多个群体传染病模型的动力学分析

本项目拟应用微分方程与动力系统中的定性理论和分岔理论，结合数值模拟方法，针对多个群体传染病模型，包括媒介传染病的自治模型、非自治模型，以及多斑块的传染病模型，进行动力学分析。这些模型的建立来源于传染病的传播机理，模型含有多个变量、多个独立参数及多个非线性项。本项目拟研究这些模型的平衡点和周期轨的存在性、个数、位置、稳定性及分支问题。对媒介传染病自治模型讨论定性性质需要避开对平衡点坐标精确表达式的依赖，分支问题需要通过中心流形等降维。对非自治模型需要结合单调动力系统理论、平均法和Floquet理论进行分析。对多斑块模型需要运用矩阵理论和动力系统谱问题得到基本再生数，分析动力学性质。在模型中应用这些理论仍具有挑战性。将模型的定性分析、分支分析和数值结果给出合理的医学解释，有助于对传染病防控措施进行评估。

This proposal aims to analyze dynamical behaviors of epidemical models with multiple population by qualitative and bifurcation theory associated with numerical simulations. It includes autonomous vector-borne models, non-autonomous vector-borne models and multi-patch epidemic models. The models arising from transmission mechanism of epidemical diseases contain multiple variables, multi-parameters and multiple nonlinear functions. We intends to obtain the existence, number, location and stability of equilibria and periodic orbits, as well as bifurcations of models. We will avoid the dependence on the exact expression of equilibria when discussing the qualitative properties of vector-borne autonomous models, and use center manifold to reduce the dimension before studying bifurcations. We will analyze the non-autonomous systems combining theory of monotone dynamical systems with averaging method and Floquet theory. We will obtain the basic reproduction number and study dynamics of multi-patch models by matrix theory and spectrum of dynamical system. It is still challenging to apply them on models. Giving medical explanations on the results of qualitative analysis, bifurcation analysis and numerical simulation, is helpful for the estimation on measures of disease control.