隔离措施对COVID⁃19疫情控制的模型分析

隔离措施对COVID⁃19疫情控制的模型分析 Analysis of the Isolation Measure on the Control Model of COVID⁃19 1 first-author 朱翌民，男，硕士生，现从事控制工程方面的研究。E-mail： 1915736983@qq.com 朱翌民，男，硕士生，现从事控制工程方面的研究。E-mail： 1915736983@qq.com 1915736983@qq.com 1 corresp E-mail：huangbosues@sues.edu.cn E-mail：huangbosues@sues.edu.cn huangbosues@sues.edu.cn 1 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 考虑新型冠状病毒肺炎（corona virus disease 2019，COVID⁃19）的传播机制和隔离措施，将人群区分为Susceptible，Exposed，Infectious，Quarantine，Recovered，建立了带有隔离措施的SEIR模型，并计算了模型的基本再生数。政府采取的隔离防控手段，减少了人群间接触，使易感者的接触率降低，通过更改模型中的接触率参数模拟不同程度的防控手段下的疫情。研究结果表明，在采取严密防控措施下，潜伏期患者和传染期患者，这两种病毒携带者的人数峰值均显著减小，且疫情高峰期到来更早，说明隔离防控措施对防止疫情在更大范围的扩散起到了很好的遏制作用。 Considering the propagation mechanism and isolation measures of COVID⁃19 (corona virus disease 2019), a SEIR（Susceptible-Exposed-Infectious-Recovered） model with isolation measures is established, and the basic regeneration number of the model is calculated. In this model, the population is divided into several categories: susceptible, exposed, infectious, quarantine and recovered. The isolation and control measures adopted by the government can reduce the contact among people and reduce the contact rate of susceptible persons. The epidemic situation under different prevention and control measures is simulated by changing the contact rate parameters in the model. The results showed that, under the strict control measures, the peak number of the two virus carriers decreased significantly in the incubation period and in the infectious period, and the peak of the epidemic came earlier than expectation, indicating that the isolation and control measures played a good role in preventing the spread of the epidemic in a wider range. 新型冠状病毒肺炎 流行病学 SEIR 隔离 COVID⁃19(corona virus disease 2019) epidemiological SEIR（susceptible-exposed-infectious-recovered） isolation measure 朱翌民,黄勃,王忠震, 等.隔离措施对COVID⁃19疫情控制的模型分析[J].武汉大学学报（理学版）,2020,66(5):442-450. DOI:10.14188/j.1671-8836.2020.0146. 0　 引 言 1 新型冠状病毒肺炎（corona virus disease 2019，COVID⁃19）暴发后，中国政府迅速采取相关防疫措施，有效地遏制了国内的疫情发展。以2020年3月18日为例，来自国家卫生健康委员会官方网站信息显示当日国内新增确诊病例仅34例，国内疫情已基本得到控制。而国外，由于没有采用有效的措施，疫情仍呈上升趋势。据世卫组织公布的数据，截至欧洲中部时间2020年7月30日10时（北京时间16时），全球新冠确诊病例较前一日新增253 801例，达到16 812 763例；死亡病例较前一日增加5 999例，达到662 095例。美国约翰斯·霍普金斯大学实时统计数据显示，截至2020年7月29日，美国累计新冠确诊病例已超过440万例，累计死亡病例超过15万例。 近期国内复工人员流动增加了传播风险，对疫情控制产生了不小的挑战。为了科学指导疫情防控，需要建立合适的模型。已有很多学者 ［ 1 1 ~ 5 5 1~5 ］ 通过建立流行病动力学模型对COVID⁃19疫情进行分析，预测疫情走势。当前疫情显示，不同类型的感染者所需防控措施不同，模型拟合和疫情预测效果各异。为此，本研究考虑了隔离措施和潜伏期的感染者不易察觉且具有传染力的特点，将人群区分为Susceptible，Exposed，Infectious，Quarantine，Recovered，建立了考虑潜伏期性质的带有隔离措施的SEIR模型。 1　 基于隔离措施的SEIR传染病模型 1 1.1　模型介绍 2 依据流行病学相关理论 ［ 6 6 ~ 9 9 6~9 ］ 、COVID-19的传播特性以及当前中国的隔离防治手段，将 t 时刻的人群划分为：易感者S（Susceptible），未被感染的健康人群；潜伏期感染者E（Exposed），已感染新型冠状病毒，但未表现出病理特征的人群，具有轻微传染性；传染期感染者I（Infectious），已感染新型冠状病毒，并表现出病理特征的人群，具有高传染性；康复者R（Recovered），经医疗手段后康复的人群，不具备感染能力，且不能被再次感染；被隔离的患者Q（Quarantine），诊断为感染新型冠状病毒，并与外界人群隔离。总人口为 <math id="M1"><mi>N</mi><mo>=</mo><mi>S</mi><mo>+</mo><mi>E</mi><mo>+</mo><mi>I</mi><mo>+</mo><mi>Q</mi><mo>+</mo><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560269&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560264&type= 39.11599731 3.30199981 ，其中： S ， E ， I ， Q ， R 分别表示易感者、潜伏期感染者、传染期感染者、被隔离的患者和康复者人数。已知当前仅存在人际间传染，针对性的治疗用特效药暂时没有研制出来，康复者再次感染的几率极小，忽略交通带来的人口流动，易感者只要接触到病毒携带者，即潜伏期感染者与传染期感染者就可能被传染，成为新的潜伏期感染者，潜伏期感染者经过一段时间后转化为传染期感染者，传染期感染者可能被确诊隔离。根据上述病毒传播过程和机制，在SEIR模型基础上，建立如下基于隔离措施的SEIR传染病模型 <math display="block" id="M2"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>A</mi><mo>-</mo><mi>M</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>k</mi><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mi>μ</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd></mtr><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>M</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>k</mi><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>-</mo><mi>μ</mi><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd></mtr><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>-</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>-</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>-</mo><mi>μ</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd></mtr><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>Q</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>-</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>Q</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>-</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>Q</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>-</mo><mi>μ</mi><msub><mrow><mi>Q</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd></mtr><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>R</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>Q</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>-</mo><mi>μ</mi><msub><mrow><mi>R</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560274&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560272&type= 57.57333374 45.97399902 模型（1）涉及参数和变量意义如下： · 传染期感染者的传染率为 <math id="M3"><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560283&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560278&type= 3.13266659 4.74133348 ，潜伏期感染者的传染率为 <math id="M4"><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560289&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560286&type= 3.13266659 4.74133348 ，易感者接触到的人数期望为 <math id="M5"><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560297&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560293&type= 1.52399993 2.79399991 ，潜伏期感染者成为传染期感染者的概率为 <math id="M6"><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560303&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560300&type= 2.70933342 3.72533321 ； · 传染期感染者被确诊隔离的比例为 <math id="M7"><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560309&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560306&type= 2.87866688 3.72533321 ； · 传染期感染者成为康复者的比例为 <math id="M8"><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560316&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560312&type= 