Diffusive Epidemic Process with quenched disorder

Abstract

Epidemic spreading often occurs in spatially heterogeneous environments, yet how quenched heterogeneity reshapes its onset and critical dynamics remains poorly understood. The diffusive epidemic process, a minimal reaction-diffusion model whose absorbing-state transition is controlled by the relative diffusion of healthy and infected species, provides a natural setting for this question. Using a new single-seed algorithm that effectively simulate infinite systems for the infected individuals, we find that effective global diffusion rates can be used to predict disorder relevance and we identify two distinct infinite-disorder fixed points. Notably, we find that disorder in diffusion rates is qualitatively different from that in reaction rates as it can even induce a total suppression of the active phase, a phenomenon not observed with other types of disorder. These results establish mobility disorder as a distinct route by which quenched heterogeneity qualitatively reorganizes spreading dynamics, with implications for systems ranging from cell polarity to epidemic propagation in heterogeneous media.

Figures (5)

• Figure 1: (a) Schematic of the Diffusive Epidemic Process (DEP). $A$ (red) and $B$ (blue) particles diffuse with rates $D_A$ and $D_B$; $B$ infects $A$ at rate $\lambda$ and recovers at rate $1/\tau$. (b) Potential renormalization Group (RG) flow as a function of disorder strength and $D_B^{\text{eff}}/D_A^{\text{eff}}$ ($D_A^{\text{eff}},D_B>0$). Pure fixed points appear for $D_A>D_B$ (pink), $D_A=D_B$ (green) kree_effects_1989, and $D_A<D_B$ (yellow) van_wijland_wilson_1998. For $D_B^{\text{eff}}<D_A^{\text{eff}}$, disorder drives an infinite-disorder fixed point (IDFP, blue); for $D_A^{\text{eff}}=D_B^{\text{eff}}$, critical properties vary slowly with disorder strength up to another IDFP. (c) Introducing disorder in $\lambda(x)$, $D_A(x)$, or $D_B(x)$ creates activity-favored regions, where $B$ particles tend to stay longer, prior to the DEP dynamics. Depending on the relative difference between the effective rates $D_A^{\text{eff}}$ and $D_B^{\text{eff}}$, these activity-favored regions are either enhanced or diminished by the internal dynamics. This behavior is consistent with the Harris criterion and allows one to predict when disorder is relevant.
• Figure 2: Effects of $\lambda$-disorder for $1=D_A > D_B=0.375$ ($p_{\lambda}=0.3, c=0.2$). (a) Log-log evolution of the survival probability $P_s(t)$. A power-law decay (black dotted lines) is observed for different $\lambda$, whereas the estimated critical point (circles, blue line) exhibits upward curvature. (b) Evolution of the effective dynamical exponent $2/z(t)$. The pronounced decrease toward zero reflects the activated, slower-than-power-law growth of the mean-square displacement $R^2(t)$ (inset).
• Figure 3: Phase transitions in $D_A$-disordered systems with fixed $D_A^1=2000$. (a) Schematic phase diagram in the ($1-p_A, D_B/D_A^{\text{eff}}$) plane ($\epsilon > 0$ is a small positive value). Blue: slow-dynamics region, defined by a continuously decaying critical exponent $2/z(t)$. Red: pure critical properties. Green: suppressed active phase with density-dependent boundaries. (b) Spatial profiles of the $B$-particle density $\rho_B(x)$ (averaged over 20 neighboring sites) at several times near criticality ($\lambda_c \approx 0.22, D_A^0 > D_B$). Infection survives in rare regions (red curve) and spreads slowly. (c) Same as (b) for $D_A^0 < D_B$ and $\lambda=50$. The spreading of the infected cluster is arrested by extended low-density gaps (blue curve). Within these trapped regions, the density equilibrates; since $D_B > D_A^0$, the density progressively decays (yellow curve) toward spontaneous recovery, precluding a stable active phase (red curve precedes extinction). (d) Time evolution of the effective dynamical exponent $2/z(t)$. Continuous decay at the estimated critical point, $\lambda_c \approx 0.22$, is compatible with an IDFP. Other parameters: $p_A=0.5, D_A^0=1.4, D_B=1$. (e) Evolution of the survival probability $P_s(t)$. Despite the large $\lambda=50$, which would typically yield a constant $P_s(t)$ at late times, the observed decay for sufficiently small $D_A^0$, confirms the suppression of the active phase. Other parameters: $p_A=0.5, D_B=1$.
• Figure 4: (a) Schematic of perfectly correlated disorder ($D_{A,i}= D_{B,i} = D^{0/1}$). Sites with $D^0$ are either randomly distributed (left) or periodically spaced (right). The coarse-grained mapping retains only the state of $D^0$ sites and their mutual distances $d_{i,j}$, mapping the system to a contact process with either a random (left) or uniform (right) effective infection rate $\tilde{\lambda}$. (b) Mean-square displacement $R^2(t)$ at criticality for various $D^0$ ($p=0.1, D^1=1$) under periodic disorder, illustrating different transient growth profiles. Randomizing the position of $D^0$ sites leads to significantly slower, activated growth (dotted green line). (c) Survival probability $P_s(t)$ for periodic (dotted) and random (solid) disorder ($D^0=0.25, D^1=1, p=0.1$). The slower decay for random disorder is attributed to persistent activity within rare clusters of adjacent $D^0$ sites.
• Figure 5: $P_s$ versus $N_B$ at criticality for $D_A^{\text{eff}} > D_B^{\text{eff}}$ under varying quenched disorder. Colors: green ($D_B$ block), blue ($\lambda$-disorder), brown ($D_A$ weak variance), cyan/purple ($D_A$ strong variance for $D_A^0 > D_B$ and $D_A^0 < D_B$, respectively). Correlated disorder (red) with $D_A^{\text{eff}} = D_B^{\text{eff}}$ is included as a reference. The black dotted line represents the slope $-\overline{\Theta}/\overline{\delta}$ of the disordered contact process. Two distinct regimes emerge: a positive-slope class (resembling pure DEP) and a negative-slope class (analogous to the disordered contact process). The transition of $D_A$-disorder across both regimes highlights the non-trivial impact of diffusion variance on scaling. Insets: Qualitative evolution of total density after infection; the first class shows $\rho(t) \neq \rho_A(t_0)$, whereas the second shows $\rho(t) \approx \rho_A(t_0) \, \forall t$.