The dynamics of SARS-CoV-2 spread are influenced by population heterogeneity. This is especially true for herd immunity, which occurs when susceptible individuals in a population are indirectly protected from infection due to immunity in others. The herd immunity threshold (HIT) is the fraction of the population that is non-susceptible when an unmitigated epidemic reaches its peak, and estimating the HIT for SARS-CoV-2 is important for forecasting the harm associated with letting the epidemic spread in the absence of interventions (Randolph and Barreiro, 2020). A population that has reached the HIT is protected from a new epidemic occurring, until births or waning immunity reduce the proportion nonsusceptible below the HIT, but existing cases will still lead to some onward transmission as the epidemic declines and can often result in a final epidemic size that exceeds the HIT. In a population with homogeneous mixing, the HIT is 1–1/R₀, where R₀ is the basic reproduction number; this translates to an HIT of 67% using an R₀ of 3.

However, population homogeneity is an unrealistic assumption, and models incorporating heterogeneity in social exposure and infection susceptibility (defined as the probability of infection given exposure) generally result in lowered HITs (Hill and Longini, 2003; Britton et al., 2020; Gomes et al., 2020; Aguas et al., 2020; Tkachenko et al., 2020). The key idea behind these models is that subpopulations important for epidemic spread (i.e., those with substantially increased susceptibility or exposure) become infected – and thus develop immunity – early on in an epidemic’s course. Herd immunity for the population overall is then achieved earlier because once these individuals are no longer susceptible to infection, further epidemic spread is slowed.

Importantly, these models also imply that in locations where SARS-CoV-2 has spread there may be demographic subpopulations with particularly high cumulative incidences of infection due to increased exposure, susceptibility, or both. Seroprevalence studies – which characterize past exposure by identifying SARS-CoV-2 antibodies – can identify these subpopulations and are more reliable and unbiased than case data, which suffer from under-reporting and other biases (Metcalf et al., 2020). Identifying and building structured models with these groups in mind is important for understanding how variation in exposure or susceptibility and social disparities is interconnected. These models are also useful for designing interventions that can both reduce disparities and disrupt overall transmission by focusing efforts on groups most affected by high transmission rates (Wallinga et al., 2010; Bubar et al., 2021). Transmission models in this space have incorporated subpopulation structure by focusing primarily on accounting for variation in susceptibility and exposure by age (Britton et al., 2020; Davies et al., 2020; Miller et al., 2020). Supporting this approach, susceptibility to infection, contact rates, and cumulative incidence in some locations all appear to vary by age (Davies et al., 2020; Mossong et al., 2008). Nonetheless, serosurveys in Belgium, Spain, Iran, New York City (NYC), Brazil, and other places exhibit relatively low variation in seropositivity by age (Herzog et al., 2020; Pollán et al., 2020; Shakiba et al., 2020; Rosenberg et al., 2020; Hallal et al., 2020), indicating additional factors that govern transmission spread.

Substantial racial and ethnic disparities in infection rates, hospitalizations, and deaths have been characterized across the US (Chamie et al., 2020; Moore et al., 2020; Millett et al., 2020a; Pan et al., 2020; Chen and Krieger, 2020; Bassett et al., 2020; Hanage et al., 2020), but it is unclear how these heterogeneities in risk are expected to change over time and what implications – if any – they have on overall epidemic dynamics. Here, we aim to address these questions by fitting compartmental SEIR transmission models structured by race and ethnicity to seroprevalence data from NYC and Long Island (Rosenberg et al., 2020). We focus primarily on building and analyzing variable exposure models because observed disparities in infection rates in US cities are strongly attributable to differences in mobility and exposure (Chang et al., 2021; Zelner et al., 2020; Kissler et al., 2020). Because of the challenges in acquiring racial and ethnic COVID-19 data (Krieger et al., 2020b), including social contact data that can be used in transmission models, we analyze a range of model structures that are compatible with the data and assess how these assumptions affect estimates of HITs, final epidemic sizes, and longitudinal trends in risk across groups. These results highlight the importance of developing socially informed COVID-19 transmission models that incorporate patterns of epidemic spread across racial and ethnic groups.