3.13266659 4.57200003 ，传染期感染者的死亡率为 <math id="M9"><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560322&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560319&type= 3.13266659 4.57200003 ； · 被隔离的确诊患者成为康复者的比例为 <math id="M10"><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560328&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560325&type= 3.13266659 4.57200003 ，被隔离的确诊患者死亡率为 <math id="M11"><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560336&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560331&type= 3.13266659 4.57200003 ； · <math id="M12"><mi>k</mi><mo>=</mo><mfrac><mrow><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560344&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560341&type= 10.92199993 8.55133343 是传染期感染者I与潜伏期感染者E的传染率的比值，人口自然死亡率为 <math id="M13"><mi>μ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560350&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560347&type= 1.94733334 3.64066648 ； · 为简化符号，记 <math id="M14"><mi>M</mi><mo>=</mo><mi>r</mi><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560355&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560353&type= 18.11866570 8.38199997 ， <math id="M15"><mi>A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560362&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560358&type= 2.87866688 2.79399991 为人口的输入（出生和迁入）。 由 <math id="M16"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560368&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560365&type= 1.10066664 2.79399991 时刻的总人口数为 <math id="M17"><msub><mrow><mi>N</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560374&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560370&type= 43.26466751 4.14866638 ，将方程组（1）的所有方程同侧相加得： <math id="M18"><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>A</mi><mo>-</mo><mi>μ</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>-</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>-</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>Q</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>≤</mo><mi>A</mi><mo>-</mo><mi>μ</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560378&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560376&type= 64.93933105 7.62000036 ，可以求得： <math id="M19"><msub><mrow><mi>N</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>≤</mo><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>N</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><mo stretchy="false">)</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><mo>-</mo><mi>μ</mi><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560383&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560380&type= 50.29199982 8.38199997 。根据生物学意义，模型的最大正向不变集 <math id="M20"><mi>Ω</mi><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><mfenced separators="|"><mrow><mi>S</mi><mo>,</mo><mi>E</mi><mo>,</mo><mi>I</mi><mo>,</mo><mi>Q</mi><mo>,</mo><mi>R</mi></mrow></mfenced><mo>∈</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msubsup><mfenced open="|" close="" separators="|"><mrow><mi>S</mi><mo>+</mo><mi>E</mi><mo>+</mo><mi>I</mi><mo>+</mo><mi>Q</mi><mo>+</mo><mi>R</mi><mo>≤</mo><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560389&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560386&type= 83.14266968 10.15999985 。 1.2　模型的基本再生数和平衡点 2 基本再生数是刻画传染病发病的一个重要指标，它是指一个感染者进入到一个完全无病且易感的人群后，在具有感染性的时间内所造成的新感染的人数的期望。如果 <math id="M21"><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=4560394&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560392&type= 10.83733368 3.72533321 ，即一个感染者在其感染期内所感染的人数平均不到一个，那么疾病就无法在人群中扩散，最终趋于消亡；反之，如果 <math id="M22"><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=4560403&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560398&type= 10.75266743 3.72533321 ，即一个感染者平均感染的人数超过一个，那么疾病就会持续扩散形成地方病 ［ 10 10 ］ 。 为研究各种人群随着时间演变的过程，通过微分方程的稳定性理论，不求解微分方程，来研究平衡状态的稳定性。本文参考文献中方法 ［ 11 11 ， 12 12 ］ ，将模型中的每种人群类型看作网络中节点，不同人群类型的转化看作节点之间的连接，每个节点的度分布均匀，因此本文提出的模型可以看作是均匀网络上的传染病模型。当传染病模型的方程组中人群数量变化率均为零时，模型方程的根组成的点称为无病平衡点。通过无病平衡点处的再生矩阵导出本文模型的基本再生数 <math id="M23"><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=4560411&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560407&type= 3.81000018 3.72533321 ，并分析平衡点的存在性。 定理1 模型（1）的基本再生数为 <math display="block" id="M24"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>r</mi><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi>N</mi></mrow></mfrac><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><mfrac><mrow><mi>k</mi><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560419&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560414&type= 62.06066895 8.72066689 当 <math id="M25"><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=4560428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560422&type= 10.92199993 4.23333359 时，模型有唯一的无病平衡点 <math id="M26"><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">0,0</mn><mo>,</mo><mn mathvariant="normal">0</mn><mo>,</mo><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560435&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560431&type= 28.19400024 8.38199997 ； 当 <math id="M27"><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=4560443&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560439&type= 10.75266743 3.72533321 时，存在唯一的地方病平衡点 <math id="M28"><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>=</mo><mo stretchy="false">(</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><mo>,</mo><msup><mrow><mi>Q</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>,</mo><msup><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560448&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560446&type= 32.85066605 3.81000018 ，满足 <math display="block" id="M29"><msup><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>=</mo><mfrac><mrow><mi>A</mi><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></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560457&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560452&type= 27.09333420 8.72066689 <math display="block" id="M30"><msup><mrow><mi>I</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>=</mo><mfrac><mrow><mi>A</mi><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><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560466&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560462&type= 55.03333664 8.72066689 <math display="block" id="M31"><msup><mrow><mi>Q</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>=</mo><mfrac><mrow><mi>A</mi><mfenced separators="|"><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560475&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560470&type= 77.89333344 9.73666668 <math display="block" id="M32"><msup><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>=</mo><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560483&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560479&type= 15.32466698 8.46666718 根据方程组可知 R 的稳定性由 S 、 E 、 I 、 Q 的稳定性来确定，因此只需考虑模型的前4个方程的稳定性即可，从而可将原来的模型（1）写为 <math display="block" id="M33"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>A</mi><mo>-</mo><mi>M</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>k</mi><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mi>μ</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd></mtr><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>M</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>k</mi><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd></mtr><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd></mtr><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>Q</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560491&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560487&type= 55.20266724 36.57600021 现在不妨取 <math id="M34"><mi mathvariant="bold-italic">X</mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>I</mi><mo>,</mo><mi>Q</mi><mo>,</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560499&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560495&type= 26.75466537 3.81000018 <math display="block" id="M35"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>A</mi><mo>-</mo><mi>M</mi><mi>S</mi><mo stretchy="false">(</mo><mi>I</mi><mo>+</mo><mi>k</mi><mi>E</mi><mo stretchy="false">)</mo><mo>-</mo><mi>μ</mi><mi>S</mi><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mi>M</mi><mi>S</mi><mo stretchy="false">(</mo><mi>I</mi><mo>+</mo><mi>k</mi><mi>E</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mi>E</mi><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mi>E</mi><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mi>I</mi><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mi>I</mi><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mi>Q</mi><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560508&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560504&type= 49.27600098 22.26733398 方程组（3）有两组解。其中一组取 <math id="M36"><mi>E</mi><mo>=</mo><mi>I</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560516&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560513&type= 15.74800014 2.79399991 ，得到无病平衡点： <math id="M37"><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">0,0</mn><mo>,</mo><mn mathvariant="normal">0</mn><mo>,</mo><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560525&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560520&type= 56.81133270 8.38199997 。 将模型（2）写成如下形式 <math display="block" id="M38"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>X</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi mathvariant="bold-italic">F</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo><mo>-</mo><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560533&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560529&type= 34.79800034 7.53533316 其中， <math id="M39"><mi mathvariant="bold-italic">F</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">X</mi></mrow></mfenced><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mi>M</mi><mi>S</mi><mfenced separators="|"><mrow><mi>I</mi><mo>+</mo><mi>k</mi><mi>E</mi></mrow></mfenced></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560542&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560538&type= 37.33800125 19.72733307 ， <math id="M40"><mi mathvariant="bold-italic">V</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">X</mi></mrow></mfenced><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi></mrow></mfenced><mi>E</mi></mtd></mtr><mtr><mtd><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mi>E</mi><mo>+</mo><mfenced separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi></mrow></mfenced><mi>I</mi></mtd></mtr><mtr><mtd><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mi>I</mi><mo>+</mo><mfenced separators="|"><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>μ</mi></mrow></mfenced><mi>Q</mi></mtd></mtr><mtr><mtd><mi>M</mi><mi>S</mi><mfenced separators="|"><mrow><mi>I</mi><mo>+</mo><mi>k</mi><mi>E</mi></mrow></mfenced><mo>+</mo><mi>μ</mi><mi>S</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560552&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560546&type= 63.50000000 25.99266624 。 因为 <math id="M41"><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560559&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560555&type= 3.89466667 3.72533321 是上述方程的无病平衡点，因此 <math display="block" id="M42"><mi mathvariant="normal">d</mi><mi>F</mi><msub><mrow><mo stretchy="false">|</mo></mrow><mrow><mi mathvariant="bold-italic">X</mi><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560568&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560564&type= 29.04066658 8.55133343 , <math display="block" id="M43"><mi mathvariant="normal">d</mi><mi>V</mi><msub><mrow><mo stretchy="false">|</mo></mrow><mrow><mi mathvariant="bold-italic">X</mi><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mi>μ</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560576&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560573&type= 30.05666542 9.48266697 且 <math