Bayesian Inference of State-Level COVID-19 Basic Reproduction Numbers across the United States

Abstract

Although many persons in the United States have acquired immunity to COVID-19, either through vaccination or infection with SARS-CoV-2, COVID-19 will pose an ongoing threat to non-immune persons so long as disease transmission continues. We can estimate when sustained disease transmission will end in a population by calculating the population-specific basic reproduction number ℛ0, the expected number of secondary cases generated by an infected person in the absence of any interventions. The value of ℛ0 relates to a herd immunity threshold (HIT), which is given by 1-1/ℛ0. When the immune fraction of a population exceeds this threshold, sustained disease transmission becomes exponentially unlikely (barring mutations allowing SARS-CoV-2 to escape immunity). Here, we report state-level ℛ0 estimates obtained using Bayesian inference. Maximum a posteriori estimates range from 7.1 for New Jersey to 2.3 for Wyoming, indicating that disease transmission varies considerably across states and that reaching herd immunity will be more difficult in some states than others. ℛ0 estimates were obtained from compartmental models via the next-generation matrix approach after each model was parameterized using regional daily confirmed case reports of COVID-19 from 21 January 2020 to 21 June 2020. Our ℛ0 estimates characterize the infectiousness of ancestral strains, but they can be used to determine HITs for a distinct, currently dominant circulating strain, such as SARS-CoV-2 variant Delta (lineage B.1.617.2), if the relative infectiousness of the strain can be ascertained. On the basis of Delta-adjusted HITs, vaccination data, and seroprevalence survey data, we found that no state had achieved herd immunity as of 20 September 2021.

Keywords: Bayesian inference; basic reproduction number; coronavirus disease 2019 (COVID-19); herd immunity; mathematical model.

Figure 1

Bayesian predictive inferences for daily confirmed case counts of COVID-19 in (A) New Jersey (B) Wyoming (C) Florida (D) Alaska, from 21 January to 21 June 2020 (inclusive dates). The compartmental model [22] accounts for an initial social distancing period followed by $n$ additional periods. We considered $n=0, 1$ and $2$ and selected the best $n$ using the model selection procedure of Lin et al. [22]. Plus signs indicate daily case reports. The shaded region indicates the prediction uncertainty and inferred noise in the detection of new cases. The color-coded bands within the shaded region indicate the median and different credible intervals (e.g., the dark purple band corresponds to the median, the band with lightest shade of yellow corresponds to the 95% credible interval, and gradations of color between these two extremes correspond to different credible intervals, as indicated in the legend). In each panel, the vertical broken line indicates the onset time of the first social-distancing period. For states with $n=1$ (Alaska and Florida), there is an additional broken line, which indicates the onset time of the second social-distancing period. The model was used to make forecasts of new case detection for 14 days after 21 June 2020. The last prediction date was 5 July 2020.

Figure 2

Marginal posterior distributions of ${ℛ}_0$ (left panels) and HIT (right panels) for ancestral strains of SARS-CoV-2 in four US states: (A,B) New Jersey, (C,D) Wyoming, (E,F) Florida, and (G,H) Alaska. Inferences are based on daily reports of new cases from 21 January to 21 June 2020. Each ${ℛ}_0$ posterior was obtained from the corresponding marginal posterior for $\beta$ and Equation (1). Each HIT posterior was obtained from the relation $\mathrm{HIT}=1-1/{ℛ}_0$ and the corresponding marginal posterior for ${ℛ}_0$. The 95% credible intervals for ${ℛ}_0$ are as follows: (6.44, 7.67) for New Jersey, (2.26, 2.47) for Wyoming, (5.20, 6.41) for Florida, and (2.26, 2.45) for Alaska. The 95% credible intervals for the HIT estimates are as follows: (0.84, 0.87) for New Jersey, (0.56, 0.59) for Wyoming, (0.81, 0.84) for Florida, and (0.56, 0.59) for Alaska. For each panel, the endpoints of the corresponding credible interval are indicated with vertical broken lines.

Figure 3

Consistency of model-derived $\lambda$ estimates with empirical growth rates during initial exponential increase in disease incidence in (A) New Jersey, (B) Wyoming, (C) Florida, and (D) Alaska. In each panel, the initial slope of the solid curve corresponds to $\lambda$ (calculated as described in Materials and Methods), the crosses indicate empirical cumulative case counts, and the broken line is the model prediction based on MAP estimates for adjustable parameters. The solid curve is derived from the reduced model (Equations (1)–(8) in the Supplementary Materials Text S1). This curve shows cumulative case counts had there not been any interventions to limit disease transmission. As can be seen, the initial slopes of the solid and broken curves are comparable. We selected $n=0$ for New Jersey and Wyoming and $n=1$ for Florida and Alaska. Among 35 states with $n=0$, New Jersey had the largest inferred $\lambda$ value (0.45) and Wyoming had the smallest inferred $\lambda$ value (0.13). Among 15 states with $n=1$, Florida had the largest inferred value of $\lambda$ (0.39) and Alaska had the smallest inferred value of $\lambda$ (0.13). It should be noted that, in contrast with Figure 1, the y-axis here indicates cumulative (vs. daily) number of cases on a logarithmic (vs. linear) scale.

