Quantifying the impact of SARS-CoV-2 temporal vaccination trends and disparities on disease control

Abstract

SARS-CoV-2 vaccines have been distributed at unprecedented speed. Still, little is known about temporal vaccination trends, their association with socioeconomic inequality, and their consequences for disease control. Using data from 161 countries/territories and 58 states, we examined vaccination rates across high and low socioeconomic status (SES), showing that disparities in coverage exist at national and subnational levels. We also identified two distinct vaccination trends: a rapid initial rollout, quickly reaching a plateau, or sigmoidal and slow to begin. Informed by these patterns, we implemented an SES-stratified mechanistic model, finding profound differences in mortality and incidence across these two vaccination types. Timing of initial rollout affects disease outcomes more substantially than final coverage or degree of SES disparity. Unexpectedly, timing is not associated with wealth inequality or GDP per capita. While socioeconomic disparity should be addressed, accelerating initial rollout for all over focusing on increasing coverage is an accessible intervention that could minimize the burden of disease across socioeconomic groups.

Speeding up vaccine rollout for all socioeconomic groups surpasses the impact of eliminating disparity or increasing coverage.

INTRODUCTION

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is a respiratory virus first detected in China in late 2019 and declared a pandemic in March 2020. Three years later, SARS-CoV-2 is responsible for over 767 million confirmed cases and over 6.9 million deaths worldwide (1). Disparities have been found in coronavirus disease 2019 (COVID-19) outcomes and temporal case distribution across sociodemographic groups [e.g., (2–6)], as well as consequences of racial and socioeconomic disparities in infection fatality rates and all-cause mortality (4, 7, 8), COVID-19 testing rates (4, 9, 10), and in the ability to social distance and reduce mobility (3, 4, 11).

This pandemic also prompted an unprecedented global development of hundreds of vaccine candidates, with around two dozen authorized for use (12). Vaccination began in late 2020 and the number of administered vaccine doses totals 13.4 billion worldwide as of June 2023, exceeding the global population (1). Global vaccination has been estimated to have averted at least 14 million deaths (13), but there has been unequal distribution of the global vaccine supply between high-income countries and low- and middle-income countries (14, 15).

Mathematical models have been useful in understanding optimal SARS-CoV-2 vaccine prioritization strategies based on age (16) and vaccine sharing to places with lower vaccine availability (17, 18). At the same time, recent studies have highlighted the need to incorporate racial and socioeconomic aspects of host heterogeneity into the study of infectious disease dynamics (11, 19, 20) and into vaccine prioritization (21). While local heterogeneity in vaccine coverage has also recently been identified at limited time scales (22, 23), still little is known about the temporal trends of vaccination at the national and subnational levels, the potential differences across socioeconomic status (SES), and their consequences for incidence and deaths due to SARS-CoV-2 infections. To characterize the scale, persistence, and impact of SES vaccination disparities, we analyzed publicly available spatiotemporal vaccination data. We then used this information to inform a mechanistic model of SARS-CoV-2 transmission stratified by SES and investigate how vaccination timing and coverage influence COVID-19 incidence and mortality. We lastly discuss alternative intervention strategies for countries and territories with limited access to the vaccine.

RESULTS

Global vaccine coverage and socioeconomic disparity

We analyzed data on vaccination and gross domestic product (GDP) per capita for 160 countries and territories, finding a statistically significant positive association when comparing GDP per capita and the percentage of the population vaccinated against SARS-CoV-2 with at least one dose (Fig. 1A). As of June 2022, 97 of these countries (60%) have at least 50% of their population vaccinated, a finding that is consistent with known disparities in vaccine distribution across country income levels (14, 15). When looking at the subnational temporal data across 81 countries and states with more than 50% vaccination coverage (Fig. 1B), we found that high SES groups are experiencing higher rates of vaccination than low SES in 83% of the places analyzed (Fig. 2 and fig. S1), a disparity that persists over time in the majority of cases (figs. S2 and S3).

Fig. 1. SARS-CoV-2 vaccine coverage.

