05933285 下面，首先分析系统（4）在无病平衡点 <math id="M137"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559485&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559481&type= 3.72533321 3.72533321 处的稳定性，我们得到如下定理： 定理 2 如果 <math id="M138"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559492&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559489&type= 10.83733368 3.72533321 ，那么无病平衡点 <math id="M139"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559499&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559497&type= 3.72533321 3.72533321 是全局渐近稳定的；而如果 <math id="M140"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559507&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559504&type= 10.75266743 3.72533321 ，则 <math id="M141"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559514&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559511&type= 3.72533321 3.72533321 不稳定。 证 定义满足的初始条件，对任意的 <math id="M142"><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559521&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559518&type= 7.95866680 2.79399991 ，满足 <math id="M143"><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559529&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559524&type= 10.92199993 3.81000018 ，由无病平衡点 <math id="M144"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559534&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559533&type= 3.72533321 3.72533321 处的Jacobian矩阵可得 <math display="block" id="M145"><mi mathvariant="bold-italic">J</mi><mo>=</mo><msub><mrow><mfenced separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><mi>d</mi></mtd><mtd><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mtd><mtd><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo></mtd><mtd><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mi>γ</mi></mtd><mtd><mo>-</mo><mo stretchy="false">(</mo><mi>μ</mi><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><msub><mrow><mo stretchy="false">|</mo></mrow><mrow><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559542&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559538&type= 80.09466553 18.79599953 则 <math display="block" id="M146"><mfenced open="|" close="|" separators="|"><mrow><mi>λ</mi><mi mathvariant="bold-italic">U</mi><mo>-</mo><mi mathvariant="bold-italic">J</mi></mrow></mfenced><mo>=</mo><mo stretchy="false">(</mo><mi>λ</mi><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo><mo>⋅</mo><mfenced open="|" close="|" separators="|"><mrow><mtable><mtr><mtd><mi>λ</mi><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo></mtd><mtd><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>-</mo><mi>γ</mi></mtd><mtd><mi>λ</mi><mo>+</mo><mo stretchy="false">(</mo><mi>μ</mi><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559550&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559546&type= 98.38265991 10.58333302 整理得 <math display="block" id="M147"><mfenced open="|" close="|" separators="|"><mrow><mi>λ</mi><mi mathvariant="bold-italic">U</mi><mo>-</mo><mi mathvariant="bold-italic">J</mi></mrow></mfenced><mo>=</mo><mo stretchy="false">(</mo><mi>λ</mi><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>A</mi><mi>λ</mi><mo>+</mo><mi>B</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559556&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559555&type= 53.76333237 3.97933340 其中， <math id="M148"><mi mathvariant="bold-italic">U</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559566&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559560&type= 2.96333337 2.79399991 为单位矩阵， <math id="M149"><mi>A</mi><mo>=</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo>+</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559571&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559567&type= 54.44067001 4.82599974 ， <math id="M150"><mi>B</mi><mo>=</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559578&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559574&type= 75.01467133 4.82599974 。 显然，当 <math id="M151"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo><</mo><mn mathvariant="normal">1</mn><mi mathvariant="normal">时</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559583&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559580&type= 14.56266594 4.31799984 ， <math id="M152"><mi>A</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>B</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559588&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559586&type= 20.31999969 3.21733332 ， <math id="M153"><mo stretchy="false">(</mo><msup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>A</mi><mi>λ</mi><mo>+</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559593&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559590&type= 28.95599937 3.81000018 必有两个具有负实部的根。因此，当 <math id="M154"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo><</mo><mn mathvariant="normal">1</mn><mi mathvariant="normal">时</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559596&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559594&type= 14.56266594 4.31799984 ，方程 <math id="M155"><mfenced open="|" close="|" separators="|"><mrow><mi>λ</mi><mi mathvariant="bold-italic">U</mi><mo>-</mo><mi mathvariant="bold-italic">J</mi></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559602&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559598&type= 21.08199883 3.97933340 的所有根具有负实部，无病平衡点 <math id="M156"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi mathvariant="normal">在</mi><mi mathvariant="normal">状</mi><mi mathvariant="normal">态</mi><mi mathvariant="normal">空</mi><mi mathvariant="normal">间</mi><mi>D</mi><mi mathvariant="normal">上</mi><mi mathvariant="normal">局</mi><mi mathvariant="normal">部</mi><mi mathvariant="normal">渐</mi><mi mathvariant="normal">近</mi><mi mathvariant="normal">稳</mi><mi mathvariant="normal">定</mi><mi mathvariant="normal">。</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559606&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559603&type= 54.94866943 4.40266657 下面根据引理2证明无病平衡点 <math id="M157"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559614&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559611&type= 3.72533321 3.72533321 全局稳定性。首先，构造Lyapunov函数满足 <math display="block" id="M158"><mi>V</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559620&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559617&type= 63.07666779 3.72533321 显然， <math id="M159"><mi>V</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559626&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559623&type= 8.12800026 3.55599999 在无病平衡点处等于零，并且 <math id="M160"><mi>V</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><mn mathvariant="normal">0</mn><mtext> </mtext></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559630&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559629&type= 15.32466698 3.64066648 满足引理2的第一个条件。在状态空间 D 上求解，根据解的有界性和正定性得， <math display="block" id="M161"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>V</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>+</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559636&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559633&type= 78.40132904 7.70466709 <math display="block" id="M162"><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>S</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo>+</mo><mi>I</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mi>E</mi><mo stretchy="false">]</mo><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559641&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559638&type= 75.52266693 3.89466667 <math display="block" id="M163"><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>γ</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559646&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559644&type= 53.84799576 3.89466667 <math display="block" id="M164"><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo>+</mo><mi>I</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mi>E</mi><mo stretchy="false">]</mo><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559651&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559648&type= 76.28466797 4.82599974 <math display="block" id="M165"><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>γ</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559656&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559653&type= 49.61466599 3.72533321 因此，得 <math