1 Introduction

The spread of epidemics in complex networks such as biological populations and computer networks is of great current interest, both for practical applications and from a fundamental point of view.

This is one of the issues of the theory of non-equilibrium phase transitions [ 1 ] and the theory of complex networks [ 2 ,  3 ] . The problems of interest include the question about the existence of a critical regime separating invasive (active) and non-invasive (absorbing) states of the system and, if such a transition exists, how it depends on internal and external parameters and also what the universal features of the transition are (see e.g. [ 4 ] ).

In one of the simplest models of epidemics, all the nodes are divided into two classes: infectious (I) and and susceptible (S) [ ? ] . The epidemic spreads by a contact process according to which an infected node can transfer infection to another susceptible node with typical infection rate $w$ and recover with typical recovery rate $\varepsilon$ becoming again susceptible (the SIS model). The system undergoes a phase transition with variation of the dimensionless parameter, $\eta =w∕\varepsilon$ , form the absorbing ( $\eta <{\eta }_{\mathrm{\text{c}}}$ ) to active state ( $\eta >{\eta }_{\mathrm{\text{c}}}$ ). The critical value is ${\eta }_{\mathrm{\text{c}}}\sim Z^{-1}$ [ 4 ,  2 ] , with $Z$ being the typical number of links per node (coordination number).

Usually, the infection and recovery rates are assumed to be node independent. However, in real systems, the values of $w$ and $\varepsilon$ can vary from node to node (quenched disorder). Investigations of contact processes in the systems with quenched disorder over recent years [ 1 ,  5 ,  ? ,  6 ,  7 ,  ? ,  ? ,  8 ,  ? ,  ? ,  9 ] have resulted in some rather intriguing findings. For example, it has been suggested that the disorder can change the universality class of the model [ 10 ,  11 ] . However, the situation is far from being completely understood, and the aim of this paper is to investigate the influence of a general form of quenched disorder on the dynamics of the contact process in the absorbing state. Using a combination of the simple epidemiological model with methods from condensed matter physics, we show how disorder in the infection or recovery rates, influences the long-time dynamics (decay time) of epidemics in the absorbing state. This is of practical importance in determining the time to extinction of epidemics within this state. We also identify a lower bound for ${\eta }_{\mathrm{\text{c}}}$ and show how the degree of disorder influences the magnitude of the extinction rate.

We consider the dynamics of the contact process far in the absorbing state when the problem can be mapped to the quantum-mechanical one described by the disordered Hamiltonian of the Anderson-type (see e.g. [ 12 ] ) and an approximate method (self-consistent mean-field) can be applied. The spectrum of the Hamiltonian under this approximation is then used in the analysis of the long-time dynamics of the system. The advantage of the approach is in the possibility of incorporating a general type of disorder in the analysis while the disadvantage is due to the rather severe restriction of being in the absorbing state (dilute regime for concentration of infected nodes). Our main result is that the disorder slows down the long-time dynamics of the system given the mean values of the random values stay the same as in ordered systems. The approximate analytical results are supported by exact numerical analysis using cellular automata approach.

2 Formulation of the problem

Consider a set of $N$ nodes (sites) connected to each other by links (infection paths). Each node, $i$ , can be in one of two states: infected (occupied by an “excitation” and characterized by occupation number, $n_i=1$ ) or not infected (empty with $n_i=0$ ). The occupation number $n_i$ changes from $0$ to $1$ as a result of infection from an occupied node $j$ occurring with infection rate $w_{ji}$ , and from $n_i=1$ to $n_i=0$ due to natural recovery with rate ${\varepsilon }_i$ . Any state of the system is characterized by the set of occupation numbers, $\left\{n\right\}\equiv \left\{n_1,\dots ,n_N\right\}$ . Bearing in mind the stochastic nature of the infection and recovery processes it is convenient to characterize the system by the state vector $\left|P\left(t\right)\right.⟩$ , the components of which are the probabilities of finding the system in different states at time $t$ , $\left|P\left(t\right)\right.⟩=\left|P_{\left\{n\right\}}\left(t\right)\right.⟩$ , where $n=1,\dots ,2^N$ runs over all the possible states of the system. The time evolution of the state vector is governed by the master equation describing the conserved probability flow [ 4 ] ,

where $\widehat{\mathcal{ℒ}}$ stands for the non-Hermitian Liouville operator, the non-zero elements of which describe the transitions between the states with different number of occupied nodes.