(A) Association between population vaccinated with at least one dose and GDP per capita for 160 countries and territories, computed with Pearson’s correlation coefficient. Vaccine and population data were compiled by the Financial Times as of 22 June 2022 (42). GDP data are sourced from World Bank (43) and PCBS (44). (B) Subnational vaccination data resolution. On the basis of publicly available data for places with 50% or more of their population vaccinated with at least one dose, we identified seven countries/territories with vaccination rates for 100 or more locations, 17 for 21 to 99, and 10 for 12 to 20 locations. Spatial resolutions for all countries analyzed are included in table S1. We excluded 78 countries that have greater than 50% vaccination coverage due to a lack of high-resolution spatial data, which are referenced as “not included” in the legend. Details of the data sources and dates of data collection are reported in tables S2 and S3, respectively.

Fig. 2. Subnational SARS-CoV-2 vaccination over time.

Observed (points) and predicted (lines) temporal trends of first-dose vaccine coverage at the subnational level for high (solid) and low (dashed) SES groups in 81 places, spanning 26 countries from 5 continents, 33 states of the United States, and 24 states of Brazil. Week 0 is the first week available in the data. SES groups represent subnational administrative units, which have been aggregated into quantiles by SES. Remaining nine places are analyzed in fig. S1, and full list of excluding factors are described in the Supplementary Text.

We further analyzed the degree of disparity using two metrics, the ratio and the difference, for the average, the maximum, and the percentage of the population vaccinated at the final week (figs. S4 and S5 and table S4). Across these six metrics, we found that California (United States, CA), Florida (United States, FL), Colombia, Malaysia, and Israel are places that have the highest disparity, while Argentina, Amapa (Brazil, AP), Republic of Korea, and Finland have some of the least disparity observed. We also identified places with disparities that favor low over high SES groups, as is the case of Amazonas (Brazil, AM) and Nevada (United States, NV). To evaluate possible biases of the quantile-based analysis due to the relative inequality within each location, we analyzed the degree of disparity as a function of the spatial resolution of the quantile-based analysis, and found no association (fig. S6).

We also estimated the Lorenz curves for 83 places, their corresponding Gini coefficient of vaccine inequality, and compared it with publicly reported Gini indices of income inequality. These analyses show that vaccines are unequally distributed, with a wide range of Gini values (fig. S7), and that Gini coefficients of vaccine disparity have a statistically significant association with the Gini index of income inequality and with the quantile-based analysis (figs. S8 to S10), suggesting that socioeconomic inequality is a likely driver of vaccine inequity across countries/states.

Distinct vaccination temporal trends

Inspired by a method traditionally used in ecology (24), enzyme kinetics (25), and life expectancy (26), we fit a functional response f(t) to the percentage of the population vaccinated over time t (Eq. 1) to obtain three parameters: the maximum percentage of the population vaccinated (V_m), the week at which half of the vaccination potential is reached (W_h), and a standard parameter that relates to the shape of the temporal trends (k). Our results not only show that the fitting successfully captured the vaccination trends in most places, seen by comparing fit lines to the observed points in Fig. 2, but we also identified distinct temporal dynamics that vary in shape from a very fast initial rollout, as is the case for Cape Verde, to sigmoidal trends, like those observed in Malaysia. Using a clustering algorithm, we identified two types of dynamics shown in Fig. 3A: a rapid initial rollout, and then slowing down until reaching a steady state (concave type), and a slow beginning translated into several weeks of delay in reaching half of the vaccination potential, but with the potential to reach a higher percentage of the population vaccinated (sigmoidal type).

Fig. 3. Characterization of subnational vaccination trends.

Parameters from the functional response fitting and the two temporal trends identified using the Gaussian mixture model (45). (A) Vaccination trends over time for concave (teal) and sigmoidal (purple) types. The percentage of the cumulative population vaccinated over time for each type and SES (low in dashed line and high in solid line) was calculated using the average parameters for each type based on the classification of the 81 locations shown in Fig. 2. Parameters for each place can be found in table S5. (B) Values for low and high SES of the shape parameter k, the week at which half of the vaccination potential is reached W_h, and of the potential maximum percentage of the population vaccinated V_m, colored by the two temporal trends. Average parameters are shown with cross icons. Clustering is still apparent when displaying the United States and Brazil as countries rather than states (fig. S11).