display="block" id="M166"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>V</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>≤</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo>⋅</mo><mfenced separators="|"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559660&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559658&type= 123.18999481 19.13466644 <math display="block" id="M167"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>V</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>≤</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559666&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559663&type= 77.04666901 7.70466709 很显然，由 <math id="M168"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559671&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559668&type= 10.83733368 3.72533321 时， <math id="M169"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>V</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>≤</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559677&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559674&type= 18.11866570 7.70466709 成立，当且仅当 <math id="M170"><mi>E</mi><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>I</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559681&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559679&type= 19.38866615 3.21733332 时等号成立。因此，根据引理2，在系统（4）的状态空间 D 中，可得无病平衡点 <math id="M171"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559686&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559685&type= 3.72533321 3.72533321 是全局渐近稳定。 证毕。 下面分析模型（4）正平衡点 <math id="M172"><msup><mrow><mi>E</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>=</mo><mo stretchy="false">(</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>,</mo><msup><mrow><mi>E</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>,</mo><msup><mrow><mi>I</mi></mrow><mrow><mi>*</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559692&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559689&type= 25.06133270 3.81000018 的全局稳定性。首先，讨论传染病持续存在的条件。这在证明模型（4）正平衡点的全局稳定性时是重要的一部分。为了定义疾病持续存在的不变集，定义系统（4）上的连续流，即 <math display="block" id="M173"><mi>Φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mtext> </mtext><msup><mrow><mi mathvariant="normal">ℝ</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>×</mo><mi>D</mi><mo>→</mo><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559697&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559695&type= 33.61266708 3.81000018 对系统（4），满足 <math id="M174"><mi>Φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559703&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559700&type= 35.55999756 4.48733330 ， <math id="M175"><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559707&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559705&type= 8.04333305 3.55599999 ， <math id="M176"><mi>t</mi><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo><mtext> </mtext><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>∈</mo><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559711&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559710&type= 20.15066719 4.31799984 。进一步的，定义 <math display="block" id="M177"><mi>Y</mi><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∈</mo><mi>D</mi><mfenced open="|" close="" separators="|"><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>></mo><mn mathvariant="normal">0</mn></mrow></mfenced><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559717&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559714&type= 60.28266525 4.31799984 , <math display="block" id="M178"><mo>∂</mo><mi>Y</mi><mo>=</mo><mi>D</mi><mo>∖</mo><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559721&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559719&type= 17.69533348 3.55599999 下面，在（ <math id="M179"><mi>Y</mi><mo>,</mo><mo>∂</mo><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559727&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559724&type= 9.31333351 3.21733332 ）上研究系统（4）疾病的持续性。根据命题1得状态空间 D 是正不变集。对任意的初始条件 <math id="M180"><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>∈</mo><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559732&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559730&type= 10.66799927 4.06400013 ，即满足 <math id="M181"><mi>I</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559737&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559735&type= 13.46199894 3.55599999 ，根据系统（4）的第三个方程得 <math id="M182"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>γ</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559742&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559739&type= 46.99000168 7.70466709 ≥ <math id="M183"><mo>-</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559744&type= 22.60599899 3.72533321 。根据比较法则得 <math id="M184"><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>I</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mfenced separators="|"><mrow><mi>d</mi><mo>+</mo><mi>μ</mi></mrow></mfenced><mi>t</mi></mrow></msup><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559751&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559748&type= 36.32199860 4.23333359 。因此，子集 <math id="M185"><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559757&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559754&type= 2.87866688 2.79399991 在半流 <math id="M186"><mi>Φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559761&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559758&type= 12.27666664 3.55599999 下是正不变集。同理可得子集 <math id="M187"><mo>∂</mo><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559766&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559764&type= 4.57200003 2.79399991 在半流 <math id="M188"><mi>Φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559771&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559768&type= 12.27666664 3.55599999 下是正不变集，即满足 <math id="M189"><mi>Φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559775&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559773&type= 19.98133469 3.55599999 ， <math id="M190"><mi>Φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mo>∂</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⊂</mo><mo>∂</mo><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559782&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559779&type= 23.36800003 3.55599999 ， <math id="M191"><mi>t</mi><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559787&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559784&type= 7.95866680 3.30199981 。 此外，命题2也是证明疾病的一致持续性的重要内容。 命题 2 对于定义在 <math id="M192"><mo>∂</mo><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559791&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559790&type= 4.57200003 2.79399991 上半流 <math id="M193"><mi>Φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559797&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559794&type= 12.36133385 3.47133350 ，无病平衡 <math id="M194"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559800&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559798&type= 3.72533321 3.72533321 在 <math id="M195"><mo>∂</mo><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559806&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559803&type= 4.57200003 2.79399991 上全局渐近稳定。 证 令初始条件 <math id="M196"><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>I</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>E</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∈</mo><mo>∂</mo><mi mathvariant="normal">Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559810&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559809&type= 38.26933289 3.55599999 ，由系统（4）可得 <math id="M197"><mi>S</mi><mo>≤</mo><mfrac><mrow><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow><mrow><mi>d</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559816&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559813&type= 14.47799969 7.78933334 （t→∞） 。