It is convenient to make a linear transformation of the state coordinates (change of basis) from $P_{n_1,n_2,\dots ,n_N}$ to the $n$ -site probabilities,

where ${\overline{P}}_i\left(t\right)$ is the probability of finding node $i$ in an occupied ( $n_i=1$ ) state independent of the occupation of all other nodes. This allows the master equation ( 1 ) to be recast in the following form:

where ${\overline{P}}_j\left(t\right)-{\overline{P}}_{ji}\left(t\right)$ is the probability of finding the system with the occupied $j$ -th node and the unoccupied $i$ -th node, independent of the state of all other nodes. The single-site probability ${\overline{P}}_i\left(t\right)$ in Eq. ( 3 ) is coupled with the double-site probabilities ${\overline{P}}_{ij}\left(t\right)$ . A similar probability-balance equation for ${\overline{P}}_{ij}\left(t\right)$ contains the three-site probabilities and so on. This makes the set of simultaneous equations to be coupled and thus be non-trivial for analysis.

The lowest level of approximations in decoupling schemes involves a complete ignorance of the double-site occupations, ${\overline{P}}_{ij}$ , in comparison with other terms in the master equation ( 3 ), which is possible if

for each pair of communicating sites $i-j$ so that the master equation under these approximations transforms to the following form,

and is hereafter referred to as the approximate master equation. The inequalities given by Eq. ( 4 ) are valid if the typical recovery rate is much greater than the typical infection rate, $\varepsilon \gg w$ , and an epidemic dies out very quickly over the typical time, ${\varepsilon }^{-1}$ . In such a regime, the single-site probabilities are small for the majority of sites practically for all times, i.e. ${\overline{P}}_i\ll 1$ , and this regime can be called a dilute regime for the concentration of infected sites.

Therefore, making approximations given by Eq. ( 4 ) we focus on the dynamics of the system far in the absorbing state ( $\eta \ll {\eta }_c$ ), i.e. where an epidemic will certainly become extinct. Bearing in mind that the terms $\propto {\overline{P}}_{ij}$ (entering Eq. ( 3 ) with a minus sign) reduce the infection rate due to a possible simultaneous occupation of both communicating sites $i$ and $j$ (the transmission of infection cannot occur between two sites if both of them are already infected) we might expect that the solution of approximate rate equation ( 5 ) exhibits the enhancement of an epidemic in the dilute regime. In fact, the approximate master Eq. ( 5 ) on its own describes the spread of an epidemic in the system of $N$ nodes with multiple reinfection of infected nodes where each node can be multiply occupied by the “excitations” (infection) and ${\overline{P}}_i$ has the meaning of an occupation number which can be larger than one.

The approximate master equation (Eq. ( 5 )) similarly to the exact one can also have solutions which behave differently as $t\to \infty$ depending on the typical value of parameter $\eta$ . In fact, there exists a critical value ${\eta }_c^{\ast }$ for the approximate master equation which separates the absorbing and active states and this critical value ${\eta }_c^{\ast }$ is smaller than the critical value ${\eta }_c$ for the exact master equation due to the nature of the approximations made. This allows the lower bound estimate, ${\eta }_c^{\ast }$ , for ${\eta }_c$ to be found by solving the approximate problem for a quite general type of disorder.

The state of the system in the dilute regime can be defined in the subspace of the singly-occupied sites spanned by the orthonormal site basis $\left|i\right.⟩$ , and is characterized by the state vector $\left|\overline{P}\right.⟩=\left|{\overline{P}}_1,{\overline{P}}_2,\dots ,{\overline{P}}_N\right.⟩$ with components ${\overline{P}}_i\left(t\right)=⟨i\mid \overline{P}\left(t\right)⟩$ . The master equation ( 5 ) can be recast as

where $\widehat{\mathcal{\mathscr{H}}}$ stands for the Liouville operator which now (under assumption of symmetric infection rates, $w_{ij}=w_{ji}$ ) is Hermitian and can be associated with the Anderson-like Hamiltonian (see e.g. [ 12 ] ),

