具有跟踪隔离措施的新冠肺炎传播模型分析及应用

报道于2019年12月底的新型冠状病毒肺炎(COVID-19)疫情, 由于2020年春运期间人口的大规模流动, 使得其迅速蔓延.自2020年1月23日起, 我国采取了各种措施使得疫情得到了有效的控制, 例如武汉封城、确诊病例的密切接触者跟踪隔离、湖北人员的居家隔离等.该文基于COVID-19在山西省的实际传播情况, 建立了具有输入病例和确诊病例密切接触者跟踪隔离的动力学模型.在不考虑输入病例的情况下, 分析了模型的动力学行为.利用山西省COVID-19病例数据, 计算了实时再生数, 发现山西省2020 年1月25日全省封村封街道有效控制了COVID-19疫情的传播, 即实时再生数小于1, 从宏观角度验证了防控措施的有效性.进一步通过模型的数值拟合得到: 早期染病者隔离14天的防控策略是合理有效的; 武汉封城时间越早, 染病者的规模越小; 跟踪隔离到大量确诊病例的接触者时, 染病者的规模越小.     

具有治疗控制的传染病模型分析

针对筛查和药物治疗对染病者传染性产生的影响,本文考虑了具有两种不同传染水平染病者的仓室数学模型.分析了模型平衡点的稳定性态,结果表明,当基本再生数小于1时,模型的无病平衡点全局稳定;当基本再生数大于1时,地方病平衡点在一定条件下也是全局稳定的.同时利用控制理论本文也研究了药物治疗的实施对染病者进行干预和影响的最优控制措施,寻找到了使目标函数值最小的治疗控制方法,并用数值模拟显示了模型解的动力学性态及治疗措施对防止疾病蔓延所起的作用.     

一类具有一般发生率的阶段结构传染病模型的稳定性

建立了一类具有一般发生率的阶段结构传染病模型,利用Hurwitz判据和极限系统理论知识等,分别讨论了染病者无输入和染病者有输入时,疾病消除平衡点和地方病平衡点的局部和全局稳定性,得到了一些重要结论,并对所得结果进行了数值模拟.     

一类具有常数输入和垂直传染的SIRI传染病模型

建立了一类易感者及染病者均有常数输入,疾病具有垂直传染以及一般形式饱和接触率的SIRI传染病模型,分别研究了p=0,0相似文献   

H7N9禽流感的数学模型

为了研究H7N9禽流感病毒的传播过程,考虑到有染病禽类输入和无染病禽类输入以及禽类的因病死亡率对模型的影响,得到禽类-人类动力学模型.针对两种不同的情况,得到了系统平衡点的存在性及基本再生数,并通过构造Lyapunov函数及利用Bendixson-Dulac定理给出了系统平衡点的稳定性条件.最后通过数值模拟验证了理论结果并给出了预防禽流感的有效措施.     

具有染病者输入的离散SIR传染病模型分析

引入相应的概率建立了具有染病者输入的离散SIR传染病模型,确定了决定其动力学性态的阂值.在阈值之下模型仅存在无病平衡点,且无病平衡点是全局渐近稳定的;在阈值之上模型是一致持续的,有唯一的地方病平衡点存在,且地方病平衡点是局部渐近稳定的.     

一类禽流感模型的全局稳定性

在饲养业中不可避免着人类行为的干涉,基于饲养业实际研究了一类染病者即刻被隔离的SEQS模型,其中常数输入是必须的且输入者中必含潜伏期者.利用三维空间的Bendixson-Dulac定理证明系统不存在周期解,从而唯一的正平衡点一旦存在即是全局稳定的.     

具有常数输入的SEIR模型的稳定性分析

讨论了易感者类和潜伏者类均为常数输入,潜伏期、染病期和恢复期均具有传染力,且传染率为一般传染率的SEIR传染病模型.利用Hurwitz判据证明了地方病平衡点的局部渐近稳定性,进一步利用复合矩阵理论得到了地方病平衡点全局渐近稳定的充分条件.     