因此，不妨设 E （ t ）≤ <math id="M198"><mover><mrow><mi>E</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559820&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559818&type= 7.53533316 3.97933340 ， I （ t ）≤ <math id="M199"><mover><mrow><mi>I</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559826&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559823&type= 6.34999990 3.97933340 ，其中 <math id="M200"><mover><mrow><mi>E</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559832&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559829&type= 9.82133293 3.97933340 <math id="M201"><mover><mrow><mi>I</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559837&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559834&type= 6.34999990 3.97933340 满足下面的系统 <math display="block" id="M202"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><mover><mrow><mi>E</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfenced separators="|"><mrow><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mfrac><mrow><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow><mrow><mi>d</mi></mrow></mfrac></mrow></mfenced><mo stretchy="false">(</mo><mover><mrow><mi>E</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mover><mrow><mi>I</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mover><mrow><mi>E</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><mover><mrow><mi>I</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>γ</mi><mover><mrow><mi>E</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mover><mrow><mi>I</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559841&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559839&type= 73.66000366 18.37266731 在正不变集 <math id="M203"><mo>∂</mo><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559847&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559844&type= 4.57200003 2.79399991 上，由 <math id="M204"><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>I</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559851&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559849&type= 18.20333290 3.55599999 ， <math id="M205"><mi>E</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∈</mo><mo>∂</mo><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559857&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559854&type= 17.94933319 3.55599999 ，可得 <math id="M206"><mover><mrow><mi>E</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559861&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559858&type= 14.05466652 3.97933340 ， <math id="M207"><mover><mrow><mi>I</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn><mtext> </mtext><mo stretchy="false">(</mo><mi>t</mi><mo>→</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559865&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559864&type= 26.24666595 3.97933340 。从而 <math id="M208"><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559871&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559868&type= 14.05466652 3.55599999 ， <math id="M209"><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559878&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559874&type= 12.86933327 3.55599999 <math id="M210"><mtext> </mtext><mo stretchy="false">(</mo><mi>t</mi><mo>→</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559885&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559881&type= 13.46199894 3.55599999 。很显然 <math id="M211"><mi>S</mi><mo>=</mo><mfrac><mrow><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow><mrow><mi>d</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559891&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559887&type= 14.47799969 7.78933334 ， <math id="M212"><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559899&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559895&type= 14.05466652 3.55599999 ， <math id="M213"><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559904&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559901&type= 12.86933327 3.55599999 是系统在 <math id="M214"><mo>∂</mo><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559910&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559907&type= 4.57200003 2.79399991 上的唯一的平衡点。因此，可证得无病平衡点 <math id="M215"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559914&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559912&type= 3.72533321 3.72533321 在 <math id="M216"><mo>∂</mo><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559918&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559917&type= 4.57200003 2.79399991 上全局渐近稳定。 证毕。 下面依据文献［ 14 14 ~ 16 16 14~16 ］的结果，得到系统（4）对应的传染病一致持续性命题。 命题 3 如果 <math id="M217"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559923&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559920&type= 10.75266743 3.72533321 ，那么系统（4）对应的传染病是持续生存的，即存在一个常数 <math id="M218"><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559930&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559927&type= 1.43933344 2.79399991 > 0，使得 <math display="block" id="M219"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>t</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mtext> </mtext><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">f</mi><mo stretchy="false">{</mo><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>≥</mo><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559935&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559933&type= 46.05867004 4.65666676 对状态空间 D 中的任何一个满足条件的解都成立。 证 令 <math id="M220"><msup><mrow><mi>W</mi></mrow><mrow><mo>+</mo></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mi>x</mi><mo>∈</mo><mi>Y</mi><mo>:</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>t</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mtext> </mtext><mi>Φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559942&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559939&type= 58.16600037 4.65666676 。根据命题2，可知无病平衡点 <math id="M221"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559947&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559945&type= 3.72533321 3.72533321 在正不变集 <math id="M222"><mo>∂</mo><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559956&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559952&type= 4.57200003 2.79399991 上全局渐近稳定。下面只需证明存在 <math id="M223"><mi>T</mi><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559960&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559958&type= 9.56733322 3.30199981 ， <math id="M224"><mi>ε</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559967&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559965&type= 8.12800026 2.79399991 ，使得 <math id="M225"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559972&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559970&type= 3.72533321 3.72533321 对 Y 是一致弱排斥的，即对任意的 <math id="M226"><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>∈</mo><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559977&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559975&type= 10.66799927 4.06400013 及 t > T ，满足 <math id="M227"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>t</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mtext> </mtext><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">p</mi><mtext> </mtext><mo stretchy="false">(</mo><mi>Φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>≥</mo><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559983&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559980&type= 39.96266937 4.65666676 ，即只需证明 <math id="M228"><mtext> </mtext><msup><mrow><mi>W</mi></mrow><mrow><mo>+</mo></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>⋂</mo><mi>Y</mi><mo>=</mo><mi mathvariant="normal">∅</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559987&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559984&type= 29.63333321 4.99533367 。 利用反证法，如果 <math id="M229"><msup><mrow><mi>W</mi></mrow><mrow><mo>+</mo></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>⋂</mo><mi>Y</mi><mo>≠</mo><mi mathvariant="normal">∅</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559993&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559990&type= 28.87133217 4.99533367 ，则必存在 <math id="M230"><mi>y</mi><mo>∈</mo><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559996&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559995&type= 8.55133343 3.72533321 ，使得 <math id="M231"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>t</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>Φ</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560000&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4559998&type= 25.56933403 4.65666676 。