Epidemic threshold and localization of the SIS model on directed complex networks

Abstract

We study the susceptible-infected-susceptible (SIS) model on directed complex networks within the quenched mean-field approximation. Combining results from random matrix theory with an analytic approach to the distribution of fixed-point infection probabilities, we derive the phase diagram and show that the model exhibits a nonequilibrium phase transition between the absorbing and endemic phases for $c \geq λ^{-1}$, where $c$ is the mean degree and $λ$ the average infection rate. Interestingly, the critical line is independent of the degree distribution but is highly sensitive to the form of the infection-rate distribution. We further show that the inverse participation ratio of infection probabilities diverges near the epidemic threshold, indicating that the disease may become localized on a small fraction of nodes. These results provide a systematic characterization of how network heterogeneities shape epidemic spreading on directed contact networks within the quenched mean-field approximation.

Figures (9)

• Figure 1: Comparison between the solutions of Eq. (\ref{['hhjj2']}) (solid lines) with the fixed-point solutions of Eq. (\ref{['fixed']}) (symbols) for directed networks with a Poisson degree distribution and a $\Gamma$-distribution of infection rates or coupling strengths. The fixed-point solutions are derived from an ensemble of $50$ networks with $N=10^5$ nodes, while the population dynamics results are obtained from $5$ independent runs of the algorithm with $M=10^6$ stochastic variables. (a) The prevalence $\langle \rho \rangle$ as a function of the mean infection rate $\lambda$ for standard deviation $\sigma=0.2$ of the infection rates. (b) and (c): distribution $\mathcal{P}(\rho)$ of the infection probabilities for $\lambda=1/2$, $\sigma=0.2$, and two different $c$.
• Figure 2: Phase diagram of the SIS model on directed networks in terms of the mean degree $c$ and the standard deviation $\sigma$ of the infection rates or coupling strengths (the mean infection rate is $\lambda=1/2$). The indegrees follow a Poisson distribution, while the infection rates follow a $\Gamma$-distribution. The model exhibits an endemic phase ($\langle \rho \rangle >0$) and an absorbing phase ($\langle \rho \rangle=0$). The standard deviations at the dots are, respectively, $\sigma_{*} = \sqrt{\lambda (1 - \lambda) }$ and $\sigma_{4} \simeq 0.36$. For $\sigma > \sigma_{*}$, the critical line is obtained by solving Eq. (\ref{['hhjj2']}) using the population dynamics algorithm with $M= 10^6$ stochastic variables. The colour scale shows the inverse participation, Eq. (\ref{['huhu55']}), which quantifies the spatial fluctuations of the infection probabilities. The fourth moment $\mu_{4}$ of the infection rates is defined in Eq. (\ref{['hhjj']}).
• Figure 3: Prevalence $\langle \rho \rangle$ as a function of the mean degree $c$ for different distributions of infection rates and indegrees in the regime $\sigma < \sigma_{*}$ (see the main text). The infection rates or coupling strengths have mean $\lambda = 1/2$ and standard deviation $\sigma = 0.2$. The results are obtained by solving Eq. (\ref{['hhjj2']}) using the population dynamics algorithm with $M= 10^5$ stochastic variables. The inset shows the prevalence near $c=\lambda^{-1}$ in logarithmic scale. The colours in the inset correspond to the same distributions as in the main panel.
• Figure 4: Prevalence $\langle \rho \rangle$ as a function of the standard deviation $\sigma$ of the infection rates for different distributions of indegrees and infection rates in the regime $\sigma > \sigma_{*}$ (see the main text). The average indegree is $c \simeq 2.7$, and the infection rates have mean $\lambda = 1/2$. For power-law distributed indegrees, the smallest indegree is $k_{\rm min}=2$. The results are obtained by solving Eq. (\ref{['hhjj2']}) using the population dynamics algorithm with $M=10^6$ stochastic variables.
• Figure 5: Dimensionless second moment $\mathcal{I}^{(2)}(\rho)$ [Eq. (\ref{['huhu56']})] and inverse participation ratio $\mathcal{I}^{(4)}(\rho)$ [Eq. (\ref{['huhu55']})] as functions of the standard deviation $\sigma$ of the infection rates for different mean degrees $c$ close to the epidemic threshold $c=\lambda^{-1}$. These results are for directed networks with Poisson indegrees and a $\Gamma$-distribution of infection rates or coupling strengths with mean $\lambda = 1/2$. Symbols represent numerical results obtained from Eq. (\ref{['hhjj2']}) using the population dynamics algorithm with $M=10^{6}$ (vertical bars indicate the standard deviation of the mean computed over $10$ independent runs). Solid lines are analytic predictions derived from the moments of the leading eigenvector [Eqs. (\ref{['r1']}) and (\ref{['r2']})].