The $P_3$-hull number of a graph is the minimum cardinality of an infecting set of vertices that will eventually infect the entire graph under the rule that uninfected nodes become infected if two or more neighbors are infected. In this paper, we study the $P_3$-hull number for generalized Petersen graphs and a number of closely related graphs that arise from surgery or more generalized permutations. In addition, the number of components of the complement of an infecting set of minimum cardinality is calculated for the generalized Petersen graph and shown to always be $1$ or $2$. Moreover, infecting times for infecting sets of minimum cardinality are studied. Bounds are provided and complete information is given in special cases.