由极限定义可知，必存在足够小的 <math id="M232"><mi>ε</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560003&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560002&type= 8.12800026 2.79399991 ，使得 <math id="M233"><msup><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msup><mo>-</mo><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560006&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560005&type= 9.56733322 3.04799986 ≤ S （ t ）≤ <math id="M234"><msup><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msup><mo>+</mo><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560010&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560008&type= 9.56733322 3.13266659 ，0≤ E （ t ） <math id="M235"><mo>≤</mo><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560015&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560013&type= 5.75733376 3.30199981 ， 0≤ I （ t ） <math id="M236"><mo>≤</mo><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560019&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560017&type= 5.75733376 3.30199981 （ t > T ）成立。 由 E （ t ）+ I （ t ）= ρ （ t ），由引理4可知，存在 <math id="M237"><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560023&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560022&type= 2.28600001 4.57200003 使得 <math id="M238"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">p</mi><mtext> </mtext><mi>E</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560027&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560025&type= 30.22600174 4.65666676 且 <math id="M239"><msup><mrow><mi>E</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">(</mo><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mi mathvariant="normal">。</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560032&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560030&type= 33.86666870 4.91066647 根据系统（4）的第二个方程可得 <math display="block" id="M240"><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><mi>E</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>S</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mi>E</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>≥</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msup><mo>-</mo><mi>ε</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>ρ</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mi>E</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560036&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560034&type= 95.41933441 14.47799969 由引理4得 <math id="M241"><msup><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup><mo>≥</mo><mfrac><mrow><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msup><mo>-</mo><mi>ε</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mo>+</mo><mi>γ</mi></mrow></mfrac><msup><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560039&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560038&type= 39.28533173 8.63599968 。同理，由系统（4）的第三个方程及引理3，得 <math id="M242"><msup><mrow><mi>I</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup><mo>≥</mo><mfrac><mrow><mi>γ</mi><msup><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup></mrow><mrow><mi>d</mi><mo>+</mo><mi>μ</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560042&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560041&type= 19.04999924 8.63599968 。由 <math id="M243"><msup><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup><mo>=</mo><msup><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup><mo>+</mo><msup><mrow><mi>I</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560047&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560044&type= 20.74333382 3.89466667 ，可知： <math id="M244"><msup><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup><mo>≥</mo><mfrac><mrow><mi>d</mi><mo>+</mo><mi>μ</mi><mo>+</mo><mi>γ</mi></mrow><mrow><mi>d</mi><mo>+</mo><mi>μ</mi></mrow></mfrac><mo>⋅</mo><msup><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup><mo>≥</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mo>+</mo><mi>μ</mi></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msup><mo>-</mo><mi>ε</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mo>+</mo><mi>γ</mi></mrow></mfrac><mo>⋅</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560052&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560050&type= 92.45600128 8.63599968 ，即得到 <math id="M245"><msup><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mi>d</mi><mo>+</mo><mi>μ</mi><mo>+</mo><mi>γ</mi></mrow><mrow><mi>d</mi><mo>+</mo><mi>μ</mi></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msup><mo>-</mo><mi>ε</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mo>+</mo><mi>γ</mi></mrow></mfrac></mrow></mfenced><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560058&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560056&type= 65.02400208 9.48266697 。因为 Y 是正不变集， <math id="M246"><msup><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560063&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560060&type= 10.75266743 4.06400013 ，上述不等式成立需满足 <math id="M247"><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mi>d</mi><mo>+</mo><mi>μ</mi><mo>+</mo><mi>γ</mi></mrow><mrow><mi>d</mi><mo>+</mo><mi>μ</mi></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msup><mo>-</mo><mi>ε</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mo>+</mo><mi>γ</mi></mrow></mfrac><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560069&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560066&type= 57.91199875 8.63599968 ，即 <math id="M248"><mfrac><mrow><mi>d</mi><mo>+</mo><mi>μ</mi><mo>+</mo><mi>γ</mi></mrow><mrow><mi>d</mi><mo>+</mo><mi>μ</mi></mrow></mfrac><mo>⋅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560073&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560070&type= 20.15066719 8.46666718 <math id="M249"><mfrac><mrow><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msup><mo>-</mo><mi>ε</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mo>+</mo><mi>γ</mi></mrow></mfrac><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560079&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560076&type= 31.75000000 8.63599968 。 然而，由 <math id="M250"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560083&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560081&type= 10.75266743 3.72533321 ，则对于足够小的 ε > 0，必会满足 <math id="M251"><mfrac><mrow><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msup><mo>-</mo><mi>ε</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560089&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560086&type= 54.61000061 8.72066689 。因此，产生矛盾，所以假设不成立，即 <math id="M252"><msup><mrow><mi>W</mi></mrow><mrow><mo>+</mo></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>⋂</mo><mi>Y</mi><mo>=</mo><mi mathvariant="normal">∅</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560094&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560091&type= 28.87133217 4.99533367 成立。从而得到系统（4）对应的传染病在 <math id="M253"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560099&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560097&type= 10.75266743 3.72533321 时持续存在。 证毕。 最后，根据命题3和引理1，利用构造迭代序列的方法证明正平衡点 <math id="M254"><msup><mrow><mi>E</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560106&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560101&type= 3.81000018 3.04799986 的全局稳定性。 定理 3 如果 <math id="M255"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560110&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560107&type= 10.75266743 3.72533321 ，那么正平衡点 <math id="M256"><msup><mrow><mi>E</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560114&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560112&type= 3.81000018 3.04799986 是全局渐近稳定的。 证 由引理2可知，存在足够小的 ε 使得 <math id="M257"><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560120&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560117&type= 31.41133308 3.64066648 。