  In this paper we establish a connection between epidemic models on random
networks with general infection times considered in Barbour and Reinert 2013
and first passage percolation. Using techniques developed in Bhamidi, van der
Hofstad, Hooghiemstra 2012, when each vertex has infinite contagious periods,
we extend results on the epidemic curve in Barbour Reinert 2013 from bounded
degree graphs to general sparse random graphs with degrees having finite third
moments as the number of vertices tends to infinity. We also study the epidemic
trail between the source and typical vertices in the graph. This connection to
first passage percolation can be also be used to study epidemic models with
general contagious periods as in Barbour Reinert 2013 without bounded degree
assumptions.