in which the recovery rates, ${\varepsilon }_i$ , play the role of the on-site energies and the infection rates, $w_{ij}$ , can be associated with the transfer (hopping) integrals. Both of these values are random and this makes the further analysis non-trivial even in the dilute regime. The topology of the underlying network of sites, in principle, can be arbitrary but the simplest choice is a regular $D$ -dimensional lattice with the nearest-neighbour interactions only. Furthermore, for simplicity, we consider a square lattice and thus the second sum in Eq. ( 7 ) runs for each site over $Z=4$ its nearest neighbours only. It is worth mentioning that if the on-site matrix elements satisfy the sum rule, ${\varepsilon }_i={\sum }_j{w}_{ij}$ , then the number of occupied sites is conserved and the problem is equivalent to the random walk problem on a lattice with random transition rates [ 13 ] , or to the scalar vibrational problem for a lattice with force-constant disorder [ 14 ] . Notice also that decoupling of the single-site probabilities can also be made in the non-dilute regime by assuming that ${\overline{P}}_{ij}={\overline{P}}_i{\overline{P}}_j$ , i.e. ignoring possible correlations in occupation of the communicating sites. This brings a non-linearity to the problem which can be treated within the mean-field approach for ideal lattices [ 4 ] .

3 Solution

The formal solution of the problem given by Eq. ( 5 ) is straightforward,

where $\left|e^j\right.⟩=\left|e_1^j,\dots ,e_N^j\right.⟩$ and ${\lambda }_j$ are the eigenvectors and eigenvalues of the Hamiltonian, respectively, $\widehat{\mathcal{\mathscr{H}}}\left|e^j\right.⟩={\lambda }_j\left|e^j\right.⟩$ . Equivalently, this solution can be written via the Laplace transform of the state vector, $\left|\overline{P}\left(\lambda \right)\right.⟩={\int \; <!--nolimits-->}_0^{\infty }\left|\overline{P}\left(t\right)\right.⟩e^{-\lambda t}\mathrm{\text{d}}t$ ,

where $\widehat{\mathcal{G}}={\left(\lambda -\widehat{\mathcal{\mathscr{H}}}\right)}^{-1}$ is the resolvent operator.

We are interested in the time evolution of the total number of infected sites,

averaged over different realizations of disorder (angular brackets) and/or over initial conditions (for concreteness, a single site $i_0$ is infected at $t=0$ , i.e. ${\overline{P}}_i\left(0;i_0\right)={\delta }_{i{i}_0}$ ) and its Laplace transform,

The other quantity of interest is the mean-squared displacement of the epidemic,

where $R_{i_{0}i}$ is the vector connecting site $i_0$ with site $i$ .

As follows from Eq. ( 8 ) the dynamics of the system in the dilute regime is defined by the eigenspectrum of the Hamiltonian. The set of characteristic times (inverse eigenvalues) controls the evolution of the system with different eigenvalues being important for different time scales. The long-time dynamics of the system is defined by the maximum eigenvalue of the Hamiltonian, ${\lambda }_{\mathrm{\text{max}}}$ ( ${\lambda }_{\mathrm{\text{max}}}<0$ in dilute regime) and our aim is to find the estimate for ${\lambda }_{\mathrm{\text{max}}}$ and how it depends on degree of disorder. The maximum eigenvalue depends on all the recovery and infection rates, ${\lambda }_{\mathrm{\text{max}}}\left({\varepsilon }_i,w_{ij}\right)$ , and this obviously complicates its analytical evaluation. The exact analytical solution of the problem in the general case is not currently known and numerous approximate analytical (see e.g. [ 12 ,  15 ] ) and numerical (see e.g. [ 16 ,  17 ] ) methods have been developed for evaluation of the spectrum of disordered Hamiltonians. Below, we use one of the well-developed self-consistent mean-field approaches (the homomorphic cluster approximation within the coherent potential approximation [ 18 ,  19 ] ) to find the estimates for ${\lambda }_{\mathrm{\text{max}}}$ in the case of off-diagonal disorder and compare these results with the exact numerical calculations both for the Hamiltonian used in the approximate approach and for original problem (cellular automata (CA) calculations). Before analysing the disordered system, we start, however, with a trivial case of an ideal crystalline lattice in order to illuminate our approach.