Figure 4

Onset times of COVID-19 disease transmission for ancestral strains of SARS-CoV-2 in all 50 US states. The onset time $\sigma$ is defined as the first day on which the cumulative reported case count was 200 cases or more. Dates corresponding to $\sigma$ values on the vertical axis are indicated above each bar. States are indicated using two-letter US postal service state abbreviations (https://about.usps.com/who-we-are/postal-history/state-abbreviations.pdf (accessed on 19 September 2021)).

Figure 5

MAP estimates of the basic reproduction number ${ℛ}_0$ for ancestral strains of SARS-CoV-2 in all 50 US states. The different symbols refer to different training datasets used to estimate ${ℛ}_0$. Open triangles correspond to surveillance data collected from 21 January to 21 May 2020, filled circles correspond to surveillance data collected from 21 January to 21 June 2020, and open squares correspond to surveillance data collected from 21 January to 21 July 2020. Estimates of ${ℛ}_0$ are sorted by state from largest to smallest values according to the ${ℛ}_0$ estimates derived from the surveillance data collected for 21 January to 21 June 2020. The whiskers associated with each filled circle indicate the 95% credible interval (inferred from the 5-month dataset). States are indicated using two-letter US postal service state abbreviations (https://about.usps.com/who-we-are/postal-history/state-abbreviations.pdf (accessed on 19 September 2021)).

Figure 6

Perturbation analysis using the full model of Lin et al. [22] for the states of New York (panels (A,B)) and Washington (panels (C,D)). In each panel, the black solid line represents the number of infectious persons (initially 1), the black broken line represents the threshold number of persons required for herd immunity (i.e., $S_h$), and the gray broken line represents the number of recovered persons (initially $S_h-\epsilon$, obtained as described in Results). Simulations are based on MAP estimates for model parameters obtained using surveillance data collected from 21 January to 21 June 2020.

Figure 7

Dependence of disease burden on key model parameters for the states of New York and Washington. In each panel, the solid line corresponds to New York and the broken line corresponds to Washington. In Panel (A), rate of decline in infections is plotted as a function of the mean duration of the incubation period (in days), which is obtained as $m/k_L$, where $m$ is the number of stages in the incubation period and $k_L$ characterizes disease progression from one stage to the next. We take $m=5$ as in the study of Lin et al. [22]. In Panel (B), the rate of growth in infections is plotted as a function of the relative distance from the herd immunity threshold number of persons, which is defined as $\epsilon /S_h$.

Figure 8

Percent progress toward herd immunity in each of the 50 US states. Percent progress $𝒫$ indicates the fraction of immune persons required for herd immunity. $𝒫$ was calculated using Equation (2). Black bars (Panel (A)) correspond to the first scenario (i.e., $f_r$ estimated as the number of detected cases divided by population size), gray bars (Panels (A,C)) correspond to the second scenario (i.e., $f_r$ estimated as the number of detected cases within a population divided by the population size, adjusted for lack of detection of undiagnosed SARS-CoV-2 infections), black bars (Panel (B)) correspond to the third scenario (i.e., $$f_r$$ given by seroprevalence survey results), and gray bars (Panels (B,D)) correspond to the fourth scenario (i.e., $f_r$ given by seroprevalence survey results adjusted for lack of detection of asymptomatic cases). Estimates for $𝒫$ are sorted by state from largest to smallest values according to the second scenario (Panels (A,C)) and the fourth scenario (Panels (B,D)). North Dakota was omitted from Panels (B,D) because a recent estimate of seroprevalence was not available at Ref. [25]. States are indicated using two-letter US postal service state abbreviations (https://about.usps.com/who-we-are/postal-history/state-abbreviations.pdf (accessed on 19 September 2021)).

Figure 9

Vaccine eligibility and vaccine coverage in each of the 50 US states on 20 September 2021. Purple bars correspond to vaccine coverage, i.e., the population fraction that is fully vaccinated [24]. Teal bars correspond to vaccine eligibility, i.e., the population fraction that is eligible for vaccination. We estimated the eligible population fraction as the adult fraction of the population [47], i.e., the population fraction 18 years or older. Yellow bars correspond to Delta-adjusted HIT estimates from Supplementary Materials Table S2. States are indicated using two-letter US postal service state abbreviations (https://about.usps.com/who-we-are/postal-history/state-abbreviations.pdf (accessed on 19 September 2021)).