基于复杂网络的SEIR周期传染病传播模型的动力学分析

Dynamic Analysis of SEIR Periodic Infectious Disease Transmission Model Based on Complex Networks

摘要

基于复杂网络的平均场理论,建立具有周期性传染率的SEIR(susceptible,exposed,infectious,removed)模型.首先,确定系统的正不变集,利用线性积分算子的谱半径方法给出基本再生数R₀的表达式.其次,利用比较原理证明当R₀<1时,系统的无病平衡点是全局渐近稳定的;当R₀>1时,系统至少存在一个正周期解,且系统是一致持续的.最后,用数值模拟验证理论分析的正确性,结果表明,网络中节点的最大度越大,感染者的绝对密度越大,说明网络结构对传染病传播有显著影响.此外,通过数值模拟得到当R₀>1时,系统存在唯一一个全局渐近稳定的正周期解,充实了理论分析结果.

Abstract

Based on the mean field theory of complex networks, we established a SEIR (susceptible, exposed, infectious, removed) model with periodic contagion rates. Firstly, we determined the positive invariant set of the system and used the spectral radius method of the linear integral operator to give the expression of the basic reproduction number R₀. Secondly, by using the principle of comparison, we prove that the disease-free equilibrium point of the system is globally asymptotically stable when R₀<1, and there exists at least one positive periodic solution in the system and the system is uniformly persistent when R₀>1. Finally, the correctness of the theoretical analysis is verified by using numerical simulations, and the results show that the greater the maximum degree of nodes in the network, the greater the absolute density of infected people, indicating that the network structure has a significant impact on the spread of infectious diseases. In addition, through numerical simulations, we obtain that when R₀>1, the system has a unique globally asymptotically stable positive periodic solution, which enriches the theoretical analysis results.

关键词

周期性 / 基本再生数 / 正周期解 / 稳定性

Key words

periodicity / basic reproduction number / positive periodic solution / stability

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基金资助

国家自然科学基金(12001501)

国家自然科学基金(12071445)

国家自然科学基金(11571324)

国家自然科学基金(12101574)

山西省自然科学基金(20210302124621)