  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.
