基于复杂网络的传染病数学模型研究

A Study of Mathematical Models of Infectious Diseases Based on Complex Networks

摘要: 科学家们为了能够更好地预测以及防范传染病的爆发，建立了传染病模型以及复杂网络，本文在对经典的传染病模型进行简单的描述后，对不同模型的传播动力学进行数学研究。本文介绍了复杂网络的基本概念，以及复杂网络中的几种基本网络模型。从SI模型、SIS模型、SIR模型三个经典模型出发，对基本再生数以及传播特征进行动力学分析。通过研究不同拓扑结构的复杂网络模型，分析传染病的传播临界值。动力学计算结果显示，基本再生数R₀的数值对于传染病的传播能力以及危害大小有着重要影响。通过对模型的深入分析，进而可预测传染病的影响范围，并给出针对性的预防和防疫措施。

Abstract: To better predict and prevent infectious disease outbreaks, scientists have established mathematical models of infectious diseases and complex networks. This paper provides a brief description of classical epidemic models and conducts a mathematical study of the transmission dynamics of different models. It introduces the basic concepts of complex networks and several fundamental network models. Starting from three classical models—the SI model, SIS model, and SIR model, it analyzes the basic reproduction number and transmission characteristics through dynamic analysis. By investigating complex network models with different topological structures, the paper examines the transmission thresholds of infectious diseases. Dynamic calculations show that the value of the basic reproduction number R₀ significantly influences the transmission capacity and hazard level of infectious diseases. Through in-depth model analysis, the research enables predictions of the impact scope of infectious diseases and proposes targeted prevention and control measures.