因此，根据系统（4）第一个方程可得 <math display="block" id="M258"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo><mo stretchy="false">(</mo><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mi>p</mi><mi>b</mi><mi>ε</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>β</mi><mi>ε</mi><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560124&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560123&type= 131.23332214 7.70466709 由比较法则可知，存在常数 <math id="M259"><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560130&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560127&type= 2.79399991 4.57200003 ，0 < <math id="M260"><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mo stretchy="false">{</mo><mi>ε</mi><mo>,</mo><mfrac><mrow><mi>p</mi><mi>b</mi><mi>ε</mi></mrow><mrow><mi>β</mi><mi>ε</mi><mo>+</mo><mi>d</mi></mrow></mfrac><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560136&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560133&type= 32.51200104 8.63599968 ，在 <math id="M261"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560139&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560138&type= 9.22866726 3.72533321 ，当 <math id="M262"><mi>t</mi><mo>></mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560143&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560141&type= 8.38199997 3.72533321 ，使得 <math display="block" id="M263"><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mi>p</mi><mi>b</mi><mi>ε</mi></mrow><mrow><mi>β</mi><mi>ε</mi><mo>+</mo><mi>d</mi></mrow></mfrac><mo>+</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560149&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560145&type= 55.62600327 8.88999939 根据系统（4）第二个方程可得 <math display="block" id="M264"><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560154&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560150&type= 79.07866669 13.54666710 <math display="block" id="M265"><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560160&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560158&type= 95.41933441 4.14866638 <math display="block" id="M266"><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560166&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560163&type= 68.49533081 4.31799984 <math display="block" id="M267"><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mo stretchy="false">(</mo><mi>β</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>+</mo><mi>d</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560172&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560169&type= 54.01733780 4.14866638 由比较法则可知，存在常数 <math id="M268"><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560176&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560173&type= 2.79399991 4.57200003 ，0 < <math id="M269"><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo><</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mo stretchy="false">{</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mfrac><mrow><mi>γ</mi><msup><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></mrow><mrow><mi>β</mi><msup><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>+</mo><mi>d</mi><mo>+</mo><mi>γ</mi></mrow></mfrac><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560182&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560180&type= 43.77266693 9.05933285 ，在 <math id="M270"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560187&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560184&type= 9.22866726 3.72533321 ，当 <math id="M271"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>></mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560193&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560190&type= 9.56733322 3.72533321 ，使得 E （ t ）≤ <math id="M272"><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><msup><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></mrow><mrow><mi>β</mi><msup><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>+</mo><mi>d</mi><mo>+</mo><mi>γ</mi></mrow></mfrac><mo>+</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo><</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560200&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560195&type= 50.20733261 9.05933285 根据系统（4）第三个方程可得 <math display="block" id="M273"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>γ</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>μ</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>γ</mi><mo stretchy="false">(</mo><msup><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>-</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>μ</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>γ</mi><msup><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>-</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mi>μ</mi><mo>+</mo><mi>γ</mi><mo stretchy="false">)</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560205&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560202&type= 142.32467651 7.70466709 由比较法则可知，存在常数 <math id="M274"><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560212&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560210&type= 2.79399991 4.57200003 ，0 < <math id="M275"><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo><</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">{</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mfrac><mrow><mi>μ</mi><msup><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></mrow><mrow><mi>μ</mi><mo>+</mo><mi>d</mi><mo>+</mo><mi>γ</mi></mrow></mfrac><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560218&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560215&type= 39.70866776 8.63599968 ，在 <math id="M276"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560223&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560220&type= 9.22866726 3.72533321 ，当 <math id="M277"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>></mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560229&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560226&type= 9.56733322 3.72533321 ，使得 <math display="block" id="M278"><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560235&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560231&type= 10.58333302 3.64066648 <math display="block" id="M279"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>γ</mi><msup><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></mrow><mrow><mi>μ</mi><mo>+</mo><mi>d</mi><mo>+</mo><mi>γ</mi></mrow></mfrac><mo>+</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo><</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560241&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560237&type= 43.26466751 8.63599968 另一方面，将系统（4）的第一个方程放缩可得 <math display="block" id="M280"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><mo stretchy="false">(</mo><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mi>p</mi><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>β</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560245&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560243&type= 131.74133301 7.70466709 由比较法则可知，存在常数 <math id="M281"><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560250&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560247&type= 2.79399991 4.57200003 ， <math id="M282"><mn mathvariant="normal">0</mn><mo><</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo><</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mo stretchy="false">{</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>,</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>-</mo><mi>p</mi><mi>b</mi><mo stretchy="false">)</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>β</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560254&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560252&type= 49.27600098 9.05933285 ，在 <math id="M283"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560259&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560256&type= 9.22866726 3.72533321 ，当 <math id="M284"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>></mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560265&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560261&type= 9.56733322 3.72533321 ，使得 S （ t ）≥ <math id="M285"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>-</mo><mi>p</mi><mi>b</mi><mo stretchy="false">)</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow><mrow><mi>β</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>+</mo><mi>d</mi></mrow></mfrac><mo>-</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560271&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560268&type= 42.41799927 9.05933285 根据系统（4）第二个方程可得 <math display="block" id="M286"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>γ</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>ε</mi><mo stretchy="false">)</mo><mi>ε</mi><mo>-</mo><mi>d</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>γ</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560279&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560275&type= 136.65200806 7.70466709 由比较法则可知，存在常数 <math id="M287"><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560287&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560284&type= 2.79399991 4.57200003 ，0 < <math id="M288"><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo><</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mo stretchy="false">{</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>,</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>ε</mi><mo