| Popis: |
In this article, we study an interacting particle system in the context of epidemiology where the individuals (particles) are characterized by their position and infection state. We begin with a description at the microscopic level where the displacement of individuals is driven by mean field interactions and state-dependent diffusion, whereas the epidemiological dynamic is described by the Poisson processes with an infection rate based on the distribution of other nearby individuals, also of the mean-field type. Then under suitable assumptions, a form of law of large numbers has been established to show that the associated empirical measure to the above system converges to the law of the unique solution of a nonlinear McKean-Vlasov equation. As a natural follow-up question, we study the fluctuation of this stochastic system around its limit. We prove that this fluctuation process converges to a limit process, which can be characterized as the unique solution of a linear stochastic PDE. Unlike the existing literature using a coupling approach to prove the central limit theorem for interacting particle systems, the main idea in our proof is to use a semigroup formalism and some appropriate estimates to directly study the linearized evolution equation of the fluctuation process in a suitable weighted Sobolev space and follows a Hilbertian approach. |