id="M44"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mi>M</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><mi>k</mi></mtd><mtd><mi>M</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560583&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560579&type= 40.21666718 19.13466644 ， <math id="M45"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub></mtd><mtd><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub></mtd><mtd><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>μ</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560592&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560587&type= 73.40600586 16.17133331 可以求得再生矩阵为 <math display="block" id="M46"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mi>M</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><mfenced separators="|"><mrow><mi>k</mi><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mfenced></mtd><mtd><mi>M</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi></mrow></mfrac></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560600&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560596&type= 135.29733276 20.48933411 因此，基本再生数 <math id="M47"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mi>ρ</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560608&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560604&type= 24.21466637 4.91066647 ，即为 <math id="M48"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560616&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560612&type= 10.32933331 4.06400013 的谱半径 <math display="block" id="M49"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>r</mi><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi>N</mi></mrow></mfrac><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><mfrac><mrow><mi>k</mi><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560627&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560620&type= 62.06066895 8.72066689 其中， <math id="M50"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560635&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560631&type= 10.24466705 8.46666718 为潜伏期的平均时间， <math id="M51"><mfrac><mrow><mi>A</mi><mi>r</mi><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mi>N</mi><mi>μ</mi></mrow></mfrac><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560642&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560638&type= 19.04999924 8.55133343 表示一个潜伏期感染者在潜伏期内所能感染的平均人数； <math id="M52"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560650&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560645&type= 26.33133316 8.46666718 为传染期的平均时间， <math id="M53"><mfrac><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560658&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560654&type= 10.24466705 8.55133343 为传染期患者的比例， <math id="M54"><mfrac><mrow><mi>A</mi><mi>r</mi><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi>N</mi><mi>μ</mi></mrow></mfrac><mfrac><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi></mrow></mfrac><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560666&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560662&type= 46.14333344 8.55133343 表示传染期患者在其传染期内所能感染的平均人数。 方程组（3）的第二组解为地方病平衡点 <math id="M55"><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>=</mo><mo stretchy="false">(</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><mo>,</mo><msup><mrow><mi>Q</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>,</mo><msup><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560674&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560670&type= 32.85066605 3.81000018 ，其中 <math display="block" id="M56"><msup><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>=</mo><mfrac><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub></mrow></mfrac><msup><mrow><mi>I</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>=</mo><mfrac><mrow><mi>A</mi><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></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560682&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560678&type= 62.48400116 8.72066689 ， <math display="block" id="M57"><msup><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>=</mo><mfrac><mrow><mi>N</mi></mrow><mrow><mi>r</mi><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mfrac><mrow><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>k</mi><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560690&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560686&type= 69.59600067 8.72066689 ， <math display="block" id="M58"><msup><mrow><mi>I</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>=</mo><mfrac><mrow><mi>A</mi><mo>-</mo><mfrac><mrow><mi>μ</mi><mi>N</mi></mrow><mrow><mi>r</mi><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mfrac><mrow><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>k</mi><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mrow><mfrac><mrow><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub></mrow></mfrac></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>A</mi><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><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560696&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560693&type= 118.02532959 18.37266731 ， <math display="block" id="M59"><msup><mrow><mi>Q</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>=</mo><mfrac><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub></mrow><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>μ</mi></mrow></mfrac><msup><mrow><mi>I</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>=</mo><mfrac><mrow><mi>A</mi><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><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560702&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560699&type= 105.57933807 8.72066689 故定理1得证。 