具有一般接触率和治疗的SIS传染病模型的后向分支

建立了具有一般传染率函数和治疗的SIS模型并分析了其动力学性态.通过分析得到,当基本再生数小于1时,系统存在无病平衡点,并且无病平衡点是局部渐近稳定的,当染病者数量较少,发现系统在基本再生数大于1时,系统存在惟一的正平衡点且是局部渐近稳定的;当染病者数量超过医院的最大承受能力时,当基本再生数小于1时,系统可能存在两个正平衡点或无正平衡点.当存在两个正平衡点时,其中染病者数量较小的是鞍点,染病者数量较大的为结点或焦点,且是局部渐近稳定的.当治疗能力较弱时,模型会出现后向分支.     

离散的SI和SIS传染病模型的研究

为了描述个体的死亡、染病者的恢复以及疾病的传染,引入了相应的概率.基于总种群中个体数量为常数的假设,根据染病者能否恢复分别建立了具有生命动力学的离散SI和SIS传染病模型.所得到的结果显示:它们具有与相应连续模型相同的动力学性态,并确定了各自的阈值.在它们的阈值之下,传染病最终将灭绝;在它们的阈值之上,传染病将会发展成为地方病,染病者的数量将趋向于一确定的正常数.     

Filling in solvable groups and in lattices in semisimple groups

We prove that the filling order is quadratic for a large class of solvable groups and asymptotically quadratic for all -rank one lattices in semisimple groups of -rank at least 3. As a byproduct of auxiliary results we give a shorter proof of the theorem on the nondistorsion of horospheres providing also an estimate of a nondistorsion constant.     

An Outlook in Teaching and Research in Mathematics in Educational Institutions

The present paper outlines some basic principles upon which an outlook in teaching and research in mathematics in educational institutions can be developed. Defining applied mathematics, the author analyses the basic topics of mathematics which permeate throughout the wider fields and diversified aspects of this branch of mathematical science.

For the institutions of learning, a course curriculum has been proposed which will gradually motivate, orient and prepare individuals through the process of learning to understand the methods and techniques, so as to apply them to physical problems. This has been illustrated in the fields of fluid dynamics and aerodynamics through the analysis of complex function theory and differential equations.

Because of the frustrating situations prevailing amongst students and teachers in the institutions of learning in the present day, a critical analysis has been made to identify the respective roles of the academicians and administrators, and with a view to improving the situation, some remedy has been suggested regarding teaching and research in educational institutions.     

Introducing technology in algebra in Hong Kong: addressing issues in learning

A review of recent studies in Hong Kong suggests that possible problems in secondary school algebra may be due to the procedural paradigm orientation in the curriculum and the conventional style of teaching in the classroom which do not provide sufficient opportunities for students to develop conceptual understanding. Based on the works of a number of projects in the West, it is hypothesized that the introduction of technology in lessons which embody a cognitive model in their design and delivery will provide a viable alternative for enhancing algebraic thinking. The key features of the cognitive model are concrete preparation, cognitive conflict, construction, metacognition and bridging, imbedded in a ‘mediation’ style of teaching. These features are exemplified by detailed accounts of classroom episodes from a Hong Kong lesson supported by the use of graphics calculators. Results indicate that while the approach places particular demands on teachers (a tension between a transmission style of teaching and mediating) there is real potential for supporting more dynamic student constructions.     

Operational research in development banking in India

The paper defines development banking as practised by major Indian commercial banks since 1969. It then goes on to discuss how OR can be useful in tackling problems of development banking. Planning, operational and organisational design problems in relation to development banking are discussed. Several case studies in each area where OR has been applied are discussed. Potential areas of application of OR in each problem class are also identified. The paper then summarizes the experiences gained so far and recommends the action to be taken for the more effective application of OR in developing economies.