Chen Tingting, Liu Guirong, Jin Zhen
School of Mathematics and Statistics, Shanxi University, Taiyuan, Shanxi, 030006, China.
Key Laboratory of Complex Systems and Data Science of Ministry of Education, Shanxi University, Taiyuan, Shanxi, 030006, China.
Infect Dis Model. 2025 Apr 5;10(3):875-896. doi: 10.1016/j.idm.2025.03.008. eCollection 2025 Sep.
Due to the heterogeneity of contact structure, it is more reasonable to model on networks for epidemics. Because of the stochastic nature of events and the discrete number of individuals, the spread of epidemics is more appropriately viewed as a Markov chain. Therefore, we establish stochastic SIRS models with vaccination on networks to study the mean and variance of the number of susceptible and infected individuals for large-scale populations. Using van Kampen's system-size expansion, we derive a high-dimensional deterministic system which describes the mean behaviour and a Fokker-Planck equation which characterizes the variance around deterministic trajectories. Utilizing the qualitative analysis technique and Lyapunov function, we demonstrate that the disease-free equilibrium of the deterministic system is globally asymptotically stable if the basic reproduction number < 1; and the endemic equilibrium is globally asymptotically stable if > 1. Through the analysis of the Fokker-Planck equation, we obtain the asymptotic expression for the variance of the number of susceptible and infected individuals around the endemic equilibrium, which can be approximated by the elements of principal diagonal of the solution of the corresponding Lyapunov equation. Here, the solution of Lyapunov equation is expressed by vectorization operator of matrices and Kronecker product. Finally, numerical simulations illustrate that vaccination can reduce infections and increase fluctuations of the number of infected individuals and show that individuals with greater degree are more easily infected.
由于接触结构的异质性，在网络上对流行病进行建模更为合理。由于事件的随机性和个体数量的离散性，流行病的传播更适合被视为一个马尔可夫链。因此，我们建立了网络上带疫苗接种的随机SIRS模型，以研究大规模人群中易感个体和感染个体数量的均值和方差。利用范坎彭的系统规模展开法，我们推导出一个描述均值行为的高维确定性系统和一个刻画确定性轨迹周围方差的福克 - 普朗克方程。利用定性分析技术和李雅普诺夫函数，我们证明如果基本再生数 < 1，确定性系统的无病平衡点是全局渐近稳定的；如果 > 1，地方病平衡点是全局渐近稳定的。通过对福克 - 普朗克方程的分析，我们得到了地方病平衡点周围易感个体和感染个体数量方差的渐近表达式，它可以由相应李雅普诺夫方程解的主对角线元素近似。这里，李雅普诺夫方程的解由矩阵的向量化算子和克罗内克积表示。最后，数值模拟表明疫苗接种可以减少感染并增加感染个体数量的波动，并且表明度数较大的个体更容易被感染。
