# Epidemic spreading

#### Marián Boguñá and M. Ángeles Serrano

Networks are often the substrate over which spreading processes take place. Our research in this area is mainly focused on the understanding of the interplay between the network structure and spreading dynamics.

Relevant references

### Nature of the Epidemic Threshold for the Susceptible-Infected-Susceptible Dynamics in Networks

Marian Boguña, Claudio Castellano, Romualdo Pastor-Satorras,

PHYSICAL REVIEW LETTERS
(2013)

abstract

We develop an analytical approach to the susceptible-infected-susceptible epidemic model that allows us to unravel the true origin of the absence of an epidemic threshold in heterogeneous networks. We find that a delicate balance between the number of high degree nodes in the network and the topological distance between them dictates the existence or absence of such a threshold. In particular, small-world random networks with a degree distribution decaying slower than an exponential have a vanishing epidemic threshold in the thermodynamic limit.

### Epidemic spreading on interconnected networks

Saumell-Mendiola, Anna M. Angeles Serrano, Marian Boguña,

PHYSICAL REVIEW E
(2012)

abstract

Many real networks are not isolated from each other but form networks of networks, often interrelated in nontrivial ways. Here, we analyze an epidemic spreading process taking place on top of two interconnected complex networks. We develop a heterogeneous mean-field approach that allows us to calculate the conditions for the emergence of an endemic state. Interestingly, a global endemic state may arise in the coupled system even though the epidemics is not able to propagate on each network separately and even when the number of coupling connections is small. Our analytic results are successfully confronted against large-scale numerical simulations.

### Percolation and epidemic thresholds in clustered networks

M. Angeles Serrano, Marian Boguña,

PHYSICAL REVIEW LETTERS
(2006)

abstract

We develop a theoretical approach to percolation in random clustered networks. We find that, although clustering in scale-free networks can strongly affect some percolation properties, such as the size and the resilience of the giant connected component, it cannot restore a finite percolation threshold. In turn, this implies the absence of an epidemic threshold in this class of networks, thus extending this result to a wide variety of real scale-free networks which shows a high level of transitivity. Our findings are in good agreement with numerical simulations.

### Absence of epidemic threshold in scale-free networks with degree correlations

Marian Boguña, Romualdo Pastor-Satorras, A. Vespignani,

PHYSICAL REVIEW LETTERS
(2003)

abstract

Random scale-free networks have the peculiar property of being prone to the spreading of infections. Here we provide for the susceptible-infected-susceptible model an exact result showing that a scale-free degree distribution with diverging second moment is a sufficient condition to have null epidemic threshold in unstructured networks with either assortative or disassortative mixing. Degree correlations result therefore irrelevant for the epidemic spreading picture in these scale-free networks. The present result is related to the divergence of the average nearest neighbor's degree, enforced by the degree detailed balance condition.

### Epidemic spreading in complex networks with degree correlations

Marián Boguñá; Pastor-Satorras, R; Vespignani, A

STATISTICAL MECHANICS OF COMPLEX NETWORKS. LECTURE NOTES IN PHYSICS
(2003)

abstract

We review the behavior of epidemic spreading on complex networks in which there are explicit correlations among the degrees of connected vertices.

### Epidemic spreading in correlated complex networks

Marian Boguña, Romualdo Pastor-Satorras,

PHYSICAL REVIEW E
(2002)

abstract

We study a dynamical model of epidemic spreading on complex networks in which there are explicit correlations among the node's connectivities. For the case of Markovian complex networks, showing only correlations between pairs of nodes, we find an epidemic threshold inversely proportional to the largest eigenvalue of the connectivity matrix that gives the average number of links, which from a node with connectivity k go to nodes with connectivity k'. Numerical simulations on a correlated growing network model provide support for our conclusions.