longtable

# Understanding Complex Patterns in Social, Geographic, and Economic
Inequities in COVID-19 Mortality at the County Level in the US Using
Generalized Additive Models

Christian Testa1,*

(November 29, 2022)

###### Abstract

I present three types of applications of generalized additive models (GAMs) to
COVID-19 mortality rates in the US for the purpose of advancing methods to
document inequities with respect to which communities suffered
disproportionate COVID-19 mortality rates at specific times during the first
three years of the pandemic. First, GAMs can be used to describe the changing
relationship between COVID-19 mortality and county-level covariates
(sociodemographic, economic, and political metrics) over time. Second, GAMs
can be used to perform spatiotemporal smoothing that pools information over
time and space to address statistical instability due to small population
counts or stochasticity resulting in a smooth, dynamic latent risk surface
summarizing the mortality risk associated with geographic locations over time.
Third, estimation of COVID-19 mortality associations with county-level
covariates conditional on a smooth spatiotemporal risk surface allows for more
rigorous consideration of how socio-environmental contexts and policies may
have impacted COVID-19 mortality. Each of these approaches provides a valuable
perspective to documenting inequities in COVID-19 mortality by addressing the
question of which populations have suffered the worst burden of COVID-19
mortality taking into account the nonlinear spatial, temporal, and social
patterning of disease.

Abbreviations used: United States (US), Coronavirus Disease 2019 (COVID-19),
Generalized Additive Model (GAM), Centers for Disease Control and Prevention
(CDC), Index of Concentration at the Extremes (ICE), Confidence Interval (CI)

1 Department of Social and Behavioral Sciences, Harvard T.H. Chan School of
Public Health

* Correspondence: Christian Testa<EMAIL_ADDRESS>
## Introduction

As we enter the third winter with the novel coronavirus disease COVID-19 in
the United States, evidence documenting the intense disparities in COVID-19
mortality rates comparing socially advantaged and disadvantaged populations
continues to mount. Eliminating inequities in health outcomes has been stated
as a major policy goal of the Biden administration (The White House, 2021,
2022), representing a revitalized commitment to health equity and underlining
the importance of adequate data reporting systems that report timely estimates
of prevalent health inequities. To this end, I use generalized additive models
(GAMs) as a flexible regression framework to illustrate the evolving roles and
relationships sociodemographic, geographic, and economic conditions have with
respect to trends in COVID-19 mortality. The code, data, and documentation
necessary to reproduce the analyses contained in this paper are online and
free to access at https://github.com/ctesta01/covid.gradient.estimation.

#### Background

Having passed over 1 million COVID-19 deaths in the United States in 2022
(Donovan, 2022), and facing uncertain prospects for the third COVID-19 winter
looming even as new iterations on the COVID-19 vaccines become available, it
remains critical that inequities in COVID-19 outcomes are documented and
analyzed to reckon with the unjust and unfair burden of preventable illness.
Even though the first vaccines were granted emergency use authorization by the
U.S. Food and Drug Administration in 2020 (Mayo Clinic, 2022), with the first
shots going in arms in December 2020, COVID-19 is still continuing to cause
hundreds of deaths a day in the US in the fall of 2022 (“United States COVID -
Coronavirus Statistics - Worldometer,” 2022). The new bivalent vaccines
released at the end of August 2022 contain mRNA sequences from both the
original strain as well as the recently emergent BA.4 and BA.5 lineages in an
effort to make the nation’s immunity more up-to-date and robust against the
myriad of phylogenetic directions the COVID-19 virus is evolving to explore
(Office of the Commissioner, 2022). Despite the updated bivalent boosters
representing a significant step forward in prevention strategy, less than 4%
of eligible Americans had taken the booster in the first month after it became
available (Bendix, 2022; Lambert, 2022). As such, and with an enduring history
of inequities in health care access in the US (Bailey et al., 2021; Blendon et
al., 2002; Carpenter, 2021; Chrisler et al., 2016; Feldman et al., 2021;
Okonkwo et al., 2021; Ortega & Roby, 2021; Rapp et al., 2022; Whitehead et
al., 2016), it is clear that without further intervention not all communities
will be equally able to benefit from the updated vaccines and inequities in
COVID-19 illness and mortality may persist despite the technological
innovations in vaccine technology.