stretchy="false">)</mo><mi>ε</mi></mrow><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>γ</mi><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560294&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560290&type= 41.06333542 8.63599968 ，在 <math id="M289"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560302&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560299&type= 9.22866726 3.72533321 ，当 <math id="M290"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>></mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560308&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560305&type= 9.56733322 3.72533321 ，使得 E （ t ）≥ <math id="M291"><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>ε</mi><mo stretchy="false">)</mo><mi>ε</mi></mrow><mrow><mi>γ</mi><mo>+</mo><mi>d</mi></mrow></mfrac><mo>-</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560314&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560311&type= 41.65599823 8.55133343 根据系统（4）第三个方程可得 <math display="block" id="M292"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>γ</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>μ</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>γ</mi><mi>ε</mi><mo>-</mo><mi>d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>μ</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560320&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560317&type= 83.56600189 7.70466709 由比较法则可知，存在常数 <math id="M293"><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560326&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560323&type= 2.79399991 4.57200003 ，0 < <math id="M294"><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mo><</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mo stretchy="false">{</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>,</mo><mfrac><mrow><mi>γ</mi><mi>ε</mi></mrow><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi>μ</mi><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560333&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560329&type= 38.26933289 8.55133343 ，在 <math id="M295"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560339&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560337&type= 9.22866726 3.72533321 ，当 <math id="M296"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mo>></mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560346&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560343&type= 9.56733322 3.72533321 ，使得 I （ t ）≥ <math id="M297"><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>γ</mi><mi>ε</mi></mrow><mrow><mi>μ</mi><mo>+</mo><mi>d</mi></mrow></mfrac><mo>-</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560352&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560349&type= 32.68133163 8.46666718 由此得到，由于 <math id="M298"><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560356&type= 2.37066674 4.57200003 （ i =1，2，3，4，5，6）足够小，因此，下面不等式成立 <math display="block" id="M299"><mtable columnalign="left"><mtr><mtd><mn mathvariant="normal">0</mn><mo><</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>≤</mo><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mo><</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>≤</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mo><</mo><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>≤</mo><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560367&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560363&type= 40.21666718 15.74800014 进一步的，我们对系统（4）重复这个过程，根据系统（4）第一个方程可得 <math display="block" id="M300"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mi>p</mi><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">[</mo><mi>d</mi><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560373&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560371&type= 153.50067139 7.70466709 由比较法则可知，存在常数 <math id="M301"><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560379&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560377&type= 2.79399991 4.57200003 ， <math id="M302"><mn mathvariant="normal">0</mn><mo><</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub><mo><</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mo stretchy="false">{</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mo>,</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mfrac><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560385&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560381&type= 32.17333221 7.53533316 ，在 <math id="M303"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560391&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560388&type= 9.22866726 3.72533321 ，当 <math id="M304"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub><mo>></mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560401&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560395&type= 9.56733322 3.72533321 ，使得 <math display="block" id="M305"><mtable columnalign="left"><mtr><mtd><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">{</mo><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn><mtext> </mtext></mrow></msub><mo>,</mo><mfrac><mrow><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>-</mo><mi>p</mi><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">[</mo><mi>d</mi><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></mfrac><mo>+</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub><mo stretchy="false">}</mo><mo><</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560410&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560405&type= 69.08799744 15.40933418 根据系统（4）第二个方程可得 <math display="block" id="M306"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>γ</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn><mtext> </mtext></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>γ</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560420&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560415&type= 155.87133789 7.70466709 由比较法则可知，存在常数 <math id="M307"><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">8</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560427&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560424&type= 2.79399991 4.57200003 ，0 < <math id="M308"><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">8</mn></mrow></msub><mo><</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mo stretchy="false">{</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub><mo>,</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">8</mn></mrow></mfrac><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560436&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560432&type= 25.56933403 7.53533316 ，在 <math id="M309"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">8</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560444&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560440&type= 9.22866726 3.72533321 ，当 <math id="M310"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">8</mn></mrow></msub><mo>></mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560451&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560447&type= 9.56733322 3.72533321 ，使得 <math display="block" id="M311"><mtable columnalign="left"><mtr><mtd><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mo stretchy="false">{</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><mi>γ</mi><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><msub><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac><mo>+</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">8</mn></mrow></msub><mo stretchy="false">}</mo><mo><</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560455&type= 73.91400146 15.32466698 根据系统（4）第三个方程可得： <math display="block" id="M312"><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>γ</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>μ</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo></mtd></mtr><mtr><mtd><mi>γ</mi><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mi>d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>μ</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560468&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560464&type= 53.67866516 14.13933372 由比较法则可知，存在常数 <math id="M313"><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">9</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560476&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560472&type= 2.79399991 4.57200003 ， <math id="M314"><mn mathvariant="normal">0</mn><mo><</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">9</mn></mrow></msub><mo><</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mo