1.3　无病平衡点的稳定性 2 定理2 对于模型（2），当 <math id="M60"><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=4560709&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560705&type= 10.83733368 3.72533321 时，无病平衡点 <math id="M61"><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560715&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560713&type= 3.89466667 3.72533321 局部渐进稳定；当 <math id="M62"><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=4560723&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560720&type= 10.75266743 3.72533321 时，无病平衡点 <math id="M63"><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560731&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560726&type= 3.89466667 3.72533321 不稳定。 模型（2）在无病平衡点 <math id="M64"><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=4560738&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560734&type= 3.81000018 3.72533321 处的Jacobian矩阵为 <math display="block" id="M65"><mi mathvariant="bold-italic">J</mi><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><mi>μ</mi></mtd><mtd><mo>-</mo><mi>M</mi><mi>k</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac></mtd><mtd><mo>-</mo><mi>M</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mi>M</mi><mi>k</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>-</mo><mi>μ</mi></mtd><mtd><mi>M</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub></mtd><mtd><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub></mtd><mtd><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560742&type= 107.86532593 31.58066750 对应的特征多项式为 <math display="block" id="M66"><mfenced open="|" close="|" separators="|"><mrow><mi>λ</mi><mi mathvariant="bold-italic">E</mi><mo>-</mo><mi mathvariant="bold-italic">J</mi></mrow></mfenced><mo>=</mo><mfenced open="|" close="|" separators="|"><mrow><mtable><mtr><mtd><mi>λ</mi><mo>+</mo><mi>μ</mi></mtd><mtd><mi>M</mi><mi>k</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac></mtd><mtd><mi>M</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mi>λ</mi><mo>-</mo><mi>M</mi><mi>k</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><mo>+</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi></mtd><mtd><mo>-</mo><mi>M</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub></mtd><mtd><mi>λ</mi><mo>+</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub></mtd><mtd><mi>λ</mi><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>μ</mi></mtd></mtr></mtable></mrow></mfenced><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560752&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560749&type= 132.24932861 31.41133308 <math display="block" id="M67"><mo stretchy="false">(</mo><mi>λ</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>λ</mi><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mfenced open="|" close="|" separators="|"><mrow><mtable><mtr><mtd><mi>λ</mi><mo>-</mo><mi>M</mi><mi>k</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><mo>+</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi></mtd><mtd><mo>-</mo><mi>M</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub></mtd><mtd><mi>λ</mi><mo>+</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi></mtd></mtr></mtable></mrow></mfenced><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560758&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560755&type= 113.87666321 14.98599911 <math display="block" id="M68"><mo stretchy="false">(</mo><mi>λ</mi><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>λ</mi><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>μ</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><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>λ</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560765&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560762&type= 63.16133499 4.91066647 其中， <math id="M69"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi><mo>-</mo><mi>M</mi><mi>k</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><mo>+</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560771&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560768&type= 62.39933395 8.38199997 ， <math id="M70"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi><mo>-</mo><mi>M</mi><mi>k</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo>-</mo><mi>M</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560785&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560782&type= 79.75600433 8.38199997 。 如果 <math id="M71"><mi mathvariant="bold-italic">J</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560791&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560788&type= 1.77800000 2.79399991 的所有特征值都具有负实部，那么模型（2）的无病平衡点 <math id="M72"><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560794&type= 3.89466667 3.72533321 局部渐进稳定。根据 <math id="M73"><mfenced open="|" close="|" separators="|"><mrow><mi>λ</mi><mi mathvariant="bold-italic">E</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=4560804&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560801&type= 20.48933411 3.97933340 显然存在负根 <math id="M74"><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mo>-</mo><mi>μ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560810&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560807&type= 13.54666710 4.57200003 ， <math id="M75"><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560812&type= 32.08866501 4.65666676 ，并且由Routh-Hurwitz判据，欲使 <math id="M76"><msup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>λ</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560823&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560818&type= 20.40466690 4.06400013 的根均具有负实部，只需 <math id="M77"><msub><mrow><mi>a</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=4560829&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560826&type= 9.99066734 3.72533321 ， <math id="M78"><mfenced open="|" close="|" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560836&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560832&type= 19.30399895 8.55133343 ，整理可得： <math id="M79"><mi>M</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><mfrac><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>k</mi><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><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=4560841&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560839&type= 67.22533417 8.72066689 。 