Nonuniversal power-law dynamics of susceptible infected recovered models on hierarchical modular networks

Power-law (PL) time-dependent infection growth has been reported in many COVID-19 statistics. In simple susceptible infected recovered (SIR) models, the number of infections grows at the outbreak as $I\left(t\right)\propto t^{d-1}$ on $d$-dimensional Euclidean lattices in the endemic phase, or it follows 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, and 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 those of regular graphs occur, 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 it increases the magnitude of the epidemic. The addition of a superspreader hot spot also does not change the growth exponent and the exponential decay in the herd immunity regime.

Figure 1

Plot of the adjacency matrix of an $N=1024$ sized sample of the HMN2D graphs used for the simulations. Black dots denote edges between nodes $i$ and $j$. The $l_{\text{max}}=4$-level structure is clearly visible by the blocks near the diagonal. Low density, scattered points away from the diagonal represent long-range links. Inset: scheme of the lowest three levels of a hierarchical, four-block structure of nodes, embedded in $2d$ space with additional long links. Black solid lines: $l=1$ links; red, dashed lines, $l=2$ links; $l=0$ edges are not shown.

Figure 2

The average number of nearest neighbors measured by the BFS algorithm as a function of graph distances from randomly selected initial seeds. Different $b$ and $s$ values and sizes $l_{\text{max}}$ were investigated as shown by the legends. Lines correspond to least-squared PL fits for $10<r<r_{\text{cut}}$, where $r_{\text{cut}}$ was estimated visually.

Figure 3

Effective exponents ${\eta }_{\mathrm{eff}}\left(t\right)$ in two dimensions for $\lambda =0.4$, 0.406, 0.407, 0.408, 0.41, 0.42, 0.44, and 0.5 (bottom to top curves). Inset: initial time evolution of $I\left(t\right)$, averaged over runs from ${10}^4$ randomly selected initial random sites. The two distinct fixed point behaviors can be seen at ${\lambda }_c=0.4059\left(1\right)$, with $\eta =0.59\left(1\right)$ and the supercritical phase, characterized by $\eta =1$.

Figure 4

Density of infected sites in different graphs for $i_0=2$ by varying $s$ and the size with $b=1$ and $\lambda =\nu =1$ fixed. Thin lines $l_{\text{max}}=7$, thick lines $l_{\text{max}}=8$ data, multiplied by a factor of 4. Only the $s=4$ curves exhibit PL initially, and exponential decay is observable following finite-size cutoff, corresponding to herd immunity. The dashed line corresponds to the single seed case: $i_0=1$, multiplied by a factor 2.

Figure 5

Density of infected sites in graphs with $s=4$ and $b=1$ for $\lambda =$ 0.28, 0.29, 0.3, 0.32, 0.35, 0.4, 0.45, 0.5, and 0.6 (bottom to top curves). Inset: local slopes of the same curves as well as for $\lambda =0.7$, 0.8. One may think of continuously changing exponents above ${\lambda }_c=0.310\left(5\right)$, because short times hinder our ability to see the supercritical point scaling behavior with $\eta \simeq 2.5$.

Figure 6

Effective exponents ${\eta }_{\mathrm{eff}}$ as in Fig. 5, for $s=4$ and $b=0.4$ for $\lambda =0.47$, 0.473, 0.475, 0.48, 0.49, 0.5, 0.55, 0.6, 0.7, and 0.8 (bottom to top curves). The two distinct fixed point behaviors can be seen at ${\lambda }_c=0.480\left(5\right)$, with $\eta =0.8\left(1\right)$ and the supercritical phase, characterized by $\eta \simeq 2$.

Figure 7

Spatiotemporal size distribution of infection avalanches for HMN2Ds with $s=4$ and $b=0.4$ at different $\lambda$'s and $\nu =1-\lambda$. Lines present PL fits for the $s>10$ region before the bump.

Figure 8

The effect of diffusion on the local slopes of $I\left(t\right)$ in $2d$ lattices of sizes $L=1000,2000$ for different $\lambda =0.4125$, 0.413, 0.4135, 0.414, 0.415, 0.42, 0.43, 0.45, and 0.5 (for curves from bottom to top). $L=2000$ results are plotted near the critical point: $\lambda =0.413,0.4135$ (thinner lines).

Figure 9

The effect of diffusion on $I\left(t\right)$ in HMN2D graphs with $s=4$, $b=1$, and $\lambda =\nu =0.5$ fixed. The diffusion (blue boxes) increases the size of the epidemic, but it does not alter the exponents. Inset: local slopes of $I\left(t\right)$ in HMN2D graphs with $s=4$, $b=1$, for $\lambda =0.22$, 0.23, 0.24, 0.25, 0.30, 0.37, 0.44, 0.5, 0.6, and 0.7 (bottom to top curves) in the presence of diffusion. The critical and supercritical $\eta$ are the same as without diffusion.

Figure 10

Spatiotemporal size distribution of infection avalanches for HMN2Ds at $s=4$ and $b=1$ in the presence of diffusion for different $\lambda =0.23$ (circles), 0.24 (squares), and 0.25 (rhombuses). Lines present PL fits for the $s>10$ region before the bump. Inset: The effect of hot spots ($H$) in $2d$ at $\lambda =0.5$. Brown, dashed line: $10\%$ quenched, random distributed H's; black, solid line: homogeneous $2d$; blue line: diffusion ($D$) $+$ $H$, red line $D$ only (top to bottom curves).

Figure 11

The effect of a single hot spot for the diffusive model in HMN2D graphs with $s=4$, $b=1$ and $\lambda =0.22$, 0.23, 0.235, 0.24, 0.245, 0.25, 0.26, 0.4, and 0.5 (bottom to top curves). Inset: local slopes of the same. The asymptotic critical and supercritical effective exponents are roughly the same as in the nondiffusive homogeneous SIR.

Figure 12

Effective exponents ${\eta }_{\mathrm{eff}}$ as in Fig. 5 for SIR on the $d=2$ lattices, with bimodal quenched disorder for $\lambda =0.35$, 0.36, 0.365, 0.3655, 0.3557, 0.366, 0.37, 0.38, and 0.40 (bottom to top curves).

Figure 13

Spatiotemporal size distribution of infection avalanches in $2d$ in the presence of bimodal quenched disorder for different $\lambda$'s and $\nu =1-\lambda$. Lines present PL fits for the $s>10$ region before the bump.