【作者】 王琪；

【导师】 窦霁虹；

【作者基本信息】 西北大学 ， 应用数学， 2021， 硕士

【摘要】 针对日益严峻的传染病形势,本文基于经典仓室模型,综合部分疾病具有垂直传播的特点以及可接种疫苗进行干预的实际情况,建立了两类微分方程动力学模型.通过理论研究和数值模拟,为控制疾病提供了有效的理论依据,并进一步丰富了现有考虑垂直传播和疫苗接种的传染病研究课题.第一部分,研究了一类具有双垂直传播且病毒发生变异的媒介传染病模型,并考虑对部分人群的新生儿接种疫苗,假设此类接种完全有效.根据第二代生成矩阵法计算出疾病的基本再生数;利用李雅普诺夫相关方法得到各类平衡点存在及稳定的结论;并通过数值模拟验证了理论结果的有效性,同时基于已有文献对加入接种和病毒变异因素后的理论结果异同点进行了比较;得到最终变异病毒将占据主导地位,以及通过提高预防接种比例、减少人群和媒介之间的有效接触和改善治疗手段增加治愈率的措施可以有效控制疾病流行的结论.进一步延伸了白伟等对与病毒变异有关的媒介传染病的研究.第二部分,研究了一类考虑垂直传播、接种及人均病床数的SIVS传染病模型,假设此类接种不完全有效.利用第二代生成矩阵法得到疾病基本再生数的表达式;分别采用几何法和特征值法探讨了两类平衡点的存在性及稳定性;通过讨论发现一定条件下模型会发生后向分支,并结合理论证明和数值模拟对其进行验证;同时基于已有文献对加入垂直传播和新生儿接种因素后的理论结果异同点进行了比较;认为极小的有效接触和丰富的医疗资源可以避免发生后向分支,使疾病走向消亡,同时还可以通过提高疫苗接种比例和接种效率控制疾病.进一步推广了白禅等对与疫苗接种和医疗策略有关的传染病的研究.更多还原

【Abstract】 In view of the increasingly severe situation of infectious diseases,two kinds of differential equation dynamic models are established based on the classical warehouse model and the characteristics of vertical transmission of some diseases and the actual situation of vaccination in some diseases.Through theoretical research and numerical simulation,theoretical results that can provide effective help to control the diseases are obtained and that further enrich the existing infectious disease research subject of vertical transmission and vaccination.In the first part,a vector-borne infectious disease model with dual vertical transmission and virus mutation is studied,and vaccination of neonates in some populations is considered.Meanwhile the model suppose such vaccination is fully effective.The expression of the the basic reproduction is obtained with the next generation matrix method.Some conclusions about the existence and stability of every kinds of equilibrium points are obtained by Liapunov methods and Matlab software is used to verify the validity of the theoretical results.At the same time,the similarities and differences of the theoretical results after adding the factors of vaccination and virus mutation are compared based on the existing literature.It is concluded that the mutated virus will dominate eventually and the epidemic disease could be effectively controlled by increasing vaccination rates,reducing effective contact between the vector with the borne and improving treatment to increase cure rates.This part further improves the study of Bai Wei et al.for a kind of vector-borne infectious disease with virus mutation.In the second part,a type of SIVS infectious disease models with vertical infection,vaccination and the number of beds per capita are studied,the model suppose such vaccination is not fully effective meanwhile.The expression of the the basic reproduction is obtained with the next generation matrix method.The existence and stability of every points are discussed by geometric method and eigenvalue method respectively.At the same time,it is found that it will appear backward bifurcation in such model and the condition of the backward bifurcation is verified by theoretical proof and numerical simulation through the discussion of the equilibrium points.In addition,the similarities and differences of the theoretical results after adding the factors of vertical infection and neonatal vaccination are compared based on the existing literature.It is shown that minimal effective contact and abundant medical resources could avoid the appearance of backward bifurcation,leading to disease extinction finally.Meanwhile the epidemic diseases can be also controlled by increasing vaccination rates and improving the effectiveness of vaccination.This part further improves the study of Bai Chan et al.for a kind of SIVS epidemic model with vaccination and limited medical resources.更多还原