While the cluster algorithm is informed by all three parameters, we found that the values of k are the most segregated into the two types of temporal trends identified (Fig. 3B). The concave type includes values of k with ranges of 0.8 to 2.7, while the range for the sigmoidal type has ranges of 2.4 to 5.5. While the distribution of W_h has some degree of overlap, the countries/states classified as sigmoidal types tend to have, on average, a halfway-week falling 8 weeks later than for the concave types, with little difference among high and low SES within each type. In contrast, the potential population vaccinated V_m shows a less clear partitioning, with average values of 75, 82, 82, and 90%, for concave low SES, concave high SES, sigmoidal low SES, and sigmoidal high SES, respectively. When looking at the association of these two temporal classifications with GDP per capita and the Gini coefficient, the maximum vaccinated V_m had a statistically significant association with wealth inequality (fig. S12). Unexpectedly, when comparing both the timing W_h and shape (given by the parameter k) to GDP per capita and the Gini coefficient, we found no statistically significant relationship (fig. S12). This suggests that while overall vaccine coverage depends on socioeconomic disparity, the initial speed and shape of the vaccination temporal trends are independent of it.

Impact of SES and vaccination temporal trends on disease transmission

We used the outputs of the functional response fitting to calculate an average daily vaccination rate, finding that under the concave type, the daily vaccination rate peaks around 3 months earlier than the sigmoidal type (Fig. 4A and fig. S13). We then implemented these average daily rates in an SES-stratified SEIR model, with the vaccine intervention starting immediately or at month 7. If vaccination is started immediately, there is a larger impact than if started later at 7 months, and a concave strategy is optimal for the first 12 months, until cases begin to rise (fig. S14B), likely due to the waning of immunity. Still, deaths remain lower in the concave type (Fig. 4). Since the vaccines for SARS-CoV-2 were not available until several months into the pandemic, we consider a scenario where vaccination starts at 7 months to be more realistic, in which a faster initial rollout (concave type) also reduces the incidence and mortality the most (Fig. 4C and fig. S14C). For the remaining analyses, we assume the latter scenario.

Fig. 4. Effect of vaccination scenarios on mortality.

(A) Average daily vaccination rates for each type and SES. Parameters for the sigmoidal type are k = 3.23 (averaged across SES), V_m = 89% and W_h = 23 weeks for high SES, and V_m = 82.5% and W_h = 24 weeks for low SES. Parameters for the concave type are k = 1.58 (averaged across SES), V_m = 82.2% and W_h = 14 weeks for high SES, and V_m = 75.5% and W_h = 15 weeks for low SES. Per-country rates are shown in fig. S13. (B and C) Cumulative deaths over time with vaccination starting immediately or after 7 months (t = 211 days). (D) Cumulative deaths after 14 months, with vaccination starting after 7 months, under no vaccination and four scenarios of socioeconomic disparity. (E) Deaths averted by vaccination. Keeping k = 1.58 for concave and k = 3.23 for sigmoidal, low SES parameters are fixed at a constant disparity from high SES based on 95th percentiles of the observed data (ΔV_m = 21%, ΔW_h = 6 weeks, worst-case disparity), or low SES parameters equal to high SES (best-case equity). Effect of varying parameter k can be found in fig. S16. (F) Difference in deaths averted under the best-case equity scenario and worst-case disparity, for concave and sigmoidal types. (G) Difference in deaths averted under concave versus sigmoidal types, for both best-case equity and worst-case disparity. Crosses denote average high SES parameters and country-specific parameters for 77 high SES places are shown with dots. Model parameters are listed in table S6.

We then examined concave and sigmoidal vaccination strategies under four different scenarios of disparities against a no-vaccination scenario considering: (i) the average parameters for each socioeconomic group (disparity: average), (ii) an extreme case of disparity between high and low SES (disparity: worst case), (iii) the average parameters disregarding socioeconomic group (equity: average), and (iv) having the low SES group be vaccinated at the same rate as the high SES group (equity: best case). We found that a substantial percentage of deaths could be averted when compared to a no-vaccination case, by moving from the worst-case disparity to any other scenario (Fig. 4D), with a reduction of up to 16% in the “equity: best case” strategy. Moreover, we observed that a faster initial rollout is always better to avert deaths, reflected in an additional 25 to 27% of deaths averted when moving from sigmoidal to concave type. This effect is slightly weaker when considering cases averted, with 24 to 25% of cases averted (fig. S14D).

