Critical properties of the susceptible-exposed-infected model with correlated temporal disorder

In this paper we study the critical properties of the nonequilibrium phase transition of the susceptible-exposed-infected (SEI) model under the effects of long-range correlated time-varying environmental noise on the Bethe lattice. We show that temporal noise is perturbatively relevant changing the universality class from the (mean-field) dynamical percolation to the exotic infinite-noise universality class of the contact process model. Our analytical results are based on a mapping to the one-dimensional fractional Brownian motion with an absorbing wall and is confirmed by Monte Carlo simulations. Unlike the contact process, our theory also predicts that it is quite difficult to observe the associated active temporal Griffiths phase in the long-time limit. Finally, we also show an equivalence between the infinite-noise and the compact directed percolation universality classes by relating the SEI model in the presence of temporal disorder to the Domany-Kinzel cellular automaton in the limit of compact clusters.

Figure 1

Reactions of the susceptible-exposed-infected (SEI) model. In the presence of an infected nearest neighbor, susceptible individuals becomes either exposed or infected with rates $\mu$ and $\lambda$, respectively.

Figure 2

Simulations of the SEI model with uncorrelated temporal disorder ($\gamma =1$) at the critical point $v=(z-2)\lambda -\mu =0$. Here $\lambda =64$, $\mu =128$, $\sigma =8$, $\mathrm{\Delta }t=16,$ the initial cluster has size $N_{IS}^0={10}^2$, and the data is averaged over $\mathcal{N}={10}^5$ temporal disorder configurations and over $N_{\text{avr}}$ realizations of the stochastic noise. In panel (a) the uncertainties are of the order of the dispersion of the data and in panel (b) of the order of the symbol size.

Figure 3

Simulations of the SEI model with power-law correlated temporal disorder at the critical point. In all panels the uncertainties are of the order of the symbol size, and the solid lines are power-law fits. (a) $ln(\left[N_{IS}\right]/N_{IS}^0$) vs $t$, for many values of the correlation exponent $\gamma$ [see Eq. (14)]. These simulations used the parameters $\sigma =1$, $\mathrm{\Delta }t=4$, and initial condition $N_{IS}^0=4$. (b, c) The survival probability $\mathcal{P}$ and the average position $\left[x\right]=ln(1+N_{IS})$ as a function of time for $\sigma =8$, $\mathrm{\Delta }t=16$, and initial condition $N_{IS}^0=4$. (d) The critical exponents extracted by fitting power laws to the data according to Eqs. (19), (17), and (20), for $N_{IS}$, $\mathcal{P}$, and $x$, respectively. All simulations average over ${10}^7$ disorder configurations.

Figure 4

Off-critical properties of the SEI model. Uncertainties, if not shown and except for $\beta$, are of the order of the symbol size. The temporal disorder strength is $\sigma =8$ and the time interval is $\mathrm{\Delta }t=16$. (a) The survival probability (solid lines) in the active phase and at the critical point for $\gamma =1.3$. The bias velocities $v$ from bottom to top are 0, 0.000 4, 0.000 8, 0.001 562 5, 0.003 125, 0.006 25, 0.012 5, 0.025, and 0.05. The black dashed curves represent the confidence interval of $P_{\text{crit}}$ vertically shifted which intercepts the off-critical curves at the crossover time $t_x$ (black circles). (b) The stationary survival probability $P_{\text{st}}$ and (c) the crossover time $t_x$ vs the distance to the critical point $v$ for many values of $\gamma$. The solid lines are power-law fits. (d) The critical exponents $\beta$ and ${\nu }_{\parallel }$ for many values of $\gamma$. The solid lines are the predictions to $\beta$ and ${\nu }_{\parallel }$, Eqs. (22) and (21). The uncertainties of $\beta$ are of order ${10}^{-3}.$ The data are averaged over ${10}^7$ disorder configurations.

Figure 5

Tilted lattice of the Domany-Kinzel cellular automaton. The horizontal axis represents the index $i$ of the lattice sites and the vertical is the time $t$. Notice that all site indices are even (odd) for even (odd) time slices $t$.