  We couple a multi-type stochastic epidemic process with a directed random
graph, where edges have random lengths. This random graph representation is
used to characterise the fractions of individuals infected by the different
types of vertices among all infected individuals in the large population limit.
For this characterisation we rely on theory of multi-type real-time branching
processes. We identify a special case of the two-type model, in which the
fraction of individuals of a certain type infected by individuals of the same
type, is maximised among all two-type epidemics approximated by branching
processes with the same mean offspring matrix.