stretchy="false">{</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">8</mn></mrow></msub><mo>,</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">9</mn></mrow></mfrac><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560484&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560480&type= 32.17333221 7.53533316 ，在 <math id="M315"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">9</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560493&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560488&type= 9.22866726 3.72533321 ，当 <math id="M316"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">9</mn></mrow></msub><mo>></mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">8</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560501&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560497&type= 9.56733322 3.72533321 ，使得 <math display="block" id="M317"><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn><mtext> </mtext></mrow></msub><mo>,</mo><mtext> </mtext><mfrac><mrow><mi>γ</mi><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi>μ</mi><mo>+</mo><mi>d</mi></mrow></mfrac><mo>+</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">9</mn></mrow></msub></mrow></mfenced><mo><</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560509&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560505&type= 63.50000000 9.48266697 另一方面，再次循环，据系统（4）第一个方程可得 <math display="block" id="M318"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>≥</mo><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560518&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560512&type= 75.60733032 7.70466709 由比较法则可知，存在常数 <math id="M319"><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">10</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560526&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560522&type= 3.81000018 4.57200003 ，0 < <math id="M320"><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">10</mn></mrow></msub><mo><</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mo stretchy="false">{</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">9</mn></mrow></msub><mo>,</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">10</mn></mrow></mfrac><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560535&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560531&type= 28.44799995 7.53533316 ，在 <math id="M321"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">10</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560545&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560539&type= 10.15999985 3.72533321 ，当 <math id="M322"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">10</mn></mrow></msub><mo>></mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">9</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560553&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560549&type= 10.58333302 3.72533321 ，使得 <math display="block" id="M323"><mtable columnalign="left"><mtr><mtd><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mo stretchy="false">{</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn><mtext> </mtext></mrow></msub><mo>,</mo><mfrac><mrow><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>-</mo><mi>p</mi><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac><mo>-</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">10</mn></mrow></msub><mo stretchy="false">}</mo><mo>></mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560562&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560557&type= 69.34200287 15.40933418 根据系统（4）第二个方程可得 <math display="block" id="M324"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>≥</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mi>d</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>γ</mi><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560570&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560566&type= 77.97799683 7.70466709 由比较法则可知，存在常数 <math id="M325"><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560581&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560577&type= 3.81000018 4.57200003 ，0 < <math id="M326"><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo><</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mo stretchy="false">{</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">10</mn></mrow></msub><mo>,</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">11</mn></mrow></mfrac><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560589&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560586&type= 29.37933350 7.45066643 ，在 <math id="M327"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560597&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560593&type= 10.15999985 3.72533321 ，当 <math id="M328"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>></mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">10</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560605&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560601&type= 11.59933376 3.72533321 ，使得 <math display="block" id="M329"><mtable columnalign="left"><mtr><mtd><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mo stretchy="false">{</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn><mtext> </mtext></mrow></msub><mo>,</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><mi>γ</mi><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac><mo>-</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo stretchy="false">}</mo><mo>></mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560614&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560610&type= 72.47466278 15.32466698 根据系统（4）第三个方程可得 <math display="block" id="M330"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>≥</mo><mi>γ</mi><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mi>d</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>μ</mi><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560622&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560618&type= 45.38133240 7.70466709 由比较法则可知，存在常数 <math id="M331"><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560630&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560626&type= 3.81000018 4.57200003 ， <math id="M332"><mn mathvariant="normal">0</mn><mo><</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo><</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mo stretchy="false">{</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">12</mn></mrow></mfrac><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560639&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560634&type= 35.98333359 7.45066643 ，在 <math id="M333"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560648&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560643&type= 10.15999985 3.72533321 ，当 <math id="M334"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo>></mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560656&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560651&type= 11.59933376 3.72533321 ，使得 <math display="block" id="M335"><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mtext> </mtext><mo stretchy="false">{</mo><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mfrac><mrow><mi>γ</mi><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi>μ</mi><mo>+</mo><mi>d</mi></mrow></mfrac><mo>-</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo stretchy="false">}</mo><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560663&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560659&type= 58.33533096 8.46666718 同理，依次迭代 m 次后，得到6个序列，满足 <math display="block" id="M336"><mtable columnalign="left"><mtr><mtd><mn mathvariant="normal">0</mn><mo><</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>≤</mo><mi>S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub><mo><</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mo><</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>≤</mo><mi>E</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub><mo><</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mo><</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>≤</mo><mi>I</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>m</mi></mrow></msub><mo><</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560671&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560668&type= 41.23266602 15.74800014 其中， <math