I'll try to be a bit more specific, although it's difficult since it's been a while.
The survival model is the Poisson process, which is not to be confused with the Poisson distribution.
Poission process is a discrete state, continuous time model.
Here we could have the states being the number of lives present.
The Poission distribution is the distribution of a random variable, being the number of states that you move. The number of states is discrete, and Poisson distribution is discrete random variable.
So the movement is continuous (in time), but the counting of the movements is discrete. That's the link between the two "Poisson"
The other link with the Poisson process is that the times between jumps is an exponential distribution (and each time is independant).
I'm sure there is something in the notes about the poisson process. Have a read of that (including the maths involved), then try to apply such a model to lives, thinking about how to measure the Exposed to Risk.