Abstract

We propose a mathematical model of infection spreading among the adult population of certain region. The model is constructed on the basis of some delay differential equations that are supplemented with integral equations of convolution type and the initial data. The variables included in the integral equations and the delay variables take into account the number of individuals in different groups and the transition rate of individuals between the groups which reflects the stages of the disease. Some properties of the solutions of the model are under study including the existence, uniqueness, and nonnegativity of the solution components on the half-axis, as well as the presence and stability of the equilibrium states. We formulate and solve the problem of eliminating infection during finite time. The time for infection eradication is estimated on using the exponentially decreasing component-by-component estimates of the solution. Also we present the results of computational experiments on estimating the eradication time and evaluating the effectiveness of the process of diagnosis and identification of sick (infected) individuals through the procedure of regular medical examinations.

中文翻译：

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基于时滞微分方程的流行病数学模型分析

摘要

我们提出了在特定区域的成年人口中传播的感染的数学模型。该模型是在一些延迟微分方程的基础上构建的，其中补充了卷积型积分方程和初始数据。积分方程中包括的变量和延迟变量考虑了不同组中个体的数量以及个体在组之间的转变速率，这反映了疾病的阶段。正在研究模型的解的一些属性，包括半轴上解分量的存在，唯一性和非负性，以及平衡状态的存在和稳定性。我们制定并解决了在有限时间内消除感染的问题。使用解决方案的逐个组件估计以指数方式减少，可以消除感染的根除时间。此外，我们还提供了计算实验的结果，这些计算是通过定期体检程序来估计根除时间并评估诊断（识别）患病（感染）个体过程的有效性的结果。