导　　师： 刘兴波

学科专业： 070104

授予学位： 硕士

作　　者： ;

机构地区： 华东师范大学

摘　　要： 本文研究了两类传染病模型： 首先,研究了一类具有一般非线性发生率的SEIR传染病模型.该模型始终存在无病平衡点,当且仅当基本再生数R0>1时系统存在地方病平衡点,且是唯一的.当R0≤1时,无病平衡点是全局渐近稳定的；当R0>1时,无病平衡点是不稳定的.当Ro>1且发生率函数满足2.2条件H时,唯一的地方病平衡点存在而且在可行域内是全局渐近稳定的.最后给出了数值模拟来证明结论的正确性,另外还通过改变接触率来说明对感染者进行一定的隔离对控制传染病有重要的作用. 其次,研究了具有饱和治疗率的SEIR传染病模型,饱和治疗率是用来描述由于感染者人数的增加和医疗条件的限制而造成的延迟治疗的影响.此模型存在两个阈值,并得到后向分支产生的条件.另外,利用几何逼近的方法得到了平衡点的全局动力学性质.最后对得到的结论进行了数值模拟,并建议给病人及时的治疗,提高治疗的有效性及降低接触率系数都是有效控制疾病的方法. In this paper, two kinds of epidemic models are studied. Firstly, we study an SEIR epidemic models with general nonlinear incidence rate. The model always exhibits the disease-free equilibrium, and the unique endemic equilibrium turns up if and only if the basic reproduction number R0≤1.If R0≤1, the disease-free equilibrium is globally asymptotically stable. And if R0>1, the disease-free equilibrium is unstable. Moreover, we show that if R0>1, a unique endemic equilibrium exists and is globally asymptotically stable in the interior of the feasible region under certain condition H which we will give in2.2. Numerical simulations are carried out to illustrate the feasibility of the obtained results and the effect of quarantine to control the disease. Secondly, we consider an SEIR epidemic model with saturated recovery rate. The recovery rate can describe the effect of delayed treatment when the number of infected individuals is getting larger and the medical condition is limited. It is shown that this model has two thresholds and a backward bifurcation will take place for some conditions. Furthermore, global dynamics are shown by geometric approaches. Numerical simulations are presented to illustrate the results and suggest that giving the patients timely treatment, improving the cure efficiency and decreasing the infective coefficient are all valid methods for the control of disease.

关 键 词： 传染病模型 稳定性 垂直传播 非线性发生率 饱和治疗率 后向分支 几何逼近

分 类 号： [O175]

领　　域： [理学] [理学]