This is a whole class of social science research! The Granovetter Model, Lyapunov Functions, Markov Processes, etc…. None of these really give us inferential power – that is, the spectrum of ability to make inference – they just help us refine the logic of our assumptions. We won’t know if our model is misbehaving if we only name its formal properties and don’t go out and test it. I prefer the empirical commitment of science using statistical frameworks because a statistical test is a pointer towards more empiricism. I don’t think mathematical modelling will get us to understand deeper behavior of systems.
However, to “problematize” this issue a bit, I can think of many justifications for mathematical models not reliant upon statistical inference. Last week, I spoke with an MSU Mathematics major who fielded a job offer from Disneyland. This person specialized as an undergraduate in stochastic processes with a focus on queuing theory. This seems reasonable because your model is based on a game that people ‘subscribe’ to, i.e. waiting in line for a ride. I’m sure the model worked just fine for this system.