脉冲效应下传染病模型的研究

【作者】 张菊平；

【导师】 靳祯；

【作者基本信息】 中北大学 ， 应用数学， 2007， 硕士

【摘要】 近些年来,脉冲微分方程引起了许多学者的关注并得到了深入的发展.它被广泛应用于生物技术、药物动力学、物理、经济、种群动力学、流行病学等领域.流行病学中有许多自然现象和人为干预因素的作用用脉冲来描述更为精确.本文考虑了脉冲作用下的传染病学模型,给出了在脉冲作用下系统无病周期解的稳定性、一致持续性、正周期解的存在性.描述了具有垂直传染的离散SI和SIS流行病模型平衡点的稳定性.第二章中先利用频闪映射及Floquet定理证明了具有脉冲出生、脉冲接种、脉冲常数移民且传染率为标准的SIR传染病模型的无病周期解的存在性及局部渐近稳定性,并多次利用比较原理和脉冲微分不等式证明了无病周期解的全局渐近稳定性,同时,采用了适当的变量代换,证明了疾病一致持续生存.最后,利用数值模拟从几何上验证了我们所作证明的正确性.第三章中考虑了具有脉冲常量接种的SIR传染病模型,利用频闪映射、Floquet定理及比较原理证明了无病周期解的存在性、局部及全局渐近稳定性,并利用标准分支理论原理说明了正周期解的存在性.第四章中利用离散系统原理证明了具有垂直传染的离散SI和SIS流行病模型的无病平衡点和正平衡点的稳定性,同时,在SIS系统中传染率取成Poission分布时出现了双稳定性,即平衡点的稳定性在某种条件下出现交替现象.更多还原

【Abstract】 The impulsive di?erential equations have obtained much attention from many authors,and deeply developed during the past a few years. It is widely applied in various domains suchas biological technology, medicine dynamics, physics, economy, population dynamics andepidemiology and so on. It is well-known that many natural phenomena and human activitiesdo exhibit impulsive e?ects in the fields of epidemiology. In this paper, we investigate someepidemic models with impulsive e?ects. In the chapter 1, 2, We give the stability of periodicinfection-free solution, the uniform persistence of the infectious disease, and the existence ofepidemic periodic solution. In the chapter 3, we study the stability of equilibrium of discretetime SI and SIS with vertical transmission.In chapter 2, we study SIR epidemic model with birth pulse, impulsive vaccination,impulsive recruitment and standard incidence. Using the stroboscopic map and Floquettheory , we obtain the existence of periodic infection-free solution and local asymptoticstability. Above this threshold, we prove global asymptotic stability of periodic infection-freesolution by comparison theorem and impulsive di?erence inequation .At the same time,theuniform persistence of the infectious disease is proved. At last, numerical simulation showsthat our proof is correct.In chapter 3, we consider SIR epidemic model with impulsive constant vaccination.We obtain the existence of periodic infection-free solution and local and global asymptoticstability. Further, the existence of a nontrivial periodic solution (endemic) is considered byusing bifurcation theory.In chapter 4, by using the theories of discrete dynamics, we study and deal with thestability of free and endemic equilibrium in discrete time SI and SIS with vertical transmis-sion. At the same time, when the force of infection arises from the Poisson distribution inthe discrete time SIS epidemic model, the system can exhibit bistability(alternative stableequilibria)over a wide range of parameter values.更多还原