Table 1.

Quantitative approaches to unveil the epidemic role of coupled heterogeneities.

Analytic approach	Analytic goals and challenges.	Example approaches and potential insights into coupled heterogeneities
Theoretical (Mathematical)	GOALS: Quantification of general properties (e.g., R0) derived from dynamical systems adjusted for coupled heterogeneities. Can capture stochastic variability. Closed form analytic solutions. CHALLENGES: Complexity of model limited by analytic tractability. Most past work limited to individual heterogeneities. Generalizability of group level results to individual dynamics with more complex transmission heterogeneities.	Patch and metapopulation models Assessment of spatial heterogeneities due to neighborhood and population-level couplings (e.g., movement as a function of disease severity class or infectiousness and severity class). Household models Allow heterogeneity in contact between risk categories (e.g., between household members and community members). Could allow assessment of impact of heterogeneities impacting the relative time spent with household members compared to community members. Extensions to other risk categories possible. Network models Allow assessment of coupling heterogeneities between risk factors and individual attributes of the contact network (e.g., edge/node weights).
Hybrid (Simulation)	GOALS: Quantify impact of changes in dynamic parameters (e.g., through heterogeneity or intervention) by simulating complex individual-level dynamics. Generates realizations of the modeled process allowing quantification of changes, variability, and patters due to predefined coupled heterogeneities. CHALLENGES: How to best define coupled heterogeneities. Simulation of correlated values? Functional associations? Must assess sensitivity to model assumptions and potential impact of model misspecification for the coupled heterogeneities. Impact of model assumptions may be difficult to assess. Complexities in parameter estimation to capture complex relationships Uncertainty quantification of impact of coupled heterogeneities.	Individual based models. Allow implementation of complex couplings of heterogeneities but require clear specification of the couplings and the heterogeneities (dependent on statistical/mathematical quantification of relationships between heterogeneities). Provide the largest flexibility in accounting for couplings. Allow for parameter uncertainty estimations. Coupling between contacts and other functional heterogeneities easily modeled through network models. Bayesian mechanistic inference (mechanistic TSIR models, approximate Bayesian computation). Allow combining equation-based simulations with data. Could address coupled heterogeneities through specific equations and mathematical structures or through multivariate random effects with coupling defined via the covariance between random effects.
Empirical (Statistical)	GOALS: Data-based estimation of associations between risk factors and probability of infection, adjusted for coupled heterogeneities. Model coupled heterogeneities through correlated error terms, often via random effect distributions. CHALLENGES: Repeated measures, temporal, spatial, and spatiotemporal correlations. Measurement error. Confounding factors (measured and unmeasured). Typical data (e.g., number of reported cases per day) may be insufficient to support accurate and precise estimates of the underlying structure of the coupled heterogeneities. Computational complexity. Limitations in data availability.	Structural equation models. Allow specification of families of coupled heterogeneities. May be quite difficult to fit. Hierarchical models (multilevel models, random effects). Allow multilevel impacts. Allows adjustment for repeated measures on the same individual. Allow use of random effects to assess both group level average associations and individual level variation from the group level average. Could address coupled heterogeneities through multivariate random effects with coupling defined via the covariance between random effects. Allows temporal, spatial, and spatiotemporal correlations in outputs. Measurement error models. Can allow coupled heterogeneities in risk factors through correlated measurement error models. Could be incorporated into hierarchical models above.