## Behavior of $SIS$ epidemics on heterogeneous networks with saturation

### Authors: Jaewook Joo and Joel Lebowitz

ABSTRACT
We investigate saturation effects in susceptible-infected-susceptible (SIS) mode ls of the spread of epidemics in heterogeneous populations. The structure of interactions in the population is represented by networks with connectivity distribution $P(k)$, including scale-free (SF) networks with power law distributions $P(k)\sim k^{-\gamma}$. Considering cases where the transmission of infection between nodes depends on t heir connectivity, we introduce a saturation function $C(k)$ which reduces the i nfection transmission rate $\lambda$ across an edge going from a node with high connectivity $k$. A mean field approximation with the neglect of degree-degree c orrelation then leads to a finite threshold $\lambda_{c}>0$ for SF networks with $2<\gamma \leq 3$.

We also find, in this approximation, the fraction of infected individuals among those with degree $k$ for $\lambda$ close to $\lambda_{c}$.

We investigate via computer simulation the contact process on a heterogeneous regular lattice and compare the results with those obtained from mean field theory with and without neglect of degree-degree correlations.

Paper Available at: ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/2004/2004-14.ps.gz