There is a purely stochastic model - a stochastic Petri net - for the SI process, which leads to the logistic equation as a result. See for example page 24 from:
* 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. Clear introduction to SI, SIR and SIRS models.