display="block" id="M337"><msub><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>-</mo><mi>p</mi><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>+</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>+</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560680&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560676&type= 40.97866821 8.97466660 <math display="block" id="M338"><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>-</mo><mi>p</mi><mi>b</mi><mo stretchy="false">(</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560687&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560684&type= 40.04733276 8.97466660 <math display="block" id="M339"><msub><mrow><mi>b</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><msub><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><mi>γ</mi><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><msub><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560694&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560691&type= 43.09533310 8.80533314 <math display="block" id="M340"><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi><mtext> </mtext></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><mi>γ</mi><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560700&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560697&type= 44.70399857 8.80533314 <math display="block" id="M341"><msub><mrow><mi>c</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>γ</mi><msub><mrow><mi>b</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow><mrow><mi>μ</mi><mo>+</mo><mi>d</mi></mrow></mfrac><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>γ</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow><mrow><mi>μ</mi><mo>+</mo><mi>d</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560707&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560704&type= 39.53933334 8.46666718 <math id="M342"><mo stretchy="false">{</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560712&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560710&type= 17.27199936 4.48733330 是有界的递增数列， <math id="M343"><mo stretchy="false">{</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560719&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560717&type= 20.15066719 4.48733330 是有界的递减序列，从而6个序列的极限存在。分别记 <math id="M344"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>m</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo stretchy="false">{</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">}</mo><mo>=</mo><mo stretchy="false">{</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560725&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560722&type= 41.99466705 4.65666676 ， <math id="M345"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>m</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo stretchy="false">{</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560733&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560729&type= 27.85533333 4.65666676 = <math id="M346"><mo stretchy="false">{</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560741&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560736&type= 15.57866669 3.55599999 。 因为 <math id="M347"><mn mathvariant="normal">0</mn><mo><</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo><</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560751&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560746&type= 19.13466644 7.53533316 ，则得到各自的极限为 <math display="block" id="M348"><mi>a</mi><mo>=</mo><mfrac><mrow><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mi>p</mi><mi>b</mi><mo stretchy="false">(</mo><mi>B</mi><mo>+</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><mi>B</mi><mo>+</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560756&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560753&type= 36.06800079 8.97466660 <math display="block" id="M349"><mi>A</mi><mo>=</mo><mfrac><mrow><mi>b</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mi>p</mi><mi>b</mi><mo stretchy="false">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mo>+</mo><mi>β</mi><mo stretchy="false">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560763&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560759&type= 34.88266754 8.97466660 <math display="block" id="M350"><mi>b</mi><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>a</mi><mo stretchy="false">)</mo><mi>c</mi></mrow><mrow><mo stretchy="false">(</mo><mi>γ</mi><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>a</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560769&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560766&type= 39.70866776 8.80533314 <math display="block" id="M351"><mi>B</mi><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>A</mi><mo stretchy="false">)</mo><mi>A</mi></mrow><mrow><mo stretchy="false">(</mo><mi>γ</mi><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>b</mi><mo>+</mo><mi>β</mi><mi>A</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560774&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560772&type= 41.31733322 8.80533314 <math display="block" id="M352"><mi>c</mi><mo>=</mo><mfrac><mrow><mi>γ</mi><mi>b</mi></mrow><mrow><mi>μ</mi><mo>+</mo><mi>d</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560778&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560776&type= 16.42533302 8.46666718 , <math display="block" id="M353"><mi>C</mi><mo>=</mo><mfrac><mrow><mi>γ</mi><mi>B</mi></mrow><mrow><mi>μ</mi><mo>+</mo><mi>d</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560784&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560781&type= 17.61066628 8.46666718 当且仅当极限满足 A=a= <math id="M354"><msup><mrow><mi>S</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560790&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560787&type= 5.24933338 3.47133350 B=b= <math id="M355"><msup><mrow><mi>E</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560796&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560793&type= 5.75733376 3.47133350 C=c= <math id="M356"><msup><mrow><mi>I</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560803&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560799&type= 2.70933342 3.04799986 等式成立。因此， <math id="M357"><msup><mrow><mi>E</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560809&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560806&type= 3.81000018 3.04799986 在状态空间 D 内全局渐近稳定。 证毕。 3　 数值模拟 1 根据针对COVID-19的已有参数估计研究结果，参考现阶段官方公布的各省确诊人数和累计治愈人数，推断出传染率 β 的估计值大约在0.12~0.25之间。人类免疫缺陷病毒（HIV）的垂直传染率（主要是血液传染）大约在0.15~0.45之间，如果进行体检和孕检等母婴阻断干预，HIV垂直传染率为0.05左右。由于COVID-19的案例较少，参照HIV的垂直传染率，数值模拟中，COVID-19的垂直传染率设定为0.01~0.10左右。 图1 图1 模拟垂直传染率 p 和潜伏者转变为染病者的转变率 γ 对基本再生数的影响。设参数 b= 0.05 ，d= 0.05， μ= 0.15。由 图1 图1 可见，垂直传染率 p 和基本再生数呈现线性正相关。因此，在考虑COVID-19潜在的垂直传染风险时，需要采用更多的医学手段降低垂直感染率，从而降低基本再生数，控制病毒传播。同时，和转变率相比，垂直传染概率对基本再生数的影响较弱。从生物学意义上看，现阶段对潜伏者（部分疑似病例）大面积集中搜寻，对这部分潜伏者采取早发现早隔离治疗的防控策略仍然是重中之重。 图2 图2 验证了模型无病平衡点的稳定性。设： b =0.05； d =0.05； p =0.01； β =0.10； μ =0.2； γ =0.25。基本再生数 <math id="M358"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560825&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560822&type= 3.81000018 3.72533321 =0.946 < 1，染病者的相对数量最终趋于零，此时传染病得到控制。 图3 图3 验证了模型正平衡点的稳定性。设定参数如下： b =0.05； d =0.05； p =0.05； β =0.20； μ =0.1； γ =0.2。基本再生数 <math id="M359"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560831&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560828&type= 3.81000018 3.72533321 =2.7 > 1，由 图3 图3 可知，此时正平衡点持续存在，此类传染病暴发，形成地方病。根据 图2 图2 和 图3 图3 显示的稳定性趋势，可知考虑垂直传染和潜伏期传染性的传染病模型的稳定性完全由基本再生数控制。因此，可通过控制基本再生数表达式中涉及的参数，使得基本再生数小于1来防控COVID-19的传播。例如，通过戴口罩、消毒、隔离等方式降低病毒的传染率 β ；通过隔离保护孕妇，降低新生儿的垂直传染率 p ；通过追踪排查潜伏期无症状患者，对其进行治疗和隔离，减少其与健康者的接触，增加转变率 γ 等，这些防控措施都可降低基本再生数，控制COVID-19的进一步传播和暴发。 4　 结 语 1 针对COVID-19疫情的发展及新生儿患病案例的出现，本文利用流行病学建模理论，基于潜在的垂直传染的风险和潜伏者的传染性，提出了一类具有垂直传染的SEIR传染病动力学模型，旨在为潜在垂直传染风险的COVID-19和类似其传播特性的传染病的防控提供理论依据。 本文提出的SEIR传染病模型考虑了可调控的垂直传染率，使得模型具有更一般的适用性，可合理有效地描述COVID-19在传播或者病毒变异中存在的垂直传染的潜在风险，也适用于具有类似传播特性的传染病的建模分析。建模过程考虑了潜伏者的感染性，克服了传统SEIR模型仅考虑易感者与有症状个体接触才被传染的局限性，为进一步研究COVID-19的传播特性、疫情预测和防控提供更合理的数学模型。本文全面分析了模型的全局稳定性的性质。在这类模型下，基本再生数完全控制传染病的传播规律和稳定性，在不考虑政府调控和人口流动的情况下，系统不会出现分支现象。同时，通过理论和数值模拟，本文得到垂直传染率对基本再生数是呈现线性正相关的关系，潜伏者的转变率与基本再生数具有非线性关系。在国家调控的情况下，采用隔离的方式明显控制了疫情的大暴发。当垂直传染风险存在的情况下，除了利用隔离的措施保证孕妇远离传染源，还应该提高对孕妇状态的监控，减少垂直传染率，从而降低基本再生数。进一步了解COVID-19的传播特性以及可能存在的传播途径，对于COVID-19疫情的控制具有重要的意义。 参考文献 车小燕 ， 郝卫 ， 丘立文 ， 等 . 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DOI:10.14188/j.1671-8836.2020.0129 陈姗姗,黄勃,方志军.传染病垂直传染的传播动力学分析：以COVID⁃19为例[J].武汉大学学报（理学版）,2020,66(5):433-441. DOI:10.14188/j.1671-8836.2020.0129 国家自然科学基金项目（61772328，61603242），上海工程技术大学科研启动基金（0232-E3-0507-19-05105） 2020-05-21 2020-10-24 2020-12-15 武汉大学学报（理学版） 5 66