These results suggest that both the temporal trend and the disparity can have a considerable impact on the outcomes. However, the effects of changes in individual parameters cannot be isolated. To this end, we independently varied the halfway week of vaccination for high SES from 5 to 30 weeks and the maximum potential vaccinated from 50 to 100%, fixing a constant disparity in the parameters for the “disparity: worst-case” and “equity: best case” scenarios (Fig. 4E). Intuitively, no matter the type of concave or sigmoidal dynamics, having a higher potential of people vaccinated with an earlier halfway week is always better for both deaths (Fig. 4E) and cases (fig. S14E). However, the halfway week has a much stronger effect on the outcomes than the maximum vaccinated. For instance, if we analyze the equity scenario for both the halfway week and maximum vaccinated (top-left corner in Fig. 4E), the deaths averted could be as low as 37% (sigmoidal) or 59% (concave) if the halfway week is 30, even if the maximum vaccinated is 100%, compared to deaths averted reaching values of 83% (sigmoidal) or 78% (concave) if the halfway week is 5 and only 50% of population is potentially vaccinated. When comparing across types, the sigmoidal dynamics are more sensitive to the halfway week, where the outcomes are worst at higher values of W_h compared to the concave type. This is illustrated in the 21% (disparity: worst case) to 22% (equity: best case) difference in deaths averted when the maximum vaccinated is 100% and half of this point is reached by week 30. The association of a lower halfway week in countries/states classified as concave may contribute to why concave dynamics are optimal in Fig. 4D. When we estimated the deaths averted in an “equity: best case” scenario, using the true halfway week and maximum vaccinated for each country (illustrated by the white dots, Fig. 4E), we found that the majority of places with concave dynamics would lead to 60 to 90% of deaths averted, compared to a majority ranging from 40 to 70% in the case of sigmoidal trends.

To further understand the impact of socioeconomic disparity, we compared the outcomes of the best-case equity and worst-case disparity scenarios, at a given set of values for the maximum vaccinated and halfway week (Fig. 4F). We found that the difference is minimal under a fast initial rollout (W_h = 5) and full coverage (V_m = 100%), reflected in only a 7 and 6% difference in the deaths between best-case equity and worst-case disparity, for concave and sigmoidal, respectively. Counterintuitively, a low coverage (V_m = 50 to 70%) together with a fast rollout (W_h = 5 to 16) could generate the biggest gap (difference > 20%) between equity and disparity scenarios. However, this scenario seems to be unlikely as only the observed parameters from North Dakota (US) would fall in this region. These analyses also show that while the values obtained from fitting the data mostly fall in areas of the parameter space with low differences between both scenarios, socioeconomic disparity could lead to higher differences in the deaths averted under the sigmoidal type, reflected in 59% of the observed parameter combinations falling in areas with a difference greater than 15%, compared to 53% for the concave type.

Last, when comparing across the two types, we observed that the concave dynamics are optimal for most of the parameter space, independent of the socioeconomic disparity (Fig. 4G), and with the biggest difference observed (22%) when the halfway week is high (30 weeks) and proportion vaccinated is maximized at 100%. However, if the initial rollout is unusually fast (e.g., W_h = 5 weeks), sigmoidal becomes optimal. Under these extreme conditions, sigmoidal and concave trends are very similar up to the halfway week, when sigmoidal rapidly overtakes concave and reaches the potential maximum vaccinated relatively fast (fig. S15). This further supports that a fast rollout in the early weeks of vaccination drives a higher percentage of cases averted. Still, the area of the parameter space where sigmoidal is optimal results in less than a 5% difference in the deaths averted, indicating that the effect is weak, especially when compared to the impact of the concave type, which has several observed parameter combinations that fall in areas with differences greater than 15% (Fig. 4G). Similar results are shown for cases averted (fig. S14G).

DISCUSSION

While the impact of SARS-CoV-2 vaccination on disease outcomes has been previously studied [e.g., (13, 16, 17)], less understood has been the extent of heterogeneity in vaccination temporal trends and its consequences for disease outcomes, especially in the context of socioeconomic disparities. By analyzing publicly available data at the national and subnational level and incorporating this information into a SES-stratified mechanistic model, we showed that the timing and shape of vaccination uptake are the most influential factors in disease outcomes, even more than mitigating disparity or increasing overall coverage. We also found that the initial vaccination speed is not associated with SES, unlike total coverage. In addition to our main findings, the implementation of a modeling approach that portrays the underlying causes of inequity in transmission as a central mechanism—by incorporating varying individual levels of vulnerability based on SES-specific values for the contact rate, infection fatality rate, and vaccination rate—partially addresses the imperative to include social aspects into infectious disease dynamics (20). Last, because we have thoroughly characterized the temporal dynamics of SARS-CoV-2 vaccination, the functional response used here can be applied in future work to understand disparities beyond the realm of socioeconomic inequality.

