Evaluating Coarse-graining of the SIR Model Across Spatial Scales

The COVID-19 pandemic underscored the critical need to model the physical and biological processes underlying disease transmission to inform effective public health measures. Traditional epidemiological models, such as the Susceptible-Infectious-Recovered (SIR) framework, often assume homogeneous mixing of individuals—a premise that may hold at local scales but breaks down across larger regions due to complex dynamics like human mobility and spatial heterogeneity. This study examines how spatial scale affects the validity of SIR model parameters, focusing on the basic reproduction number (R_0) and its time-varying counterpart (R(t)). We show that in well-mixed networks, coarse-grained models retain parameters from finer scales, while in networks with delayed propagation, they exhibit flattened epidemic curves and obscured local parameters. Using early COVID-19 data from U.S. counties and states, we observe significant discrepancies in (R(t)) curves between scales due to asynchronous outbreaks. We further examine the relationship between the number of Metropolitan Statistical Areas (MSAs) within a state and the number of epidemic clusters (as indicated by optimal lags). Our findings reveal that a strong correlation between MSA count and epidemic clusters indicates significant spatial heterogeneity, where coarse-graining could oversimplify dynamics. We emphasize the need for adaptive models that adjust spatial scales according to the epidemic phase to improve accuracy in parameter estimation and public health interventions.