#### The Role of Geography in COVID-19

Figure 1: Estimates of monthly COVID-19 mortality rates per 100,000 person-
years by county organized by Census Division. For each division, the median
trendline and quantile ranges are shown weighted by county population size.

Prior literature has demonstrated that the spread and impact of COVID-19 has
varied geography over time reflecting dynamics in the timing of introduction
and transmission events. Methods employed to highlight the geographic
patterning in COVID-19 outcomes have included quantile regression (Sigler et
al., 2021), Besag-York-Molli'e mixed models (Whittle & Diaz-Artiles, 2020),
spatial cluster analysis (Sugg et al., 2021), geographically weighted
regression (Mollalo et al., 2020; Park et al., 2021), and others.

Figure 1 shows the monthly COVID-19 mortality rates for the counties grouped
within each of the nine U.S. Census Divisions (U.S. Department of Commerce
Economics and Statistics Administration & U.S. Census Bureau, 2000). The
figure summarizes each division’s median mortality rates weighted by county
population size. Notably, the mortality associated with the early surge of
cases starting in New York City and spreading through New York, New Jersey,
and Massachusetts is visible in the Middle Atlantic and New England division
figures. The figure also illustrates how the first peak in the mortality time-
series for states in the Midwest (West North Central, East North Central)
occurred later, in late 2020 and going into early 2021.

In the US context, one of the key aspects to the geographic story of
COVID-19’s spread and diffusion was the early surge of cases and epicenter in
New York City during March 2020 (Thompson, 2020) followed by subsequent waves
of cases in the South and Midwest (Glenza, 2020; Scott, 2020; Shumaker & Wu,
2020). As Park et al. stated summarized the trends in the US from March 2020
to May 2021, “hot spots have shifted from densely populated cities and the
states with a high percentage of socially vulnerable individuals to the states
with relatively relaxed social distancing requirements, and then to the states
with low vaccination rates” (2021).

When considering the drivers of the COVID-19 pandemic, it’s necessary to note
that geography and social conditions are inextricably linked. In July 2021,
the CDC reported that “the COVID-19 cumulative death rate in non-metropolitan
areas has exceeded that of metropolitan areas since December 2020,” noting
that of the approximately 1/5th of Americans who live in rural areas, many
“are considered highly vulnerable according to CDC’s Social Vulnerability
Index (SVI), which includes factors such as housing, transportation,
socioeconomic status, race, and ethnicity” (CDC, 2021). Moreover, rural
communities often have lower health insurance rates, higher disability rates,
older populations, and limited access to health care. One of CDC’s Morbidity
and Mortality Weekly Reports found that vaccination against COVID-19 was lower
in rural communities than in urban communities between December 2020 and April
2021 (Murthy, 2021).

#### The Social Determinants of COVID-19 Mortality

Even since the beginning of the COVID-19 outbreak in the US, data reflected
sharp inequities in mortality rates. During January 22nd to May 5th 2020,
county COVID-19 mortality rates were 4.94 (95% CI 4.78, 5.09) times higher in
counties in the highest quintile of percent People of Color (61%-100%)
compared to counties in the lowest quintile of percent People of Color
(0%-17.2%) (Chen & Krieger, 2021). This was not wholly unanticipated: as
COVID-19 was beginning to take off in the US, some were already calling
attention to the fact that COVID-19 threatened to exacerbate existing
disparities (Kim et al., 2020). Kim, Marrast, and Conigliaro noted at least
three structural barriers in COVID-19 prevention and care: 1) originally
requiring residents to have a doctor’s prescription for a COVID-19 test
reduced the opportunity for healthcare for People of Color as they are less
likely to have a primary care provider; 2) drive-through testing made testing
disadvantaged those who rely on public transportation; and 3) quarantining at
home while waiting the 7-10 days originally required for COVID-19 test results
to come back posed an economic and social challenge that many in already
financially difficult situations may not have been able to take on (2020).
Others noted yet more reasons why COVID-19 threatened to worsen an already
inequitable healthcare landscape in the US: in particular, those who reside in
prisons and jails, immigrants and undocumented people, people with
disabilities, and people experiencing homeless all face additional challenges
in seeking and getting the healthcare they deserve (Okonkwo et al., 2021).
Even though stay-at-home orders designed to mitigate spread that were
prevalent in many states (Moreland et al., 2020), workers functioning in
capacities essential to the functioning of critical infrastructure operations
(later termed “essential workers”) were exposed to heightened risk of COVID-19
transmission (Hanage et al., 2020; National Bureau of Economic Research, 2021;
National Conference of State Legislatures, 2021; The Lancet, 2020; Wei et al.,
2022). It’s clear that those with more structurally enfranchised privileges
have been more able to mitigate their risk of negative health outcomes
associated with COVID-19 — during February 1 to April 1, 2020, New York City
residents who lived in more affluent neighborhoods were more likely to have
left the city, while New York City residents from more Black and Hispanic
neighborhoods were more likely to continue working (Coven & Gupta, 2020).

