分层模块网络上SIR模型的非普适幂律动力学
Non-universal power-law dynamics of SIR models on hierarchical modular networks
摘要：
在许多COVID-19统计数据中报告了幂律（PL）时间依赖性感染增长。在简单的SIR模型中，传染病的数量在爆发时以$I(t) \propto t^{d-1}$的形式在地方病阶段的$d$维欧几里德格上增长，或者在临界点遵循较慢的普适PL，直到有限的大小导致免疫和指数衰减的交叉。异质性可能会改变传播模式的动态，空间上不均匀的感染率可能会导致更慢的衰减，对大流行的长期恢复构成威胁。COVID-19统计数据还提供了几个国家的流行病规模分布和PL尾部。在这里，我调查SIR像模型分层模块化网络，嵌入在2D晶格中，除了远程链接。我表明，如果网络的拓扑维数是有限的，平均度依赖PL增长的流行出现。超临界时，出现与规则图相同的指数，但拓扑无序改变了临界行为。对于流行病规模分布也是如此。除了$d=2$晶格外，个体的流动性不影响标度行为的形式，但增加了流行病的量级。超级传播者热点的添加也不会改变群体免疫机制中的增长指数和指数衰减。
英文摘要：
Power-law (PL) time dependent infection growth has been reported in many COVID-19 statistics. In simple SIR models the number of infections grows at the outbreak as $I(t) \propto t^{d-1}$ on $d$-dimensional Euclidean lattices in the endemic phase or follow a slower universal PL at the critical point, until finite sizes cause immunity and a crossover to an exponential decay. Heterogeneity may alter the dynamics of spreading models, spatially inhomogeneous infection rates can cause slower decays, posing a threat of a long recovery from a pandemic. COVID-19 statistics have also provided epidemic size distributions with PL tails in several countries. Here I investigate SIR like models on hierarchical modular networks, embedded in 2d lattices with the addition of long-range links. I show that if the topological dimension of the network is finite, average degree dependent PL growth of prevalence emerges. Supercritically the same exponents as of regular graphs occurs, but the topological disorder alters the critical behavior. This is also true for the epidemic size distributions. Mobility of individuals does not affect the form of the scaling behavior, except for the $d=2$ lattice, but increases the magnitude of the epidemic. The addition of a super-spreader hot-spot also does not change the growth exponent and the exponential decay in the herd immunity regime.
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