Our results are in line with previous studies that have shown socioeconomic and racial disparities in SARS-CoV-2 incidence (4, 19, 27) and vaccination in a few places at the subnational (22, 23) and global (14, 18) levels. Although this work does not investigate the precise causal pathways responsible for socioeconomic disparity in vaccination, previous studies suggest potential mechanisms, including low SES being linked to general vaccine hesitancy or refusal (28), lower use of preventive health care services (29), and not seeking medical care due to limited transportation access (30). Still, when we interpret our results in the context of fundamental cause theory (31), which anticipates that individuals with lower SES may encounter limited resources in terms of knowledge, money, power, prestige, and beneficial social connections, it is clear that the observed temporal vaccination trends contribute to the production of inequality between socioeconomic groups, supporting the hypothesis that the emergence of new innovations can exacerbate inequity over time (32). However, it may be challenging to completely close the socioeconomic divide in vaccination due to educational, political, cultural, and/or economic factors, especially when considering the finding that vaccine coverage and disparity in distribution are associated with wealth inequality.

Notably, we found that the timing and shape of rollout, the two most important factors deciding the burden of mortality, are not associated with SES. Given that previous studies have mainly focused on overall coverage and vaccine hesitancy, rather than early uptake [e.g., (33–35)], these findings suggest that timing deserves further consideration when studying social and demographic heterogeneity. While it is intuitive that a faster vaccine rollout leads to a lower burden of disease, our results suggest that modifying the distribution and timing of early vaccine rollout for all groups can effectively counterbalance the negative effects of vaccine inequity. This might not only reduce the disparities caused by unequal access to vaccines but also surpass the potential benefits achieved by reaching equitable distribution, resulting in reduced mortality across all groups. It is also worth noting that when it comes to reducing mortality, the timing of vaccination plays a more substantial role than the overall coverage and that stockpiling vaccines can actually be detrimental to the entire population, as previously highlighted within the framework of vaccine nationalism (17).

Our paper should be considered in light of the following limitations. First, the association between SES and vaccination can be dependent on the SES metric selected, as previously shown for influenza (36). This study is also limited by the availability of vaccine and socioeconomic data at a high spatial resolution for many countries, especially those in Africa, and thus, it is challenging to ascertain with certainty the specific outcomes at the individual level. Second, we analyzed first dose coverage but trends may differ for second dose or booster. First dose coverage may vary in quality as a long-term vaccination strategy depending on the quality of immune response (37). Third, we did not study the causes of socioeconomic heterogeneity in vaccination. Access and intent to vaccinate can vary widely across different countries, and for the case of SARS-CoV-2, it has been linked to demographic factors, including income, race, political affiliation, and education (22, 38–41). And last, our analyses did not consider age structure in the population, nonpharmaceutical interventions, or pathogen evolution, which could affect the quantification of overall burden and the intent to vaccinate.

To summarize, we have analyzed the temporal trends of vaccination at a national and subnational scale across five continents, showing not only that vaccine inequity exists within and between countries but also that the speed of the rollout can play a crucial role in the impact on disease incidence and that unexpectedly, this speed is not associated with socioeconomic metrics. Promoting faster initial vaccine uptake, rather than aiming to reach full coverage over many months, seems to be a viable alternative across varying levels of national and subnational SES and has the potential to reduce the effect of inequitable vaccination on COVID-19 outcomes for everyone.

METHODS

Data collection and processing

We collected national statistics on the percentage of the population with at least one dose for 160 countries and territories, which were compiled by the Financial Times (42), and GDP per capita for these nations from the World Bank (43) and Palestinian Central Bureau of Statistics (PCBS) (44).

