School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, Shanxi, China.
Complex Systems Research Center, Shanxi University, Taiyuan, 030006, Shanxi, China.
J Math Biol. 2022 May 5;84(6):50. doi: 10.1007/s00285-022-01754-y.
We derive a stochastic SIS pairwise model by considering the change of the variables of this system caused by an event. Based on approximations, we construct a low-dimensional deterministic system that can be used to describe the epidemic spread on a regular network. The mathematical treatment of the model yields explicit expressions for the variances of each variable at equilibrium. Then a comparison between the stochastic pairwise model and the stochastic mean-field SIS model is performed to indicate the effect of network structure. We find that the variances of the prevalence of infection for these two models are almost equal when the number of neighbors of every individual is large. Furthermore, approximations for the quasi-stationary distribution of the number of infected individuals and the expected time to extinction starting in quasi-stationary are derived. We analyze the approximations for the critical number of neighbors and the persistence threshold based on the stochastic model. The approximate performance is then examined by numerical and stochastic simulations. Moreover, during the early development phase, the temporal variance of the infection is also obtained. The simulations show that our analytical results are asymptotically accurate and reasonable.
我们通过考虑系统变量因事件而发生的变化，推导出了一个随机 SIS 对模型。基于逼近方法，我们构建了一个低维确定性系统，可以用于描述规则网络上的传染病传播。通过对模型的数学处理，我们得到了平衡状态下每个变量方差的显式表达式。然后，我们对随机对模型和随机平均场 SIS 模型进行了比较，以表明网络结构的影响。我们发现，当每个个体的邻居数量很大时，这两个模型的感染流行率方差几乎相等。此外，我们还推导出了感染个体数量的拟平稳分布和从拟平稳开始灭绝的预期时间的逼近。我们基于随机模型分析了临界邻居数量和持久性阈值的逼近。然后通过数值和随机模拟检验了近似性能。此外，在早期发展阶段，我们还获得了感染的时间方差。模拟结果表明，我们的分析结果是渐近准确和合理的。