We model the dynamics of COVID-19 infection allowing for social exposure to infection to vary across racial and ethnic groups. Models incorporating variable susceptibility to COVID-19 are commonly used when stratifying by age because children are thought to have decreased susceptibility to infection (Davies et al., 2020). Variable susceptibility to infection across racial and ethnic groups has been less well characterized, and observed disparities in infection rates can already be largely explained by differences in mobility and exposure (Chang et al., 2021; Zelner et al., 2020; Kissler et al., 2020), likely attributable to social factors such as structural racism that have put racial and ethnic minorities in disadvantaged positions (e.g., employment as frontline workers and residence in overcrowded, multigenerational homes) (Henry Akintobi et al., 2020; Thakur et al., 2020; Tai et al., 2021; Khazanchi et al., 2020). In line with the notion that variation in exposure could instead be the main driver of observed seroprevalence differences, our primary focus is on analyzing variable exposure models; we have also analyzed variable susceptibility models for comparison (see Materials and methods section).

The simplest variable exposure models assume proportionate mixing, where the contact rate between groups is set to be proportional to the total contact rates (i.e., total number of contacts per time period per individual) of the two groups (Hethcote, 1996). We fit proportionate mixing models allowing for variable contact rates across racial and ethnic demographic groups to serosurvey data collected in late April from NYC and Long Island, comprising 5946 and 2074 adults, respectively (Rosenberg et al., 2020). The serosurvey data were compatible with proportionate mixing models in which Hispanics or Latinos, non-Hispanic Black people, non-Hispanic Asians, and multiracial or other people had 2.25, 1.62, 0.86, and 1.28 times the total contact rates relative to non-Hispanic whites in NYC. Model fits to Long Island resulted in even more pronounced exposure differences because of greater between-group differences in seropositivity (e.g., the seropositivity in Hispanics or Latinos relative to non-Hispanic whites was 1.85 times higher in Long Island than in NYC). Under proportionate mixing, Hispanics or Latinos, non-Hispanic Black people, non-Hispanic Asians, and multiracial or other people had 4.31, 1.96, 0.92, and 2.48 times the fitted total contact rates relative to non-Hispanic whites in Long Island, respectively. These differences in exposure impacted herd immunity levels and final epidemic sizes relative to the homogeneous model across a range of R₀ values (Figure 1 and Figure 1—figure supplement 4); for example, for an R₀ of 3, the HIT decreases to 58% in NYC and 40% in Long Island compared to 67% under the homogeneous model. The observed contrast in HITs and final sizes between the proportionate mixing and the homogenous model is in line with theoretical derivations (Figure 1—figure supplement 3). The HIT overall is reached in this model after cumulative incidence has disproportionately increased in certain minority groups: at the HIT, 75% of Hispanics or Latinos and 63% of non-Hispanic Black people were infected compared to 46% of non-Hispanic whites in NYC, and 77% of Hispanics or Latinos and 48% of non-Hispanic Black people were infected compared to 29% of non-Hispanic whites in Long Island (Figure 2).

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The estimated total contact rate ratios indicate increased contacts for minority groups such as Hispanics or Latinos and non-Hispanic Black people, which is in line with studies using cell phone mobility data (Chang et al., 2021); however, the magnitudes of the ratios are substantially higher than one would expect given the findings from those studies. This may reflect some of the limitations of the proportionate mixing assumption, which does not allow for preferential within-group contacts and hence must fit observed seropositivity differences solely by scaling total contact rates. To address this, we augment the model by partitioning a specified fraction ${\displaystyle ϵ}$ of contacts to be exclusively within-group, with the remaining contacts distributed proportionately. This assortative mixing model captures more realistic patterns of interactions due to neighborhood structure. After fitting the models across a range of $ϵ$ values, we observed that as ${\displaystyle ϵ}$ increases, HITs and epidemic final sizes shifted higher back towards the homogeneous case (Figure 1, Figure 1—figure supplement 4); this effect was less pronounced for R₀ values close to 1. This observation can be understood by comparing the epidemic cumulative incidence trajectories (Figure 3—figure supplement 1) and next-generation matrices (Figure 2—figure supplement 5 and 6): under proportionate mixing ($ϵ=0$), lower-risk demographic groups are protected from further infection due to built-up immunity in higher-risk demographic groups, but the magnitude of this indirect protection decreases as the proportion of exclusively within-group contacts increases and groups become more isolated.