We also collected publicly available from 34 countries, including 10,193 subnational locations, on vaccination, SES, and population size. We selected countries, territories, and states that have at least 12 locations with both vaccination and SES data to avoid single locations having an outsize impact on the dynamics. We excluded countries where less than 50% of the total population had received a first dose, to adequately capture the temporal variation across trends. Sources and dates of collection are listed in tables S2 and S3, respectively. Within each country and state, we grouped administrative units at the subnational level into quantiles based on measures representative of SES such as income or poverty rates (table S2). We then analyzed the vaccination temporal trends of 90 places by SES, spanning 26 countries from five continents, 34 states of the United States and 24 states of Brazil, and a range of national GDP per capita. For countries with 12 to 20 locations, we used three groups; for 21 to 99, we used five groups; for 100 or more, we used 10 groups. We then computed the average proportion vaccinated at each week, for the highest and lowest SES groups in each country. Figure-specific data inclusion and exclusion, as well as sample size, are described in table S7. Using the subnational data, we also analyzed the Lorenz curves for 83 places and estimated a corresponding Gini coefficient of vaccine inequality. We additionally collected publicly reported Gini indices of income inequality for countries/states (sources in table S8).

Some of the vaccination data have inconsistent reporting and apparent adjustments to the data over time. Issues with data reporting have been flagged by other authors studying vaccination data in the United States (23). We discarded data points where the cumulative vaccination was not monotonically increasing by applying the despike() function in R to the final aggregated data, which tracks a window of the median in the data and removes any points outside a specified deviation from that median.

Vaccination fitting and clustering

We fit a functional response f(t) to the percentage of the population vaccinated over time t (Eq. 1) for 81 places, starting at the first week of available data for each country. This allows us to quantify for each country/state the potential maximum percentage of the population vaccinated V_m, the week at which half of the vaccination potential is reached W_h, and a standard parameter k that relates to the shape of the temporal trend. To further characterize these temporal trends, we fit a finite Gaussian mixture model to the values of V_m, W_h, and k, and inferred two clusters using the function Mclust of the R package mclust (45).

$$f(t)=\frac{V_{\mathrm{m}}{t}^k}{W_{\mathrm{h}}^k+t^k}$$ (1)

We used the outputs of the functional response fitting to calculate an average daily vaccination rate (time derivative) for the concave and sigmoidal trends over a 14 month period. The daily vaccination rates, estimated as an approximation to the first derivative of the fit data, are given by the following expression

$$\frac{\mathrm{\%}\mathrm{vaccinated}-\mathrm{lag}(\mathrm{\%}\mathrm{vaccinated})}{\mathrm{week}-\mathrm{lag}(\mathrm{week})}$$ (2)

We implemented these average daily rates in an SES-stratified susceptible-exposed-infectious-recovered (SEIR) model, with the vaccine intervention starting immediately or at month 7.

Mechanistic model

We implemented a mechanistic SEIR model of transmission, varying the force of infection, fatality rate, and vaccination rate by SES. We used the functional response to estimate daily vaccination rates ν(t) for both high and low SES under two scenarios of parameter k, reflecting the average values of the two clusters identified in the results shown in Fig. 3. The following equations were implemented for the low SES group, and an identical set were implemented for high SES, but changing the parameter values

$$\begin{array}{rl}\frac{d{S}_{L1}}{dt}& =-{\mathrm{\lambda }}_L{S}_{L1}-{\mathrm{\nu }}_L(t)S_{L1}\\ \frac{d{E}_{L1}}{dt}& ={\mathrm{\lambda }}_L{S}_{L1}-\mathrm{\epsilon }{\mathrm{E}}_{L1}-{\mathrm{\nu }}_L(t)E_{L1}\\ \frac{d{I}_{L{1}_A}}{dt}& =a_1\mathrm{\epsilon }{E}_{L1}-{\mathrm{\gamma }}_A{I}_{L{1}_A}\\ \frac{d{I}_{L{1}_S}}{dt}& =(1-a_1)\mathrm{\epsilon }{E}_{L1}-{\mathrm{\gamma }}_S{I}_{L{1}_S}\\ \frac{d{R}_{L1}}{dt}& ={\mathrm{\gamma }}_A{I}_{L{1}_A}+(1-{\mathrm{\alpha }}_{L1}){\mathrm{\gamma }}_S{I}_{L{1}_S}-\mathrm{\omega }{R}_{L1}-\mathrm{\nu }(t)R_{L1}\\ \frac{d{S}_{L2}}{dt}& =\mathrm{\omega }{R}_{L1}+b\mathrm{\omega }{R}_{L2}-ss{\mathrm{\lambda }}_L{S}_{L2}-\mathrm{\nu }(t)S_{L2}\\ \frac{d{E}_{L2}}{dt}& =ss{\mathrm{\lambda }}_L{S}_{L2}-\mathrm{\epsilon }{E}_{L2}-\mathrm{\nu }(t)E_{L2}\\ \frac{d{I}_{L{2}_A}}{dt}& =a_2\mathrm{\epsilon }{E}_{L2}-{\mathrm{\gamma }}_A{I}_{L{2}_A}\\ \frac{d{I}_{L{2}_S}}{dt}& =(1-a_2)\mathrm{\epsilon }{E}_{L2}-{\mathrm{\gamma }}_S{I}_{L{2}_S}\\ \frac{d{R}_{L2}}{dt}& =\mathrm{\nu }(t)(S_{L1}+S_{L2}+E_{L1}+E_{L2}+R_{L1})+{\mathrm{\gamma }}_A{I}_{L{2}_A}\\ & +(1-{\mathrm{\alpha }}_{L2}){\mathrm{\gamma }}_S{I}_{L{2}_S}-b\mathrm{\omega }{R}_{L2}\\ \frac{d{D}_L}{dt}& ={\mathrm{\alpha }}_{L1}{\mathrm{\gamma }}_S{I}_{L{1}_S}+{\mathrm{\alpha }}_{L2}{\mathrm{\gamma }}_S{I}_{L{2}_S}\end{array}$$ (3)

