Mean field approximation of imprecise population processes

We consider stochastic population processes in presence of uncertainty, originating from lack of knowledge of parameters or by unpredictable eects of the environment. We set up a formal framework for imprecise population processes, where some parameters are allowed to vary in time within a given domain, but with no further constraint. We then consider the limit behaviour of these systems for an innite population size, proving it is given by a dierential inclusion constructed from the (imprecise) drift. We discuss also the steady state behaviour of such a mean eld approximation. Finally, we discuss dierent approaches to compute bounds of the so-obtained dierential inclusions, proposing an eective control-theoretic method based on Pontryagin principle for transient bounds. In the paper, we discuss separately the simpler case of models with a more constrained form of imprecision, in which lack of knowledge on parameter values allows us to assess that they belong to a given interval, albeit being constant in time. Such uncertain population models are amenable of simpler forms of analysis. The theoretical results are accompanied by an in-depth analysis of a simple epidemic model.