We assessed a range of values for ${\displaystyle ϵ}$ because the serosurvey data cannot be used to also fit the optimal $ϵ$ value; given limited numbers of data points, any value of ${\displaystyle ϵ}$ can fit exactly to the single seroprevalence time point we consider. To inform plausible assortativity levels, we instead used additional data on demographic population distributions from the American Community Survey US census stratified at the census block group level, which represents a small geographic area and population (Figure 2—figure supplement 7). We first calculated the exposure index, which represents the average neighborhood’s demographic composition from the perspective of an individual from a given racial or ethnic group; the proportion of contacts within group were elevated, suggesting assortativity in the census data (Supplementary file 1). To directly fit $ϵ$ with these data, we assumed proportionate mixing within census block groups and ran an iterative fitting approach to jointly fit $ϵ$ and the a_i total contact rates using both census and serosurvey data (see Materials and methods and Supplementary file 3). This approach accounts for contacts based on geographic proximity using strong assumptions on mixing patterns, but may not capture contacts in other settings, such as work, beyond one’s immediate neighborhood of residence. The models jointly fitted to serosurvey and census data indicated that 46 and 39% of contacts were exclusively within-group in NYC and Long Island, respectively. The same groups had elevated total contact rates as under proportionate mixing, but the magnitudes of differences were now lower and more concordant with reported mobility differences (Chang et al., 2021): model estimates indicated Hispanics or Latinos, non-Hispanic Black people, non-Hispanic Asians, and multiracial or other people had 1.62, 1.35, 0.90, and 1.17 times the total contact rate relative to non-Hispanic whites in NYC, respectively, and 2.60, 1.63, 0.93, and 1.90 times the total contact rate relative to non-Hispanic whites in Long Island, respectively. For an epidemic with $R_0=3$, the HITs for NYC and Long Island using these census-informed assortativity models were 61 and 45%, respectively. Similar to previous models, at the HIT, 76% of Hispanics or Latinos and 66% of non-Hispanic Black people were infected compared to 50% of non-Hispanic whites in NYC, and 81% of Hispanics or Latinos and 56% of non-Hispanic Black people were infected compared to 34% of non-Hispanic whites in Long Island (Figure 2).

Using these census-informed assortative mixing models, we then considered how the relative incidence rates of infection in demographic groups could change over the course of the epidemic. Early comparisons of infection and mortality rates have helped to identify racial and ethnic groups at high risk and the risk factors for infection (Chamie et al., 2020; Moore et al., 2020; Millett et al., 2020a; Pan et al., 2020; Chen and Krieger, 2020; Bassett et al., 2020; Hanage et al., 2020), but these studies often rely on cross-sectional snapshots of epidemiological patterns. The challenge is that these metrics can change over time: for instance, multiple studies indicate decreasing disparities in incidence rate over time across racial and ethnic groups relative to non-Hispanic whites (Krieger et al., 2020a; Van Dyke et al., 2021). The reasons for these changes are multifaceted, but even independent of the effect of interventions, behavioral changes, or differential access to testing, models of epidemic spread in structured populations imply that incidence rate ratios for high-risk groups can decrease substantially as the epidemic progresses because of depletion of susceptible individuals from these groups (Goldstein et al., 2017; Koopman et al., 1991). In line with this, we observe that instantaneous incidence rate ratios are elevated initially in high-contact groups relative to non-Hispanic whites, but this trend reverses after the epidemic has peaked and overall incidence is decreasing – a consequence of the fact that a majority of individuals have already become infected (Figure 3). Similarly, cumulative incidence ratios remain elevated in high-contact racial and ethnic groups throughout the epidemic, but the magnitude decreases as the epidemic progresses (Figure 3—figure supplement 2). Although these trajectories are not meant to be taken as predictive estimates, these results highlight the importance of controlling for dynamics-induced changes in epidemiological measures of disease burden when evaluating the impact of interventions for reducing inequities in SARS-CoV-2 infections (Kahn et al., 2020); otherwise, ineffective interventions – depending on their timing – might still be associated with declines in relative measures of disease burden.

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Finally, we assessed how robust these findings were to the impact of social distancing and other NPIs. We modeled these mitigation measures by scaling the transmission rate by a factor $\alpha$ beginning when 5% cumulative incidence in the population was reached. Setting the duration of distancing to be 50 days and allowing $\alpha$ to be either 0.3 or 0.6 (i.e., a 70% or 40% reduction in transmission rates, respectively), we assessed how the R₀ versus HIT and final epidemic size relationships changed. We found that the R₀ versus HIT relationships were similar comparing a mitigated to an unmitigated epidemic (Figure 1—figure supplement 5). In contrast, final epidemic sizes depended on the intensity of mitigation measures, though qualitative trends across models (e.g., increased assortativity leads to greater final sizes) remained true (Figure 1—figure supplement 6). To explore this further, we systematically varied $\alpha$ and the duration of NPIs while holding R₀ constant at 3. We found again that the HIT was consistent, whereas final epidemic sizes were substantially affected by the choice of mitigation parameters (Figure 1—figure supplement 7); the distribution of cumulative incidence at the point of HIT was also comparable with and without mitigation measures (Figure 2—figure supplement 8). The most stringent NPI intensities did not necessarily lead to the smallest epidemic final sizes, an idea which has been explored in studies analyzing optimal control measures (Neuwirth et al., 2020; Handel et al., 2007). Longitudinal changes in incidence rate ratios also were affected by NPIs, but qualitative trends in the ordering of racial and ethnic groups over time remained consistent (Figure 3—figure supplement 3).

