  A broad class of systems, including ecological, epidemiological, and
sociological ones, are characterized by populations of individuals assigned to
specific categories, e.g., a chemical species, an opinion or an epidemic state,
that are modeled as compartments. Due to interactions and intrinsic dynamics,
individuals are allowed to change category, leading to concentrations varying
over time with complex behavior, typical of reaction-diffusion systems. While
compartmental modeling provides a powerful framework for studying the dynamics
of such populations and describe the spatiotemporal evolution of a system, it
mostly relies on deterministic mean-field descriptions to deal with systems
with many degrees of freedom. Here, we propose a method to alleviate some of
the limitations of compartmental models by capitalizing on tools originating
from quantum physics to systematically reduce multi-dimensional systems to an
effective one-dimensional representation. Using this reduced system, we are
able to not only investigate the mean-field dynamics and their critical
behavior, but we can additionally study stochastic representations that capture
fundamental features of the system. We demonstrate the validity of our
formalism by studying the critical behavior of models widely adopted to study
epidemic, ecological and economic systems.
