The mathematical derivation of this deterministic mean-value process from the definitional properties of stochastic Petri nets - this gets into some deeper mathematical waters, which we won't cover here.
Here is a whole book devoted to the subject:
* John C. Baez and Jacob Biamonte, [Quantum techniques for stochastic mechanics](https://arxiv.org/abs/1209.3632), arXiv:1209.3632 [quant-ph]. Text includes treatment from the ground up of Petri nets, both stochastic and deterministic. Example includes SI, SIR and SIRS models.
We can look into this in another context - in the network theory study group!