Non-universal power-law dynamics of SIR models on hierarchical modular networks

TL;DR

This paper studies how hierarchical modular networks influence the spread dynamics of SIR models, revealing non-universal power-law behaviors and the effects of network topology, heterogeneity, and mobility on epidemic growth and decay.

Contribution

It demonstrates that finite topological dimensions in hierarchical networks lead to power-law prevalence growth, with heterogeneity and mobility affecting epidemic dynamics but not the fundamental scaling exponents.

Findings

Power-law growth depends on network topological dimension.

Heterogeneity alters critical behavior and epidemic size distribution.

Mobility increases epidemic magnitude without changing scaling exponents.

Abstract

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.