Mean Field Approximation of Uncertain Stochastic Models

Abstract

Abstract-We consider stochastic models in presence of uncertainty, originating from lack of knowledge of parameters or by unpredictable effects of the environment. We focus on population processes, encompassing a large class of systems, from queueing networks to epidemic spreading. We set up a formal framework for imprecise stochastic 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 as the population size goes to infinity. We prove that this limit is given by a differential inclusion that can be constructed from the (imprecise) drift. We provide results both for the transient and the steady state behaviour. Finally, we discuss different approaches to compute bounds of the soobtained differential inclusions, proposing an effective controltheoretic method based on Pontryagin principle for transient bounds. This provides an efficient approach for the analysis and design of large-scale uncertain and imprecise stochastic models. The theoretical results are accompanied by an in-depth analysis of an epidemic model and a queueing network. These examples demonstrate the applicability of the numerical methods and the tightness of the approximation.

Over-Approximation of Fluid Models

“…For nonlinear fluid models, the proposed approach a) is efficient; b) induces bounds that can be expected to be tight and; c) allows for an algorithmic treatment in the case where the ODE system is given by multivariate polynomials, thus covering in particular biochemical models. A comparison with the state-of-the-art tool for reachability analysis CORA [35] in the context of the well-known SIRS model from epidemiology [7] confirms the potential of the proposed technique.…”

“…|u β (·)| ≤ δ β (·), |u I (·)| ≤ ε I (·)}, (7) with π u β ,u I (0) = V u β (0) and opt ∈ {inf, sup}. By interpreting the uncertainty functions u * β and u * I as optimal controls, (7) defines an optimal control problem with cost π u β ,u I B…”

“…A major result of the paper shows that uncertainty functions u * β and u * I can be efficiently computed by relying on a strict version of Pontryagin's principle which is sufficient for optimality. Apart from solving (7) exactly, this allows one to devise an efficient procedure for the formal estimation of E whose bounds can be expected to be tight.…”

“…It may however happen that an envelope does not satisfy (A2). To see this on an example, we extend Example 4 to the multi-class SIRS model [7] in which the overall population of agents is partitioned into classes, thus providing a better picture of the actual spread dynamics [38].…”

“…Unfortunately, a precise parameterization of a fluid model is often not possible due to finite-precision measurements or stochastic noise [7], [8]. Hence, in order to verify that a nonlinear fluid model satisfies some given property in the presence of parameter functions that are subject to uncertainty, it becomes necessary to estimate the reachable set.…”