We introduce a new percolation model to describe and analyze the spread of an
epidemic on a general directed and locally finite graph. We assign a
two-dimensional random weight vector to each vertex of the graph in such a way
that the weights of different vertices are i.i.d., but the two entries of the
vector assigned to a vertex need not be independent. The probability for an
edge to be open depends on the weights of its end vertices, but conditionally
on the weights, the states of the edges are independent of each other. In an
epidemiological setting, the vertices of a graph represent the individuals in a
(social) network and the edges represent the connections in the network. The
weights assigned to an individual denote its (random) infectivity and
susceptibility, respectively. We show that one can bound the percolation
probability and the expected size of the cluster of vertices that can be
reached by an open path starting at a given vertex from above and below by the
corresponding quantities for respectively independent bond and site percolation
with certain densities; this generalizes a result of Kuulasmaa. Many models in
the literature are special cases of our general model.Comment: 15 page
