Impact of the distribution of recovery rates on disease spreading in complex networks

We study a general epidemic model with arbitrary recovery rate distributions. This simple deviation from the standard setup is sufficient to prove that heterogeneity in the dynamical parameters can be as important as the more studied structural heterogeneity. Our analytical solution is able to predict the shift in the critical properties induced by heterogeneous recovery rates. We find that the critical value of infectivity tends to be smaller than the one predicted by quenched mean-field approaches in the homogeneous case and that it can be linked to the variance of the recovery rates. Our findings also illustrate the role of dynamical-structural correlations, where we allow a power-law network to dynamically behave as a homogeneous structure by an appropriate tuning of its recovery rates. Overall, our results demonstrate that heterogeneity in the recovery rates, eventually in all dynamical parameters, is as important as the structural heterogeneity.

Figure 1

Susceptibility curves extracted from the Monte Carlo simulations for an Erdős-Rényi network with $N={10}^5$ and $\langle k\rangle \approx 10$ considering that the recovery rate distribution follows an inverse-gamma distribution, whose shape parameter $\alpha$ is color coded. The insets show the dependence of ${\lambda }_c^{\mathrm{QMF}}$ on $\alpha$ (top) and a comparison with numerical results (bottom), also for different $\alpha$.

Figure 2

Results for the susceptibility when the dynamics are on top of real networks. We use (a) the UC Irvine messages social network [32, 33] ($•$) and (b) the OpenFlights network [33, 34] ($\blacksquare$). In both cases, we considered the undirected version of the giant component. In the main figure of each panel, we present the susceptibility for different values of $\alpha$ and $\lambda$. In the bottom right insets, we present the order parameter. In the top inset of (b), we present a comparison between the QMF estimated and predicted critical points.

Figure 3

Finite-size analysis considering structure-dynamics correlations. We present the susceptibility (colors represent the sizes) in (a) the heterogeneous recovery rates, where ${\delta }_i=k_i$ on top of the power-law networks, whose exponents are $\gamma =2.1,2.7,3.5$, and (b) the SIS model with heterogeneous recovery rates, considering an Erdős-Rényi network and the recovery rates as ${\delta }_i=\frac{k_i}{k_{\text{PL}}}$, where $k_{\text{PL}}$ is a discrete power law, more specifically, the degree sequence of the networks used in (a). In the inset we present the critical point as a function of the system size in log-log scale.

Figure 4

Evaluation of $\mathbb{P}(\delta \le 1)$ and $\mathbb{P}(\delta >1)$ as a function of $\alpha$, the shape parameter of an inverse-gamma distribution with $\beta =\alpha -1$.

Figure 5

Susceptibility curves extracted from the Monte Carlo simulations for an Erdős-Rényi network with $N={10}^5$ and $\langle k\rangle \approx 10$ considering that the rate distribution follows a uniform distribution (symmetric) $U(a,2-a)$, where the parameter $a$ controls the variance and is color coded.

Figure 6

Phase diagram for (a) Erdős-Rényi ($•$) networks and (b) PL networks. In (b), from top to bottom, we have $\gamma =2.1$ ($\blacksquare$), $\gamma =2.7$ ($⧫$), and $\gamma =3.5$ ($▾$). In the inset in (a) we present the behavior for larger $\lambda$'s in all the networks. They comprise $N={10}^4$ nodes. The recovery rate distribution is given as an inverse-gamma distribution, whose shape parameter $\alpha$ is color coded.

Figure 7

Finite-size analysis for the CP for (a) susceptibility and (b) the order parameter. The colors represent the sizes and from bottom to top, the curves are grouped by the power-law exponents $\gamma =2.1,2.7,3.5$, respectively.

Figure 8

The $\mathrm{IPR}$ versus $N$ for the Erdős-Rényi, PL networks and the correlated case ${\mathrm{\Delta }}^{-1}\mathbf{A}$, where ${\mathrm{\Delta }}_i=\frac{k_i}{k_{\text{PL}}}$ for (a) $\gamma =2.1$ and (b) $\gamma =3.5$. Fifty networks were considered in each case.