Analysis of metapopulation models of the transmission of SARS-CoV-2 in the United States

During the COVID-19 pandemic, renewal equation estimates of time-varying effective reproduction numbers were useful to policymakers in evaluating the need for and impact of mitigation measures. Our objective here is to illustrate the utility of mechanistic expressions for the basic and effective (or intrinsic and realized) reproduction numbers, R-0, R-E, and related quantities derived from a Susceptible Exposed-Infectious-Removed (SEIR) model including features of COVID-19 that might affect transmission of SARS-CoV-2, including asymptomatic, pre-symptomatic, and symptomatic infections, with which people may be hospitalized. Expressions from homogeneous host population models can be analyzed to determine the effort needed to reduce R-E from R-0 to 1 and contributions of modeled mitigation measures. Our model is stratified by age, 0-4, 5-9, ..., 75+ years, and location, the 50 states plus District of Columbia. Expressions from such heterogeneous host population models include subpopulation reproduction numbers, contributions from the above-mentioned infectious states, metapopulation numbers, subpopulation contributions, and equilibrium prevalence. While the population-immunity at which R-E = 1 has captured the popular imagination, the metapopulation R-E <= 1 could be attained in an infinite number of ways even if only one intervention (e.g., vaccination) were capable of reducing R-E. However, gradients of expressions derived from heterogeneous host population models, del R-E, can be evaluated to identify optimal allocations of limited resources among subpopulations. We illustrate the utility of such analytical results by simulating two hypothetical vaccination strategies, one uniform and other indicated by del R-E, as well as the actual program estimated from one of the CDC's nationwide seroprevalence surveys conducted from mid-summer 2020 through the end of 2021.