Here, we explored how incorporating heterogeneity in SARS-CoV-2 spread across racial and ethnic groups could affect epidemic dynamics using deterministic transmission models. Models incorporating variable exposure generally decreased the HIT and final epidemic size, but incorporating preferential within-group contacts shifted HITs and final epidemic sizes higher, approaching the homogeneous case. Epidemiological measures of disease burden such as incidence rate ratios and cumulative incidence ratios also changed substantially over the course of the epidemic, highlighting the need to account for these trends when evaluating interventions (Kahn et al., 2020). These results illustrate the varied effects of different structured heterogeneity models, but are not meant to be best estimates given the limited seroprevalence data.

Across all model variants, the observed higher cumulative incidence among Hispanics or Latinos and non-Hispanic Black people compared to non-Hispanic whites led to estimates of higher estimated contacts relative to non-Hispanic whites, mirroring existing inequities in housing, education, healthcare, and beyond (Khazanchi et al., 2020; Millett et al., 2020a; Millett et al., 2020b; Boyd et al., 2020; Bailey et al., 2021). The estimated contact rate differences also concord with reports that frontline workers, who are unable to engage in physical or social distancing to the same degree as other types of workers, are disproportionately from minority backgrounds (Blau et al., 2020; Chang et al., 2021; Kissler et al., 2020). The assortativity we observed in the census data has root causes in many areas, including residential segregation arising from a long history of discriminatory practices (Yang et al., 2020; Millett et al., 2020b; Bailey et al., 2021; Benfer et al., 2021). Projecting the epidemic forward indicated that the overall HIT was reached after cumulative incidence had increased disproportionately in minority groups, highlighting the fundamentally inequitable outcome of achieving herd immunity through infection. All of these factors underscore the fact that incorporating heterogeneity in models in a mechanism-free manner can conceal the disparities that underlie changes in epidemic final sizes and HITs. In particular, overall lower HIT and final sizes occur because certain groups suffer not only more infection than average, but more infection than under a homogeneous mixing model; incorporating heterogeneity lowers the HIT but increases it for the highest-risk groups (Figure 2).

These results also suggest that public health interventions for reducing COVID-19 inequities can have synergistic effects for controlling the overall epidemic (Richardson et al., 2020). For instance, from a transmission-control perspective, age-structured models indicate that vaccination of high-contact age groups – such as young adults – is optimal for controlling the spread of SARS-CoV-2 (Bubar et al., 2021). Similar interventions are being explored for disadvantaged populations because of how increased exposure underlies much of the higher infection risk that racial and ethnic minorities experience. For instance, Mulberry et al. used a structured SEIR model to evaluate the impact of preferentially vaccinating essential workers, finding that such a strategy was more effective in reducing morbidity, mortality, and economic cost due to COVID-19 compared to age-only vaccination prioritization strategies (Mulberry et al., 2021). Because racial and ethnic minorities are overrepresented among essential workers (Blau et al., 2020; Chang et al., 2021; Kissler et al., 2020), further modeling studies could evaluate the impact of such strategies on also reducing inequities in COVID-19 infection rates. Wrigley-Field et al. took a different methodological approach, projecting demographically stratified death rates from 2020 into 2021 assuming various vaccination strategies (Wrigley-Field et al., 2021). In line with the prior study, they found that vaccination strategies that prioritized geographic areas based on socioeconomic criteria or prior COVID mortality rates outperformed age-based strategies in reducing overall mortality and inequities in mortality. Because populations here were considered independently, these results could be extended using a transmission modeling framework, such as the one we have described, that accounts for interactions between racial and ethnic groups and thus allows for synergistic benefits from vaccinating high-risk populations. All policy proposals in this space should, of course, carefully consider the legal and ethical dimensions of vaccinations or other interventions targeted by race or ethnicity (Schmidt et al., 2020).