> Contrary to the oft used phrase that the ‘virus does not discriminate’, the
> data presented here suggest that this virus, as many other infectious
> diseases, has the greatest implications for the most vulnerable people. The
> intersections between health and human rights are clear—the health of a
> society and vulnerability to a pandemic are directly related to its human
> rights track record for those who are marginalised. (Okonkwo et al., 2021)

When vaccines became available, vaccine appointments were often only available
to be scheduled through online web-portals contributing to the inequities
between those who had internet access and technological literacy and those who
didn’t (Press et al., 2021). Vaccination sites have not been equally
distributed and areas determined to be vaccine deserts have been found to have
disproportionately more Black and Hispanic residents (Rader et al., 2021). In
fact, healthcare facilities in counties with higher Black composition had 32%
(95% CI 14%-47%) lower odds of serving as vaccine sites (Hernandez et al.,
2022).

What vaccination has been administered hasn’t suddenly erased the unequal
burden of COVID-19 either; in August 2022 the New York Times was reporting
“Black death rates at this winter’s peak were greater than those of white
people by 34 percent in rural areas, 40 percent in small or medium cities and
57 percent in big cities and their suburbs” (Goldstein, 2022). As COVID-19
case and mortality rates have waxed and waned, the inequities have widen and
shrunken, often with racial/ethnic inequities growing during times when
COVID-19 rates have surged (Hill & Artiga, 2022). During 2022, the age-
standardized COVID-19 mortality rates for white people have, at times, been
slightly higher than those of Black and Hispanic people, predominantly because
the mortality rate among white people has increased (Johnson & Keating, 2022).
It’s important to note that white COVID-19 mortality rates overtaking the
Black mortality rates does not imply that equity has been established: neither
does this undo the cumulative impact of mortality (which has been twice as
high for Black people compared to white people (Hill & Artiga, 2022)), nor
does it imply the underlying systemic barriers to equity have been overturned
(Del Rios et al., 2022). As Del Rios, Chomilo, and Lewis note, instead,
COVID-19 leaves in its wake more years of life expectancy lost, wages lost,
and degradation of mental health in Communities of Color.

## Methods

### Data Sources

The following variables were retrieved at the county level:

  * •

Counts of COVID-19 deaths (The New York Times, 2021).

  * •

Population size estimates for 2020 from the U.S. Census (US Census Bureau,
2021).

  * •

Median age, median household income, racial/ethnic composition, population
density, percent below the federal poverty line, and number of households with
high ($100k+)/low (<$25k) household income by racial/ethnic group from the
2014-2019 5-year American Community Survey (US Census Bureau, 2020) through
the tidyverse R package (Walker & Herman, 2022).

  * •

Votes cast in the 2020 presidential election (MIT Election Data and Science
Lab, 2022)

### Generalized Additive Models

Generalized additive models (GAMs) improve upon generalized linear models by
allowing for the fitting of smooth functions that transform right-hand-side
variables. This is a convenient means to account for nonlinear relationships
between the outcome and predictor variables. Whereas a generalized linear
model may look like

$g(\mu_{i})=\beta_{0}+\beta_{1}x_{i1}+\beta_{2}x_{i2}+\beta_{3}x_{i3}...$

a generalized additive model could look like

$g(\mu_{i})=\mathbf{A}_{i}\mathbf{\theta}+f_{1}(x_{i1})+f_{2}(x_{i2})+f_{3}(x_{i3},x_{i4})+\dots$

where the expected value of the outcome is given as
$\mu_{i}\equiv\mathbb{E}(Y_{i})$, $\mathbf{A}_{i}$ is a row of the model
matrix for any strictly non-parametric model components, $\mathbf{\theta}$ is
the corresponding parameter vector, and the $f_{j}$ are smooth functions of
the covariates $x_{k}$ (S. N. Wood, 2017).

