Long-range epidemic spreading in a random environment

Modeling long-range epidemic spreading in a random environment, we consider a quenched, disordered, $d$-dimensional contact process with infection rates decaying with distance as $1/r^{d+\sigma }$. We study the dynamical behavior of the model at and below the epidemic threshold by a variant of the strong-disorder renormalization-group method and by Monte Carlo simulations in one and two spatial dimensions. Starting from a single infected site, the average survival probability is found to decay as $P\left(t\right)\sim t^{-d/z}$ up to multiplicative logarithmic corrections. Below the epidemic threshold, a Griffiths phase emerges, where the dynamical exponent $z$ varies continuously with the control parameter and tends to $z_c=d+\sigma$ as the threshold is approached. At the threshold, the spatial extension of the infected cluster (in surviving trials) is found to grow as $R\left(t\right)\sim t^{1/z_c}$ with a multiplicative logarithmic correction and the average number of infected sites in surviving trials is found to increase as $N_s\left(t\right)\sim {(lnt)}^{\chi }$ with $\chi =2$ in one dimension.

Figure 1

(a), (d), and (g) Logarithm of the average survival probability $P\left(t\right)$ plotted against $ln\left[t{(lnt)}^4\right]$. The data were obtained by numerical simulations of the one-dimensional model for different values of the control parameter $\lambda$. The solid line with a slope $-1/\alpha$ indicates the asymptotic behavior predicted by the SDRG approach of the model at the critical point. (b), (e), and (h) Average number $N_s\left(t\right)$ of active sites conditioned on survival plotted against ${(lnt)}^2$. (c), (f), and (i) Logarithm of the spread $R\left(t\right)$ defined in Eq. (14) plotted against $ln[t/{(lnt)}^{2\alpha -4}]$. The solid line with a slope $1/\alpha$ indicates the asymptotic behavior predicted by the SDRG approach of the model at the critical point.

Figure 2

(a) Logarithm of the average survival probability $P\left(t\right)$ plotted against $ln\left[t{(lnt)}^{-3/7}\right]$. The solid line has a slope $-1$. The data were obtained by numerical simulations of the one-dimensional model with $\alpha =3/2$ and $c=0.5$ for different values of the control parameter $\lambda$. (b) Logarithm of the average number of active sites $N\left(t\right)$ plotted against $lnt$. (c) Logarithm of the spread $R\left(t\right)$ plotted against $lnt$. The solid line has a slope $1/\sigma =2$.

Figure 3

Average mass of the last decimated cluster plotted against ${(lnL/L_0)}^2$ with $L_0=3$. The data were obtained by numerical renormalization of the two-dimensional model with decay exponent $\alpha =3$ for different values of the control parameter $\Theta$.

Figure 4

Effective dynamical exponents obtained by two-point fits using Eq. (22) as a function of the system size $L$. The straight line is a fit to the data obtained for $\Theta =2.45$.

Figure 5

(a), (d), and (g) Logarithm of the average survival probability $P\left(t\right)$ plotted against $ln\left[t{(lnt)}^4\right]$. The solid line has a slope $-d/\alpha$. The data were obtained by numerical simulations of the two-dimensional model for different values of the control parameter $\lambda$. (b), (e), and (h) Average number $N_s\left(t\right)$ of active sites conditioned on survival plotted against ${(lnt)}^2$. (c), (f), and (i) Logarithm of the spread $R\left(t\right)$ defined in Eq. (14) plotted against $ln[t/{(lnt)}^{2\alpha /d-4}]$. The solid line has a slope $1/\alpha$.

Figure 6

(a) Logarithm of the average survival probability $P\left(t\right)$ plotted against $ln\left[t{(lnt)}^{-3/7}\right]$. The solid line has a slope $-1$. The data were obtained by numerical simulations of the two-dimensional model with $\alpha =3$ and $c=0.5$ for different values of the control parameter $\lambda$. (b) Logarithm of the average number of active sites $N\left(t\right)$ plotted against $lnt$. (c) Logarithm of the spread $R\left(t\right)$ plotted against $lnt$. The solid line has a slope $1/\sigma =1$.