We note several limitations with this study. First, biases in the serosurvey sampling process can substantially affect downstream results; any conclusions drawn depend heavily on the degree to which serosurvey design and post-survey adjustments yield representative samples (Clapham et al., 2020). For instance, because the serosurvey we relied on primarily sampled people at grocery stores, there is both survival bias (cumulative incidence estimates do not account for people who have died) and ascertainment bias (undersampling of at-risk populations that are more likely to self-isolate, such as the elderly) (Rosenberg et al., 2020; Accorsi et al., 2021). These biases could affect model estimates if, for instance, the capacity to self-isolate varies by race or ethnicity – as suggested by associations of neighborhood-level mobility versus demographics (Kishore et al., 2020a; Kissler et al., 2020) – leading to an overestimate of cumulative incidence and contact rates in whites. Other sources of uncertainty, such as antibody test sensitivity and specificity, could also be incorporated into transmission models in future work (Larremore et al., 2020; Accorsi et al., 2021). Second, we have assumed that seropositivity implies complete immunity and that immunity does not wane. These are strong assumptions that can be revisited as empirical studies on the length of natural immunity are conducted. Third, we have assumed the impact of NPIs such as stay-at-home policies, closures, and the like to equally affect racial and ethnic groups. Empirical evidence suggests that during periods of lockdown certain neighborhoods that are disproportionately wealthy and white tend to show greater declines in mobility than others (Kishore et al., 2020a; Kissler et al., 2020). These simplifying assumptions were made to aid in illustrating the key findings of this model, but for more detailed predictive models, the extent to which contact rate differences change could be evaluated using longitudinal contact survey data (Feehan and Mahmud, 2020) since granular mobility data are typically not stratified by race and ethnicity due to privacy concerns (Kishore et al., 2020b). Fourth, due to data availability, we have only considered variability in exposure due to one demographic characteristic; models should ideally strive to also account for the effects of age on susceptibility and exposure within strata of race and ethnicity and other relevant demographics, such as socioeconomic status and occupation (Mulberry et al., 2021). These models could be fit using representative serological studies with detailed cross-tabulated seropositivity estimates. Finally, we have estimated model parameters using a single cross-sectional serosurvey. To improve estimates and the ability to distinguish between model structures, future studies should use longitudinal serosurveys or case data stratified by race and ethnicity and corrected for underreporting; the challenge will be ensuring that such data are systematically collected and made publicly available, which has been a persistent barrier to research efforts (Krieger et al., 2020b). Addressing these data barriers will also be key for translating these and similar models into actionable policy proposals on vaccine distribution and NPIs.

In summary, we have explored how deterministic transmission models can be extended to study the dynamics of infection in racial and ethnic groups, and how the impact of heterogeneity on the HIT and final epidemic size depends strongly on the details of how heterogeneity is modeled. We have shown that due to early infections in individuals from the most at-risk group, relative measures of incidence may decline and even reverse, but inequities in the cumulative burden of infection persist throughout the epidemic as the HIT is reached. These results describe a framework that can be extended to other cities and countries in which racial and ethnic disparities in seropositivity have been observed (Hallal et al., 2020; Flannery et al., 2020; Chan et al., 2021) and are a step towards using transmission models to design policy interventions for reducing disparities in COVID-19 and other diseases.

All data generated or analysed during this study are included in the manuscript and supporting files.

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Author details

1. Kevin C Ma

   Department of Immunology and Infectious Diseases, Harvard TH Chan School of Public Health, Boston, United States

   Contribution

   Formal analysis, Investigation, Software, Visualization, Writing – original draft, Conceptualization

   For correspondence

   kevinchenma@g.harvard.edu

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-4326-2911

2. Tigist F Menkir

   Center for Communicable Disease Dynamics, Department of Epidemiology, Harvard TH Chan School of Public Health, Boston, United States

   Contribution

   Investigation, Software, Funding acquisition, Conceptualization

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-6070-8017

3. Stephen Kissler

   Department of Immunology and Infectious Diseases, Harvard TH Chan School of Public Health, Boston, United States

   Contribution

   Formal analysis, Resources, Conceptualization

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-6000-8387

4. Yonatan H Grad

   1. Department of Immunology and Infectious Diseases, Harvard TH Chan School of Public Health, Boston, United States
   2. Division of Infectious Diseases, Brigham and Women’s Hospital and Harvard Medical School, Boston, United States

   Contribution

   Supervision, Writing – review and editing, Validation, Methodology, Conceptualization

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-5646-1314

5. Marc Lipsitch

   1. Department of Immunology and Infectious Diseases, Harvard TH Chan School of Public Health, Boston, United States
   2. Center for Communicable Disease Dynamics, Department of Epidemiology, Harvard TH Chan School of Public Health, Boston, United States

   Contribution

   Supervision, Writing – review and editing, Resources, Methodology, Conceptualization

   Competing interests

   Reviewing editor, eLife

   "This ORCID iD identifies the author of this article:" 0000-0003-1504-9213

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