Figure 2: A demonstration of fitting regression models using B-spline basis
functions. Part A. shows the simulated dataset. Part B. shows the B-spline
basis functions. Part C. shows the weighting of the B-spline basis functions
and their weighted sum with an intercept added to create the regression model
fit to the data.

The smooth functions estimated as part of fitting a GAM are constructed using
spline basis functions. These spline basis functions allow for the smooth
interpolation of trends in the data allowing for the incorporation of
nonlinearity. To avoid overfitting the data, a penalty term is introduced that
controls the degree of “wiggliness” or smoothness and this penalty term is fit
using generalized cross validation. B-splines are one kind of spline basis
function that is commonly used and are especially popular because of their
property that they are non-negative on only a finite interval.

While a variety of spline-based approaches exist (cubic splines, B- and
P-splines, thin-plate splines, etc.), our particular application setting
warrants the use of tensor-product smooths because we require anisotropic
penalization. That is to say, when we seek to create spatiotemporally smoothed
model estimates, it’s inappropriate to assume that the amount of smoothing
across space should be the same as the amount of smoothing across time because
they’re in fundamentally different units (space being measured in Cartesian
coordinates and time being measured in years, months, days, etc.). More
details about tensor-product smooths are available in Simon Wood’s
_Generalized Additive Models - An Introduction with R_ (2017).

### Overdispersion and Event Modeling

Above and beyond using the GAM framework to allow for flexible, nonlinear
relationships between our observed county-level variables and COVID-19
mortality, we must have a model specification that agrees with the data
generating process. In our case, the data generated are counts of deaths per
population, and the class of models most suited to represent counts of events
are Poisson, quasi-Poisson, and negative binomial models. Here we’ve chosen to
use the negative binomial model as it accounts for overdispersion and has the
intuitive interpretation of a Poisson model with gamma distributed underlying
rate parameter (Gelman et al., 2013).

A negative binomial model in the context of modeling count data can be written

$y_{i}\sim\text{NegativeBinomial}(u_{i}\theta_{i},\phi),$

where $y_{i}$ are the counts observed, $u_{i}$ is the “exposure”, $\theta_{i}$
are the rates, and $\phi$ determines the amount of overdispersion. The rates
are modeled as $\theta_{i}=e^{X_{i}\beta}$ where $X_{i}$ are the observed
covariates and $\beta$ are the coefficients on the covariates corresponding to
log rate ratios. The logarithm of the exposure, $\log(u_{i})$ is often called
(and later herein referenced as) the offset. In epidemiological contexts, the
offset is often representative of the amount of person-time during which
observations were recorded. Whereas the Poisson model holds that
$\text{var}(y)=\mu$ where $\mu$ is the average rate, the negative binomial
model instead assumes that $\text{var}(y)=\mu+\mu^{2}/\phi$. The Poisson model
is a special case of the negative binomial model when $\phi\to\infty$.

An alternative and equivalent formulation of the negative binomial that is
commonly used makes the connection to the Poisson model even more clear:
instead of $\theta_{i}$ and $\phi$, using $\alpha$ and $\beta$,

$y\sim\text{NegativeBinomial}(\alpha,\beta),\text{ and}$
$\text{NegativeBinomial}(y|\alpha,\beta)=\int\text{Poisson}(y|\theta)\text{Gamma}(\theta|\alpha,\beta)d\theta.$

Introductions to and additional exposition on the negative binomial model,
especially in the context of modeling counts of events outcome data, are
available in _Bayesian Data Analysis_ and _Regression and Other Stories_
(Gelman et al., 2020, 2013).

