Particle-based stochastic reaction-diffusion (PBSRD) models are a popular approach for studying biological systems involving both noise in the reaction process and diffusive transport. In this work we derive coarse-grained deterministic partial integro-differential equation (PIDE) models that provide a mean field approximation to the volume reactivity PBSRD model, a model commonly used for studying cellular processes. We formulate a weak measure-valued stochastic process (MVSP) representation for the volume reactivity PBSRD model, demonstrating for a simplified but representative system that it is consistent with the commonly used Doi Fock Space representation of the corresponding forward equation. We then prove, (a): the convergence of the general volume reactivity model MVSP to the mean field PIDEs in the large-population (i.e. thermodynamic) limit, and (b): the next order fluctuation correction to the mean field limit, which satisfies systems of stochastic PIDEs with Gaussian noise. Numerical examples are presented to illustrate how such approximations can enable the accurate estimation of higher order statistics of the underlying PBSRD model. This is joint work with Samuel Isaacson, Max Heldman, Jingwei Ma and Qianhan Liu.