非线性传播模型解析近似解的并行符号算法研究

Information propagation is an important issue of national economy and people's livelihood, and the quantitative study of propagation is also important basis for prevention and control work. Recently, a majority of the scholars mainly focus on building model of information propagation and theoretical analysis, but few study about analytic method focus on quantitative analysis and constructing analytic approximate method for the model of information propagation. Therefore, we will concentrate on information propagation. Firstly, we will find the auxiliary linear operator and the form of the solutions in the homotopy analysis method by background and theoretical analysis of information propagation. Secondly, we will determine the auxiliary linear operator by error analysis and mathematical technology, and concentrate on the relationship between mathematical theory and auxiliary linear operator. Thirdly, based on the mathematical experiments of various propagation models, a mechanized algorithm for constructing an analytical approximate solution of a kind of propagation models is proposed. With parallel programming, a software package is developed on Python platform. Finally, considering the characteristics of non-smooth factors, we will develop a mechanized algorithm of homotopy analysis method to construct analytical approximation for non-smooth propagation model, and its and analyze its propagation behavior. This project highlights the intersection and penetration of computer science, mathematics, infectious diseases and communication science. It is expected to provide theoretical support for the further development of analytical methods and effective means for symbol computation and constructing analytical approximation of dynamic systems in sciences and engineering applications.

信息传播是国计民生的重大问题，对流行规律的定量研究是防治工作的重要依据。目前，大多数学者们关注于信息传播的模型建立与稳定性分析，而较少用解析方法来定量分析与构造其解析近似解。为此，本项目拟以信息传播模型为研究对象，首先通过了解信息传播的背景和理论分析，来寻找同伦分析方法中的辅助线性算子和解的形式；其次，通过误差以及数学技巧来确定合适的线性算子以及分析稳定性与线性算子之间的联系；再次，根据大量数学实验和并行计算，提出构造一类传播模型解析近似解的并行符号算法，在Python平台上研制软件包；最后，结合非光滑因素的特点，发展和完善同伦分析方法的机械化算法来自动推导非光滑传播模型的解析近似解，并分析其传播行为。本项目突出了计算机科学、数学、传染病学以及传播学等学科间相互交叉和渗透，理论上可望为解析方法的进一步发展提供支持，应用上可望为其他科学和工程中动力系统的符号演算和自动推导提供了有效手段。

传播规律的定量研究是防治工作的重要依据，也是非线性科学重要的研究内容。本项目以传播模型为研究对象，首先了解传播模型的背景和进行理论分析，如解的存在性、唯一性、稳定性、极限环、同/异宿轨、分岔以及解和参数的依赖关系. 其次，基于理论研究和误差分析寻找到合适的辅助线性算子，通过同伦分析方法构造了解析近似解，进而去解释动力学行为和分析稳定性与线性算子之间的关系；另一方面，在极限环理论的基础上，可通过摄动增量法来构造非线性系统稳定和不稳定的极限环；再次，根据大量的数学实验和并行计算技术，提出了构造一类传播模型解析近似解的并行符号算法，在Maple平台或Python平台上研制同伦分析方法和增量摄动方法软件包；最后，结合非光滑因素的特点，构造了一些非光滑系统的解析近似解，进一步完善同伦分析方法的机械化算法和增量摄动方法的机械化算法，并分析其传播行为. 本项目揭示非线性系统的传播规律, 阐明系统运动特性与参数之间的原理, 建立新的非线性动力系统的定量分析方法.