poisson proces on random graph

hi,

i try to simulate Infection process on graph.

at first, i did it by simulate "day by day" process, while in evry step i run on all the infections nodes and checked who is recover and who infections new nodes, based on the number of his Infection susceptible neighboors.

now i been told that its better to simulate it with poisson process method:

my question is how can i simulate poisson process. based on the fact that each node of the graph has differnet number of neighboors, and the number of susceptible nodes also change by time.

also, i want the infection parametrs to be able to change all time.

thank you