  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.