故定理2得证。 定理3 对于模型（2），当 <math id="M80"><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=4560847&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560845&type= 10.83733368 3.72533321 时，无病平衡点全局渐进稳定。 由模型（2）中的方程 <math id="M81"><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>A</mi><mo>-</mo><mi>M</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>k</mi><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mi>μ</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>≤</mo><mi>A</mi><mo>-</mo><mi>μ</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560854&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560850&type= 66.80200195 7.62000036 ，可得： <math id="M82"><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>≤</mo><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><mo>+</mo><mo stretchy="false">(</mo><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><mo stretchy="false">)</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><mo>-</mo><mi>μ</mi><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560861&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560857&type= 48.93733215 8.38199997 ， <math id="M83"><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>≤</mo><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560867&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560864&type= 12.69999981 8.38199997 ，则 <math id="M84"><msub><mrow><mi>S</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>≤</mo><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560878&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560876&type= 12.27666664 8.38199997 。 构造Lyapunov函数 <math id="M85"><msub><mrow><mi>V</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>M</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><msub><mrow><mi>I</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560883&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560881&type= 56.89599991 8.38199997 ，有： <math display="block" id="M86"><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>V</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mfenced open="[" close="]" separators="|"><mrow><mi>M</mi><mi>S</mi><mo stretchy="false">(</mo><mi>I</mi><mo>+</mo><mi>k</mi><mi>E</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mi>E</mi></mrow></mfenced><mo>+</mo><mi>M</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mi>E</mi><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mi>I</mi></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mfenced open="[" close="]" separators="|"><mrow><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mi>M</mi><mi>S</mi><mi>k</mi><mo>+</mo><mi>M</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mfenced><mi>E</mi><mo>+</mo></mtd></mtr><mtr><mtd><mfenced open="[" close="]" separators="|"><mrow><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mi>M</mi><mi>S</mi><mo>-</mo><mi>M</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mfenced><mi>I</mi><mo>≤</mo></mtd></mtr><mtr><mtd><mfenced open="[" close="]" separators="|"><mrow><mi>M</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><mi>k</mi><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo>+</mo><mi>M</mi><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mfenced><mi>E</mi><mo>=</mo></mtd></mtr><mtr><mtd><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><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>+</mo><mi>μ</mi><mo stretchy="false">)</mo><mi>E</mi></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560888&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560885&type= 145.79598999 48.51399994 显然上式有 <math id="M87"><msub><mrow><mi>V</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560892&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560890&type= 10.66799927 4.14866638 ，且当 <math id="M88"><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=4560896&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560894&type= 10.83733368 3.72533321 时， <math id="M89"><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>V</mi></mrow><mrow><mi>t</mi></mrow></msub></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=4560901&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560899&type= 13.80066681 7.62000036 ，当且仅当 <math id="M90"><mi>E</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560908&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560905&type= 9.39799976 2.79399991 ，即在无病平衡点 <math id="M91"><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">0,0</mn><mo>,</mo><mn mathvariant="normal">0</mn><mo>,</mo><mfrac><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560913&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560910&type= 28.19400024 8.38199997 处等号成立，满足Lyapunov第二方法，故无病平衡点全局渐进稳定。 故定理3得证。 通过定理2和定理3可以知道，利用基本再生数就可以判断传染病的发展趋势，即当 <math id="M92"><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=4560919&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560917&type= 10.83733368 3.72533321 时，无病平衡点是全局渐进稳定的，因此传染病将会逐渐消除。 2　 模拟实验 1 本节从传染病的基本模型出发，根据文献［ 13 13 ~ 15 15 13~15 ］对实际情况进行分析，结合模型的模拟结果，向传染病的基本模型中增加新的考虑对象，逐步过渡到本文提出的模型。对于此次疫情存在的潜伏期感染，我们通过潜伏期传染性参数来考察潜伏期传染性对疫情发展的影响。隔离防控手段可减少人群间接触，相当于控制模型中易感者的接触率，因此通过更改模型中的接触率参数来模拟不同程度的防控手段下的疫情，评估应对此次疫情的防治手段的有效性。 