1. My guess is that the parameter 's' is a position value on a path length and 'time' is of course time. The exponential decay function is used to represent how the heat load moves along the path length (s) as time (time) progresses.
2. This is probably due to the function you've defined. I'm not sure what the solution is, but off the top of my head my thought is using a conditional statement to control a scalier each heat load function that controls when it's "on" (ie 0 or 1).