【作者】 张彤；

【导师】 齐龙兴；

【作者基本信息】 安徽大学 ， 应用数学， 2024， 硕士

【摘要】 本文主要研究异质性人群中传染病的动力学模型.异质性是一种广泛存在于生物种群中的典型现象,在人群中如个体的身体状况、生活习惯、与他人交往的频率等因素的差异可能导致个体在感染过程中表现出不同的特征.本文主要考虑患者的就诊情况、患者症状有无、感染延迟和环境噪声等因素,挖掘出影响传染病传播流行的关键因素.本文的主要内容如下:第一章是本文的绪论部分,主要阐述了空气污染对呼吸道传染病的影响与研究背景,异质性的研究现状和重要性等等.其次,介绍了本文模型的创新点.最后概述了本文的主要研究内容.在第二章中,针对不同的空气污染水平,我们将患者分为就诊患者、无症状的未就诊患者和有症状的未就诊患者,分别构建了呼吸道传染病模型SI_hI_sI_aS和SI_pI_hI_sI_aSP.理论分析表明,当空气污染水平较高时,决定模型平衡点存在性和稳定性的阈值与空气污染物的日排放量和人类对污染物的吸入量密切相关,模型SI_pI_hI_sI_aSP在一定条件下会在无病平衡点处发生折分支.当空气污染水平较低时,可以得到模型的基本再生数和平衡点的全局稳定性.通过比较两个模型可以发现,空气污染会导致病毒性呼吸道疾病的传播发生复杂的动力学行为.最后,敏感性分析和数值模拟结果表明,无论空气污染程度的高低,病毒性呼吸道疾病的无症状感染者比例的变化都会对病毒性呼吸道疾病患者的峰值数量产生显著影响,当空气污染程度较高时,这种影响更为明显,而患者总数与空气污染日排放量、污染物吸入量和无症状感染者比例密切相关.在第三章中,建立了一个具有感染时间延迟的异质性人群传染病模型SI_hI_sI_aS-D.讨论了平衡点的存在性、疾病的持久性和灭绝性.利用稳定性切换几何判据法对模型的动力学行为进行分析,证明了模型在特定情况下会发生稳定性切换.然后,利用规范型理论和中心流形理论,分析了地方病平衡点处Hopf分支的方向和周期解的稳定性.最后,数值模拟表明,感染延迟可以引起系统复杂的动态行为,如混沌现象.利用相空间重构和庞加莱截面法证明了模型中混沌的存在.通过最大李雅普诺夫指数和初始值敏感性分析,发现时滞、患者比例和治疗率分别存在一个阈值,使得模型从周期振荡切换到稳定状态.与此同时,过高的住院率会导致周期振荡,使疾病的传播不稳定.第四章,考虑环境噪声的影响,在第二章SI_hI_sI_aS模型的基础上建立了一个具有随机项的异质性人群随机传染病模型SI_hI_sI_aS-N.通过构造随机李雅普诺夫函数,我们证明了该随机模型非负解的全局存在唯一性.之后,通过估计其在对应确定性模型中平衡点附近的振荡研究了随机模型解的长期动力学行为.最后,通过数值模拟验证了理论结果.结果表明,随着噪声强度的降低,特别是与易感人群相关的噪声强度降低,疾病的持续性也会降低.同时,较高的就诊率和未就诊无症状患者的比例都不利于疾病的控制,应将其控制在适当的水平以更好地限制疾病的传播.更多还原

【Abstract】 This paper focuses on the dynamical models of infectious diseases in heterogeneous popu-lations.Heterogeneity is a typical phenomenon that exists in a wide range of biological popula-tions,in which differences in factors such as physical condition,living habits,and frequency of interaction with other people may lead to different characteristics of individuals in the process of infection.In this paper,the key factors affecting the spread of infectious diseases are iden-tified by considering factors such as the awareness of consultation and presence of symptoms,the delay of infection and environmental noise.The main contents of this paper are as follows:Chapter 1 is the introductory part of this paper,which mainly describes the impact of air pollution on respiratory infectious diseases and the research background,the current status and importance of heterogeneity research,etc.Secondly,it introduces the innovative points of the models of this paper.Finally,the main research content of this paper is summarized.In Chapter 2,for different air pollution levels,including patients who visited the clinic,asymptomatic patients who did not visit the clinic,and symptomatic patients who did not visit the clinic,respectively,the respiratory disease models SI_hI_sI_aS and SI_pI_hI_sI_aSP are construc-ted in this study.Theoretical analysis demonstrates that when the air pollution level is high,the thresholds that determine the existence and stability of the equilibria of the system are closely related to the daily emissions of air pollutants and the inhalation of pollutants by humans,and the system SI_pI_hI_sI_aSP will undergo fold bifurcation at disease-free equilibrium under certain condition.When the air pollution level is low,the basic reproduction number of the system and the global stability of the equilibria are obtained.Air pollution causes complex dynamical behavior in the spread of viral respiratory diseases,as it can be shown by comparing the two models.Finally,the sensitivity analysis and numerical simulation results show that regardless of the level of air pollution,the change in the proportion of symptomatic infected patients can significantly impact the peak number of patients with viral respiratory diseases.This effect is more pronounced when the level of air pollution is high,and the total number of patients is strongly correlated with daily air pollution emissions,pollutant inhalation,and the proportion of asymptomatic infected patients.In Chapter 3,the model SI_hI_sI_aS-D of heterogeneous populations of infectious dise-ases with infection delay is constructed.The existence of equilibria,disease persistence,and extinction are discussed.It is demonstrate that the system experiences stability switching under specific circumstances by analyzing the dynamical behavior of the system using the geometric stability switch criterion.Then,using the theory of normal form and center manifold theory,the direction of Hopf bifurcation at the endemic equilibrium and the stability of periodic solution are analyzed.Finally,numerical simulations have shown that the infection delay can induce complex dynamical behaviors of the system,such as chaotic phenomena.Phase space recons-truction and Poincar’e section techniques are used to demonstrate the existence of chaos in the system.It is found that the delay,the proportion of patients,and treatment rate of patients visi-ting the clinic each have a threshold that allows the system to switch from periodic oscillation to a stable state by using Largest Lyapunov Exponent and initial value sensitivity analysis.At the same time,a high hospitalization rate can actually lead to periodic oscillations,making the spread of the disease unstable.In Chapter 4,a heterogeneous population stochastic infectious disease model SI_hI_sI_aS-N with random terms is developed based on the SI_hI_sI_aS model in Chapter 2,taking into account the effect of environmental noise.The population is categorized into susceptible individuals,patients visiting the clinic,symptomatic patients not visiting the clinic,and asymptomatic pati-ents not visiting the clinic.By constructing a stochastic Lyapunov function,global existence and uniqueness of the nonnegative solution of this stochastic model are proved.Then the long-term dynamical behavior of the solutions to the stochastic model is investigated by estimating their oscillations around equilibria in the corresponding deterministic model.Finally,the theoretical results are verified by numerical simulations.The results show that the persistence of the dise-ase decreases with a decrease in noise intensity,especially the noise intensity associated with susceptible individuals.Nevertheless,the high rate of consultation and the higher proportion of asymptomatic individuals who do not visit the clinic are not conducive to disease control and they should be controlled at appropriate levels to better limit the spread of the disease.更多还原