根据2020年2月5日国家卫生健康委员会发布的《新型冠状病毒感染的肺炎诊疗方案（试行第五版）》及相关文献 ［ 16 16 ， 17 17 ］ ，取 <math id="M93"><msub><mrow><mi>t</mi></mrow><mrow><mi>e</mi><mi>i</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0.125</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560924&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560921&type= 16.17133331 3.72533321 ， <math id="M94"><msub><mrow><mi>t</mi></mrow><mrow><mi>i</mi><mi>q</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0.128</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560928&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560925&type= 16.34066582 3.72533321 ， <math id="M95"><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0.027</mn><mi mathvariant="normal">%</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560932&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560930&type= 28.53266716 4.91066647 ， <math id="M96"><mi>μ</mi><mo>=</mo><mn mathvariant="normal">0.712</mn><mi mathvariant="normal">%</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560936&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560934&type= 18.88066673 3.97933340 ， <math id="M97"><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0.007</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560941&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560939&type= 16.67933464 4.57200003 ， <math id="M98"><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0.014</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560944&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560942&type= 16.67933464 4.57200003 ， <math id="M99"><mi>r</mi><mo>=</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560948&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560946&type= 8.38199997 2.79399991 ， <math id="M100"><mi>A</mi><mo>=</mo><mn mathvariant="normal">12</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560953&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560951&type= 11.26066685 2.79399991 ， <math id="M101"><mi>M</mi><mo>=</mo><mn mathvariant="normal">5</mn><mtext> </mtext><mn mathvariant="normal">917</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560958&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560955&type= 25.99266624 3.04799986 。 2.1　基本模型 2 对于最简单的SI模型，疫情曲线如 图1 图1 （a）中实线所示，可以看到在易感者与传染者曲线的交点，即两者人数相等的时候，患者的增长率最大，意味着传染病扩散最快的时候到来，但是该模型完全没有考虑治愈病人的情况，任何人最后都会变成患者，患者却无法变回健康人，然后因为传染病的病死率，人口不断减少。 如果在SI模型的基础上考虑到患者可以被治愈，但是治愈后还是会被传染成为感染者的SIS模型，疫情曲线如 图1 图1 （a）中虚线所示。而且通过解其微分方程可知，经过无穷长的时间后，传染者和易感者两种人群的数量会趋于某一极限。特别的，当 <math id="M102"><mi>r</mi><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560971&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560968&type= 12.78466702 4.74133348 ，接触率与传染率乘积小于康复率时，相当于在传染期内新产生的传染者小于康复者时，传染者趋于消失。虽然本次疫情暂时没有确实证据显示治愈者具有免疫力，但是这种放任治愈者不断补充到易感人群，重新加入传染循环的模型，与我国现在严密的疫情防控手段下，将患者与治愈者集中隔离的情形有较大出入。 进一步使用SIR模型进行模拟，即考虑传染者康复后客观上可能具有了免疫力，虽然本次疫情并没有确切的证据显示康复者的免疫力，但根据这次的疫情控制措施，可以一定程度上将康复者看作被完好的隔离保护起来，这使得再次感染会很罕见，因此不妨将他们视作完全退出传染体系的一个单独类型，可以得到如 图1 图1 （b）所示的曲线。 2.2　考察潜伏期传染性对疫情发展的影响 2 考虑到此次疫情实际情况中存在的潜伏期，使用本文基于隔离措施的SEIR模型进行模拟，在这里分别对潜伏期具有传染性和潜伏期不具有传染性两种情形分别进行模拟，得到如 图2 图2 所示曲线，其中实线部分对应潜伏期具有传染性的结果，虚线部分对应潜伏期不具有传染性的结果。可以看到，在潜伏期具有传染性的情况下，潜伏期患者的数量将在47~51 d左右达到峰值，传染者则将在56~59 d左右达到峰值，而潜伏期不具有传染性的情况下，潜伏期患者与传染期患者的峰值分别在83~88 d和86~91 d左右到达。显然，潜伏期具有传染性的情况下，疫情会暴发的更快，虽然这从一方面意味着疫情的转折点也会更快的到来，但是潜伏者和感染者的峰值也更大了，也就是说有更多的人会受到病毒的感染。通过比较疫情实际变化曲线的走势，考虑潜伏期具有传染的模型更符合实际情况。 2.3　基于隔离措施的SEIR模型与实际情况的对比 2 本文模型是在SEIR模型的基础上，进一步考虑到实际疫情控制中采取对于患者的大力度排查，并对排查到的传染期患者严密隔离，使得原本SEIR模型中的传染期患者部分被隔离，无法再参与病毒的传播，而且隔离患者受到了集中治疗，提高了康复率。对本文提出的模型进行模拟，得到结果如 图3 图3 所示，黑点为2020年1月1日到3月3日湖北省的实际确诊人数，可以看到模型模拟传染者的走势与实际情况较为吻合。 2.4　不同接触率下的疫情模拟 2 根据湖北省2020年1月24日启动重大突发公共卫生事件Ⅰ级响应，启动了必要的防控隔离手段，如停止集市、集会、电影院等人群聚集活动，并对流动人口采取预防工作，防止病毒的人际传播。这些手段相当于控制本文模型中的病毒携带者接触到的易感者人数 <math id="M103"><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560988&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560986&type= 1.52399993 2.79399991 ，我们通过改变 <math id="M104"><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560993&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560989&type= 1.52399993 2.79399991 的数值来对疫情控制效果进行模拟，如 图4 图4 所示，图中实线分别为 <math id="M105"><mi>r</mi><mo>=</mo><mn mathvariant="normal">2,3</mn><mo>,</mo><mn mathvariant="normal">4</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560997&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4560995&type= 15.74800014 3.21733332 时传染者的人数曲线，虚线为 <math id="M106"><mi>r</mi><mo>=</mo><mn mathvariant="normal">2,3</mn><mo>,</mo><mn mathvariant="normal">4</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4561003&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4561001&type= 15.74800014 3.21733332 时潜伏者的人数曲线。 <math id="M107"><mi>r</mi><mo>=</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4561008&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4561005&type= 8.38199997 2.79399991 时，传染者人数的峰值大约为77 900； <math id="M108"><mi>r</mi><mo>=</mo><mn mathvariant="normal">3</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4561012&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4561010&type= 8.38199997 2.79399991 时，传染者人数峰值为129 700； <math id="M109"><mi>r</mi><mo>=</mo><mn mathvariant="normal">4</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4561018&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4561015&type= 8.38199997 2.79399991 时，传染者人数峰值为160 300。如果没有采取强力的防控措施，湖北省的传染者人数将会急剧增加，严重情况下人数甚至可能达到当前实际感染人数的2.05倍以上。可以看到模型预测与实际数据吻合得较好，在采取严密防控措施，降低接触率后，潜伏期患者和传染期患者，这两种病毒携带者的人数峰值明显减小，且疫情高峰期到的更早，对于防止疫情更大范围的扩散起到了很好的作用。 3　 结 语 1 本文在流行病学知识的基础上，综合考虑新型冠状病毒的传播特点，根据确诊患者以及疑似患者被隔离观察的实际情况，建立基于隔离措施的COVID-19疫情控制模型，分析隔离集中诊治的效果，并对疫情发展做出一定的预测。但是，在本模型假设中忽略掉一些因素如年龄、职业等，这些因素可能对疫情走向有着一定的影响。此外在设置模型参数时候，一些数据无法得到确切值，例如潜伏期感染者的传染率暂时没有明确的数据支持，所以部分参数进行相应的估计，来实现对疫情走势进行的仿真模拟。从仿真结果看来，在新型冠状病毒出现的初期，由于缺乏对病毒严重性和有害性的客观认识，延误了防治措施，病情呈现出S型趋势发展，患病人数已经达到了一个较大值。之后国家及时出台强力的管控举措，限制出行、居家隔离、提示及强制出行戴口罩等防护措施切断病毒传染途径，减少了病毒携带者与易感人群的接触，一定程度上减缓了疫情的传播。 参考文献 HELLEWELL J ， ABBOTT S ， GIMMA A ， et al . Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts ［J］. The Lancet Global Health ， 2020 ， 8 （ 4 ）： e488 - e496 . DOI： 10.1016/S2214-109X（20）30074-7 http://dx.doi.org/10.1016/S2214-109X（20）30074-7 . Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts ZAREIE B ， ROSHANI A ， MANSOURNIA M A ， et al . A model for COVID-19 prediction in Iran based on China parameters ［J］. MedRxiv ， 2020 ， 23 （ 4 ）： 244 - 248 . DOI： 10.1101/2020.03.19.20038950 http://dx.doi.org/10.1101/2020.03.19.20038950 . A model for COVID-19 prediction in Iran based on China parameters WU J T ， LEUNG K ， LEUNG G M . Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan， China： A modelling study ［J］. The Lancet ， 2020 ， 395 （ 10225 ）： 689 - 697 . DOI： 10.1016/S0140-6736（20）30260-9 http://dx.doi.org/10.1016/S0140-6736（20）30260-9 . Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan， China： A modelling study CHEN J L . Pathogenicity and transmissibility of 2019-nCoV： A quick overview and comparison with other emerging viruses ［J］. Microbes and Infection ， 2020 ， 22 （ 2 ）： 69 - 71 . DOI： 10.1016/j.micinf.2020.01.004 http://dx.doi.org/10.1016/j.micinf.2020.01.004 . Pathogenicity and transmissibility of 2019-nCoV： A quick overview and comparison with other emerging viruses WANG X ， TANG S Y ， CHEN Y ， et al . When will be the resumption of work in Wuhan and its surrounding areas during COVID-19 epidemic？ A data-driven network modeling analysis ［J］. Scientia Sinica Mathematica ， 2020 ， 50 （ 7 ）： 969 - 979 . DOI： 10.1360/SSM-2020-0037 http://dx.doi.org/10.1360/SSM-2020-0037 . When will be the resumption of work in Wuhan and its surrounding areas during COVID-19 epidemic？ A data-driven network modeling analysis LIU Z . Dynamics of positive solutions to sir and SEIR epidemic models with saturated incidence rates ［J］. Nonlinear Analysis： Real World Applications ， 2013 ， 14 （ 3 ）： 1286 - 1299 . DOI： 10.1016/j.nonrwa.2012.09.016 http://dx.doi.org/10.1016/j.nonrwa.2012.09.016 . Dynamics of positive solutions to sir and SEIR epidemic models with saturated incidence rates WU Q ， XIAO G . A colored mean-field model for analyzing the effects of awareness on epidemic spreading in multiplex networks ［J］. Chaos： An Interdisciplinary Journal of Nonlinear Science ， 2018 ， 28 （ 10 ）： 103116 . DOI： 10.1063/1.5046714 http://dx.doi.org/10.1063/1.5046714 . A colored mean-field model for analyzing the effects of awareness on epidemic spreading in multiplex networks CAO W J ， LIU X F ， HAN Z ， et al . Statistical analysis and autoregressive modeling of confirmed coronavirus disease 2019 epidemic cases ［J］. Acta Physica Sinica ， 2020 ， 69 （ 9 ）： 0902033 - 1 . DOI： 10.7498/aps.69.20200503 http://dx.doi.org/10.7498/aps.69.20200503 . Statistical analysis and autoregressive modeling of confirmed coronavirus disease 2019 epidemic cases MACHIDA M ， NAKAMURA I ， SAITO R ， et al . Adoption of personal protective measures by ordinary citizens during the COVID-19 outbreak in Japan ［J］. International Journal of Infectious Diseases ， 2020 ， 94 ： 139 - 144 . DOI： 10.1016/j.ijid.2020.04.014 http://dx.doi.org/10.1016/j.ijid.2020.04.014 . Adoption of personal protective measures by ordinary citizens during the COVID-19 outbreak in Japan LIU Y ， GAYLE A A ， WILDER⁃SMITH A ， et al . The reproductive number of COVID-19 is higher compared to sars coronavirus ［J］. Journal of Travel Medicine ， 2020 ， 27 （ 2 ）. DOI： 10.1093/jtm/taaa021 http://dx.doi.org/10.1093/jtm/taaa021 . The reproductive number of COVID-19 is higher compared to sars coronavirus DELAMATER P L ， STREET E J ， LESLIE T F ， et al . Complexity of the basic reproduction number （R0） ［J］. Emerg Infect Dis ， 2019 ， 25 （ 1 ）： 1 - 4 . DOI： 10.3201/eid2501.171901 http://dx.doi.org/10.3201/eid2501.171901 . Complexity of the basic reproduction number （R0） FANG Y ， NIE Y ， PENNY M . Transmission dynamics of the covid-19 outbreak and effectiveness of government interventions： A data-driven analysis ［J］. Journal of Medical Virology ， 2020 ， 92 （ 6 ）： 645 - 659 . DOI： 10.1002/jmv.25750 http://dx.doi.org/10.1002/jmv.25750 . Transmission dynamics of the covid-19 outbreak and effectiveness of government interventions： A data-driven analysis ROTHAN H A ， BYRAREDDY S N . The epidemiology and pathogenesis of coronavirus disease （COVID-19） outbreak ［J］. Journal of Autoimmunity ， 2020 ， 109 ： 102433 . DOI： 10.1016/j.jaut.2020.102433 http://dx.doi.org/10.1016/j.jaut.2020.102433 . The epidemiology and pathogenesis of coronavirus disease （COVID-19） outbreak LI R Y ， PEI S ， CHEN B ， et al . Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus （sars-cov-2） ［J］. Science ， 2020 ， 368 （ 6490 ）： 489 - 493 . DOI： 10.1126/science.abb3221 http://dx.doi.org/10.1126/science.abb3221 . Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus （sars-cov-2） LAI C C ， SHIH T P ， KO W C ， et al . Severe acute respiratory syndrome coronavirus 2 （sars-cov-2） and coronavirus disease-2019 （COVID-19）： The epidemic and the challenges ［J］. International Journal of Antimicrobial Agents ， 2020 ， 55 （ 3 ）： 105924 . DOI： 10.1016/j.ijantimicag.2020.105924 http://dx.doi.org/10.1016/j.ijantimicag.2020.105924 . Severe acute respiratory syndrome coronavirus 2 （sars-cov-2） and coronavirus disease-2019 （COVID-19）： The epidemic and the challenges SATHIAN B ， ASIM M ， MEKKODATHIL A ， et al . Impact of COVID-19 on community health： A systematic review of a population of 82 million ［J］. Journal of Advances in Internal Medicine ， 2020 ， 9 （ 1 ）： 4 - 11 . DOI： 10.3126/jaim.v9i1.29159 http://dx.doi.org/10.3126/jaim.v9i1.29159 . Impact of COVID-19 on community health： A systematic review of a population of 82 million ZU J ， LI M L ， LI Z F ， et al . Transmission patterns of COVID-19 in the mainland of China and the efficacy of different control strategies： A data and model-driven study ［J］. Infectious Diseases of Poverty ， 2020 ， 9 （ 1 ）： 1 - 14 . DOI： 10.1186/s40249-020-00709-z http://dx.doi.org/10.1186/s40249-020-00709-z . Transmission patterns of COVID-19 in the mainland of China and the efficacy of different control strategies： A data and model-driven study 10.14188/j.1671-8836.2020.0146 O29 A 1671-8836（2020）05-0442-09 ZHU Yimin,HUANG Bo,WANG Zhongzhen, et al .Analysis of the Isolation Measure on the Control Model of COVID⁃19 [J]. J Wuhan Univ （ Nat Sci Ed ）,2020,66(5):442-450. DOI: 10.14188/j.1671-8836.2020.0146 http://dx.doi.org/10.14188/j.1671-8836.2020.0146 (Ch). ZHU Yimin,HUANG Bo,WANG Zhongzhen,et al.Analysis of the Isolation Measure on the Control Model of COVID⁃19 [J].J Wuhan Univ（Nat Sci Ed）,2020,66(5):442-450. DOI:10.14188/j.1671-8836.2020.0146(Ch). 朱翌民,黄勃,王忠震, 等.隔离措施对COVID⁃19疫情控制的模型分析[J].武汉大学学报（理学版）,2020,66(5):442-450. DOI:10.14188/j.1671-8836.2020.0146 朱翌民,黄勃,王忠震, 等.隔离措施对COVID⁃19疫情控制的模型分析[J].武汉大学学报（理学版）,2020,66(5):442-450. DOI:10.14188/j.1671-8836.2020.0146 国家自然科学基金青年基金项目（61603242， 61802251）；江西省经济犯罪侦查与防控技术协同创新中心开放基金资助课题（JXJZXTCX-030） 2020-06-05 2020-10-24 2020-12-15 武汉大学学报（理学版） 5 66