  We investigate saturation effects in susceptible-infected-susceptible (SIS)
models 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 their connectivity, we introduce a
saturation function $C(k)$ 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 $2<\gamma \leq 3$. 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.
