Imagine I have an array of sites ($1\leq i \leq N$) that "activate" at given activating rates $\{a_i\}$, so that the time $t_i$ it takes for a site $i$ to activate follows an exponential distribution with parameter $a_i$, that is, $t_i\sim \text{Exp}(a_i)$. Once a site is activated, it starts two activating waves in either direction, passively activating neighbour sites at a speed $v$, so that the time it takes for this wave to activate $k$ sites follows $\text{Erlang}(k,v)$. Furthermore, once a site is activated, it stays activated (similar to crystallization or unzipping models).
Now imagine that, rather than having discrete sites, I have continuous space. I can then define the activation rate $A(x)$ as the number of times a site at $x$ would activate per time per unactivated length (meaning that the site has not been passively activated)
My question: What is the relation between $\{a_i\}$ and $A(x)$? In other words, how can the discrete case be obtained from the continuous one?
My attempt: Following this answer, we might think of $A(x)$ as an activation rate density (with dimension $L^{-1} T^{-1}$). Is it then as simple as $A(x)=\sum_{i \in I} \delta(x-y) a_i$, for a set of indexes $I$?