Abstract

Inverse problems for systems of nonlinear ordinary differential equations are studied. In these problems, the unknown coefficients and initial data must be found given additional information about the solution to the corresponding direct problems; this information is obtained by measurements made at some specified points in time. Examples of inverse immunology and epidemiology problems arising in the analysis of infectious diseases progression, in the study of HIV dynamics, and spread of tuberculosis in highly endemic regions taking treatment into account are discussed. In the case when the solution to the inverse problem is not unique, three approaches to the study of identifiability of mathematical models are considered. A numerical solution algorithm based on the minimization of a quadratic objective functional is proposed. At the first stage, neighborhoods of the global minimizers are found, and gradient methods are used at the second stage. The gradient of the objective functional is calculated by solving the corresponding adjoint problem. Numerical results are discussed.

中文翻译：

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解决免疫和流行病学逆问题的优化方法

摘要

研究了非线性常微分方程系统的逆问题。在这些问题中，必须给定有关相应直接问题解决方案的其他信息，才能找到未知系数和初始数据。该信息是通过在某些指定时间点进行的测量获得的。讨论了在分析传染病进展，研究HIV动态以及在高流行地区考虑到治疗后结核病扩散方面出现的逆向免疫学和流行病学问题的例子。在反问题的解决方案不是唯一的情况下，考虑了三种研究数学模型可识别性的方法。提出了一种基于最小二次目标函数的数值求解算法。在第一阶段 找到全局最小化器的邻域，并在第二阶段使用梯度法。通过求解相应的伴随问题来计算目标函数的梯度。讨论了数值结果。