3.1 Ideal lattice

In the case of an ideal crystalline lattice, the probability distribution functions are $\delta$ -functions, ${\rho }_{\varepsilon }\left({\varepsilon }_i\right)=\delta \left({\varepsilon }_i-{\varepsilon }_0\right)$ and ${\rho }_w\left(w_{ij}\right)=\delta \left(w_{ij}-w_0\right)$ , and the translationally invariant solutions of the eigenproblem are well known (see e.g. [ 20 ] ), so that the eigenvectors are the Bloch’s waves characterized by the wavevector $k$ ,

with $R_i$ being the position vector of site $i$ and the eigenvalues are

where $S_k={\sum }_j{e}^{-\mathrm{\text{i}}{kR}_{ij}}$ (the sum is taken over $j$ running over the nearest neighbours to arbitrary atom $i$ ) is the structure factor and the wavevector $k$ lies in the first Brillouin zone of the reciprocal space so that ${\lambda }_{\mathrm{\text{max}}}={\lambda }_{k=0}=-{\varepsilon }_0+Z{w}_0$ .

The Laplace-transform of the total number of infected states is then (see Eq. ( 11 )), $I\left(\lambda \right)={\left(\lambda -{\lambda }_{k=0}\right)}^{-1}$ , and thus

This means that in the dilute regime, $\eta =w_0∕{\varepsilon }_0\ll 1$ , the exponent in Eq. ( 15 ) is negative and the total number of infected states decays exponentially with time.

Therefore, for an ideal crystalline lattice the critical value of parameter $\eta =w_0∕{\varepsilon }_0$ obtained from equation, ${\lambda }_{\mathrm{\text{max}}}=-{\varepsilon }_0+Z{w}_0=0$ , for the system described by the approximate master equation is

which gives ${\eta }_{\mathrm{\text{cryst}}}^{\ast }=0.25$ for a square lattice with nearest-neighbour interactions only. This estimate is equivalent to the standard (not self-consistent) mean-field estimate, and, as expected, is less than the true critical value, ${\eta }_{\mathrm{\text{cryst}}}\simeq 0.4122$ [ 4 ] .

The Laplace transform of the mean-squared displacement $⟨R^2\left(\lambda \right)⟩$ for ideal crystal in the dilute regime is given by the following expression (see Eq. ( 12 )), $⟨R^2\left(\lambda \right)⟩=w_0{a}^{2}Z{\left(\lambda -{\lambda }_{k=0}\right)}^{-2}$ , with $a$ being the nearest neighbour distance, and thus

It follows from Eq. ( 17 ) that the mean squared displacement increases exponentially in the active state when $\eta >{\eta }_{\mathrm{\text{cryst}}}^{\ast }$ (i.e. ${\lambda }_{\mathrm{\text{max}}}={\lambda }_{k=0}>0$ ) and exponentially decays with time in absorbing state for $\eta <{\eta }_{\mathrm{\text{cryst}}}^{\ast }$ (i.e. ${\lambda }_{\mathrm{\text{max}}}<0$ ).

3.2 Disordered lattice

The problems becomes much harder for a disordered lattice characterized by random infection and recovery rates. In order to find the time-dependence of the number of infected sites and the mean-squared displacement for the contact process we need to evaluate the configurationally averaged resolvent operator, $⟨\widehat{\mathcal{G}}⟩$ (see Eqs. ( 9 )-( 12 )). This can be done approximately for lattice models with certain types of disorder, namely, with diagonal disorder (disorder in the recovery rates, ${\varepsilon }_i$ ), off-diagonal disorder (disorder in transfer rates, $w_{ij}$ ), and for binary systems with substitutional disorder (two species of sites randomly occupy the lattice sites [ 21 ] ). One of the successful approximate analytical approaches is the self-consistent mean-field approach (coherent potential approximation, CPA) which allows the main spectral features of the disordered Hamiltonian and its eigenfunctions to be modelled (see e.g. [ 20 ] ).

The main idea of the CPA is in replacement of the disordered lattice by the ideal crystalline one which is characterized by the effective complex parameters (complex fields), e.g. by the effective recovery, $\tilde{\varepsilon }\left(\lambda \right)={\tilde{\varepsilon }}^{\prime }\left(\lambda \right)+\mathrm{\text{i}}{\tilde{\varepsilon }}^{\prime \prime }\left(\lambda \right)$ , and transmission, $\tilde{w}\left(\lambda \right)={\tilde{w}}^{\prime }\left(\lambda \right)+\mathrm{\text{i}}{\tilde{w}}^{\prime \prime }\left(\lambda \right)$ , rates which depend on the eigenvalues, $\lambda$ , of the Hamiltonian and should be found self-consistently. The self-consistency equation follows from the requirement that a single defect placed in the effective crystal does not scatter the effective crystalline eigenfunctions if averaged over disorder.

