  A two-state epidemic model in networks with links mimicking two kinds of
relationships between connected nodes is introduced. Links of weights w1 and w0
occur with probabilities p and 1-p, respectively. The fraction of infected
nodes rho(p) shows a non-monotonic behavior, with rho drops with p for small p
and increases for large p. For small to moderate w1/w0 ratios, rho(p) exhibits
a minimum that signifies an optimal suppression. For large w1/w0 ratios, the
suppression leads to an absorbing phase consisting only of healthy nodes within
a range p_L =< p =< p_R, and an active phase with mixed infected and healthy
nodes for p < p_L and p>p_R. A mean field theory that ignores spatial
correlation is shown to give qualitative agreement and capture all the key
features. A physical picture that emphasizes the intricate interplay between
infections via w0 links and within clusters formed by nodes carrying the w1
links is presented. The absorbing state at large w1/w0 ratios results when the
clusters are big enough to disrupt the spread via w0 links and yet small enough
to avoid an epidemic within the clusters. A theory that uses the possible local
environments of a node as variables is formulated. The theory gives results in
good agreement with simulation results, thereby showing the necessity of
including longer spatial correlations.