In this SEIR model, individuals start in the S₁ class, and become exposed (E₁) through contact with infectious individuals. Once they become infectious, they can either develop a mild (I_1A) or severe (I_1S) disease. The force of infection weights mild and severe infections equally, based on the observation that mild infections are substantial drivers of transmission (46). Infectious individuals can move to the recovery class R₁ with rate γ, and only individuals with severe infections can die with probability α. We estimated infection fatality rate for low versus high SES as a weighted average, using the age-based estimates from Santiago, Chile (4) and United Nations population structure data for Chile (47). We further adjusted to account for the proportion of mild/asymptomatic cases in the model, since only severe cases can die. We ran simulations with a population of 10 million low SES and 10 million high SES individuals and assuming one exposed individual in each SES group at time 0.

Immunity from the primary infection wanes after 4 months, when individuals move from R₁ to S₂. In this second susceptible class, the probability of infection given contact is lowered compared to fully naive individuals by 50% (parameter ss). If infected again, then they are less infectious, less likely to be symptomatic, and have a reduced probability of death. They move to the R₂ class upon recovery, where the duration of immunity is longer, lasting 8 months. During vaccine intervention, individuals in S, E, and R categories are vaccinated at a rate ν determined by the functional response and move directly to R₂ even if they were not previously infected. We assume that individuals in the second compartments are less susceptible and less infectious, and there is a slightly higher proportion of mild infections, using the Centers for Disease Control and Prevention upper bound on asymptomatic cases (48). We scaled the infection fatality rate for the I_S2 class using estimates of vaccine efficacy and waned vaccine immunity against severe disease (49, 50).

Literature on contact patterns across socioeconomic groups is limited. However, it is known that workplace and housing environments differ for low and high SES (51, 52), SES groups tend to have more mobility within-group than outside of their group (3, 53), and low SES groups have not been as able to reduce their mobility during the pandemic [e.g., (4, 11)]. For these reasons, we assumed contact rates were asymmetric across SES groups.

The force of infection λ, defined as the infection risk per susceptible individual, is given by

$${\mathrm{\lambda }}_L=\mathrm{\mu }{c}_{LH}\left(\frac{I_{H1}+si{I}_{H2}}{P_H-D_H}\right)+\mathrm{\mu }{c}_{LL}\left(\frac{I_{L1}+si{I}_{L2}}{P_L-D_L}\right)$$ (4)

$${\mathrm{\lambda }}_H=\mathrm{\mu }{c}_{HL}\left(\frac{I_{L1}+si{I}_{L2}}{P_L-D_L}\right)+\mathrm{\mu }{c}_{HH}\left(\frac{I_{H1}+si{I}_{H2}}{P_H-D_H}\right)$$ (5)

where μ is the probability of infection given contact, c_ij is the contact rate among SES groups, and si is a scale parameter that reduces the infectiousness of individuals that are in the second infected class I₂. All parameter values are listed in table S6.

Supplementary Materials

This PDF file includes:

Supplementary Text

Figs. S1 to S18

Tables S1 to S8

References