Extinction of epidemics in lattice models with quenched disorder

The extinction of the contact process for epidemics in lattice models with quenched disorder is analyzed in the limit of small density of infected sites. It is shown that the problem in such a regime can be mapped to the quantum-mechanical one characterized by the Anderson Hamiltonian for an electron in a random lattice. It is demonstrated both analytically (self-consistent mean field) and numerically (by direct diagonalization of the Hamiltonian and by means of cellular automata simulations) that disorder enhances the contact process, given the mean values of random parameters are not influenced by disorder.

Figure 1

(Color online) The spectrum of the effective (dashed line) and true (solid line) Hamiltonian (density of states) defined on the square lattice with nearest-neighbor interactions $(Z=4)$ in the system with transmission rates uniformly distributed around the mean value $w_0=1$ with half-width $\Delta =1.0$ and ${\epsilon }_0=30$. The exact spectrum was obtained numerically using the kernel polynomial method [31] for a model of $N=2000\times 2000$ sites. The spectrum of the crystalline Hamiltonian with all nearest-neighbor interactions $(\Delta =0)$ is shown by the dot-dashed line. The inset magnifies the spectrum around the top of the band.

Figure 2

(Color online) The dependence of the maximum eigenvalue ${\lambda }_{\max }$ evaluated using a mean-field approach (solid curve, labeled CPA) and ${\lambda }_{\max }^{*}$ calculated by direct diagonalization (DD, circles for $4000\times 4000$ and triangles for $2000\times 2000$ lattices; the error bars represent the standard deviations of the distribution of the maximum eigenvalues) on the degree of disorder characterized by the half-width of the box distribution $\Delta$ for ${\epsilon }_0=30$ and $w_0=1$. The squares (labeled CA) represent the long-time decay rates ${\lambda }_{\mathrm{CA}}$ obtained by the CA simulations, each data point corresponding to approximately $5\times {10}^{10}$ runs, on a $5\times 5$ lattice. The inset shows a version of the CA data scaled vertically to clarify the trend.