In what follows, for concreteness, we consider the case of off-diagonal disorder, when all the recovery rates are the same, $\rho \left({\varepsilon }_i\right)=\delta \left({\varepsilon }_i-{\varepsilon }_0\right)$ ,while the transfer rates are taken from a uniform (box) distribution,

where $\Delta$ is the half-width of the distribution, $0\le \Delta \le w_0$ , and the mean value ${\overline{w}}_{ij}$ coincides with the crystalline one, ${\overline{w}}_{ij}=w_0$ . The particular form of the distribution ( 18 ) is not important for the method discussed below. The conclusions are also applicable to any well behaved distribution given the mean value coincides with the value for ordered system.

The disordered Hamiltonian ( 7 ) can be conveniently rewritten in the bond representation [ 14 ] ,

where the summation is taken over all bonds $\left(ij\right)$ in the system. Such a form of the Hamiltonian allows the single non-correlated scatters (bonds) to be introduced in the absence of the on-site disorder (the homomorphic cluster approximation [ 18 ,  19 ] ). The next step is to replace the above Hamiltonian with the effective non-Hermitian one,

where the effective fields $\tilde{\varepsilon }$ and $\tilde{w}$ are found from the following two self-consistency equations (see Appendix  A ) [ 22 ] ,

The averaging in Eqs. ( 21 ) is performed over random values of transition rates $w_{ij}$ distributed according to the probability distribution given by Eq. ( 18 ). The effective resolvent (Green’s function) elements ${\mathcal{G}}_{ii}$ and ${\mathcal{G}}_{ij}$ can be expressed via the ideal crystalline resolvent elements, ${\mathcal{G}}_{ii}^{\mathrm{\text{cryst}}}$ , of complex argument (see Appendix  A ),

which are well-known for the square lattice (see e.g. [ 20 ] ).

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-neighbour 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 ${\varepsilon }_0=30$ . The exact spectrum was obtained numerically using the kernel polynomial method [ 23 ] for a model of $N=2000\times 2000$ sites. The spectrum of the crystalline Hamiltonian with all nearest-neighbour interactions ( $\Delta =0$ ) is shown by the dot-dashed line. The inset magnifies the spectrum around the top of the band.

The self-consistency equations ( 21 ) can be solved numerically and thus both effective fields can be found. Once the complex effective fields, $\tilde{\varepsilon }\left(\lambda \right)$ and $\tilde{w}\left(\lambda \right)$ , are known then the spectrum of the effective Hamiltonian can be found (see Fig.  1 ) enabling the dynamics of the system in the dilute regime within the self-consistent mean-field approach to be studied.

It can be shown that the total number of infected states and the mean-squared displacement in the CPA approximation obey the following equations (see Appendix  B ),

and

with the effective dispersion law,

The integration in Eqs. ( 23 ) and ( 24 ) is performed over the band(s) of eigenvalues, ${\lambda }_{\mathrm{\text{min}}}\left(\Delta \right)\le \lambda \le {\lambda }_{\mathrm{\text{max}}}\left(\Delta \right)$ , where the imaginary parts of the effective fields are finite for $\Delta >0$ .

As follows from Eqs. ( 23 ) and ( 24 ) the long-time dynamics of the system both for the number of infected nodes $I\left(t\right)$ and for mean-squared displacement $⟨R^2\left(t\right)⟩$ are defined by the largest eigenvalue. The upper band edge, ${\lambda }_{\mathrm{\text{max}}}\left(\Delta \right)$ , can be found within the CPA from the self-consistency Eqs. ( 21 ) by solving them for $\lambda >{\lambda }_{\mathrm{\text{max}}}\left(\Delta \right)$ where both effective fields are real, i.e. $F_{\pm }\left({\tilde{w}}^{\prime },{\tilde{\varepsilon }}^{\prime },\lambda \right)=0$ with $F_{\pm }$ standing for the left hand-side of Eqs. ( 21 ). The analysis of the dependencies of the effective fields on $\lambda$ shows that the upper band edge corresponds to the branching point at which the following equation holds (see Appendix  C ),

Figure 2: (Color online) The dependence of the maximum eigenvalue, ${\lambda }_{\mathrm{\text{max}}}$ , evaluated using a mean-field approach (solid curve, labelled CPA), and ${\lambda }_{\mathrm{\text{max}}}^{\ast }$ 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 ${\varepsilon }_0=30$ and $w_0=1$ . The squares (labelled CA) represent the long-time