Behavior of susceptible-infected-susceptible epidemics on heterogeneous networks with saturation

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