### Variables of Interest

The following variables have been included as covariates of interest:

  * •

Median Age

  * •

Population Density per Square Mile

  * •

Median Household Income

  * •

Proportion in Poverty

  * •

the Index of Concentration at the Extremes for Racialized Economic Segregation

  * •

Political Lean in the 2020 Election (1 = 100% Republican votes, -1 = 100%
Democratic Votes)

##### Index of Concentration at the Extremes for Racialized Economic
Segregation

The Index of Concentration at the Extremes (ICE) is a measure which describes
how concentrated a given area’s population is in terms of the extreme ends of
privilege and marginalization (Krieger et al., 2016). In general, the ICE
measure is formulated as

$\text{ICE}=\frac{\text{Number of People in Most Privileged
Category}-\text{Number of People in Least Privileged Category}}{\text{Total
Population}}$

Applying the ICE approach to a specific context involves defining the axes of
privilege of interest. In this case, data on racialized economic segregation
are used from the US Census American Community Survey on white households
earning more than $100,000 a year (the most privileged group) or households of
People of Color earning less than $25,000 a year (the least privileged group).
This variable is referred to throughout as the ICE for racialized economic
segregation, or ICEraceinc in the code.

Compared with the Gini coefficient which is one of the most popular methods
for summarizing area-level rates of inequities, the ICE has the advantage that
it is suitable for describing inequities at smaller area levels (Krieger et
al., 2016). While the Gini coefficient measures within-area dissimilarity (as
in, for example, how unequal wealth is distributed within a county), the ICE
measure establishes where on a spectrum a given county’s population falls
allowing for comparison across counties. The Gini coefficient suffers from the
fact that areas which are made up of relatively homogeneous populations will
appear as having low within-area inequality (and therefore low Gini
coefficient). Instead, the ICE measures how much of the population is either
privileged or not. The Gini coefficient remains useful for reporting on the
degree of inequity in larger areal units (like countries, states, and
regions), but at smaller areal units (like counties, ZIP codes, census tracts)
can be more difficult to interpret and compare. Maps of ICE measure can
elucidate what spatial social segregation and polarization exist, and the ICE
for racialized economic segregation has been repeatedly and significantly
associated with COVID-19 outcomes (Brown et al., 2021; Chen & Krieger, 2021;
Eichenbaum & Tate, 2022; Hanage et al., 2020; Krieger et al., 2022; Saha &
Feldman, 2020; Sonderlund et al., 2022).

##### Political Lean

Political lean has been associated with COVID-19 mortality in numerous
studies, with plausible mechanisms explaining the association including
differences in non-pharmaceutical intervention uptake (mask usage, social
distancing, quarantining), differences in rates of vaccination, differences in
political leadership’s messaging, resource allocation, and the adoption of
policy interventions (Gonzalez et al., 2021; Grossman et al., 2020; Kaashoek
et al., 2021; Krieger et al., 2022; Leonhardt, 2021).

## Results

### Application 1: Non-Spatial Covariate Effects Over Time

The GAM models shown in Figure 3 are fit with the following structure using
the gam function from the mgcv package (S. Wood, 2022):

gam(

formula = deaths ~ s(median_age) + te(covariate, date, d=c(1,1)), # regression
formula

offset = log(popsize/1e5/12), # our offset represents the person-time

data = df, # our data-set of county-level observations

family = nb(link='log') # indicates negative binomial family and a log-link
function

)

The formula used puts a one-dimensional smoothing spline on median age to
represent a nonlinear age-effect and a two-dimensional tensor-product smooth
on the interaction between the given covariate and the date. The d=c(1,1)
argument provides the instruction necessary to consider covariate and date as
being on separate scales and therefore to fit the tensor-product smooth with
anisotropy — that is, to allow for independent amounts of scaling in the
dimensions of the covariate and time. The offset used structures the
regression to estimate rates in units of person-time per 100,000 person-years.
Since the death counts are aggregated to the monthly level, the person-time in
units of 100,000 person-years are calculated by taking each county’s
population size, dividing by 100,000, and dividing by 12 (for the 12 months in
a year).

The above structure is used to estimate models for our different covariate
variables of interest: median income, percent in poverty, the ICE for
racialized economic segregation, political lean. The model presenting median
age treats median age as the main covariate of interest including it as the
te(covariate, date, d=c(1,1)) and dropping the s(median_age) term which
otherwise becomes redundant. Results of these models are summarized in Figure
3 where the additional COVID-19 mortality associated with each covariate is
visualized.

Figure 3: Additional COVID-19 mortality associated with covariates over time.
A) age and time, B) log (base 10) population density and time, C) median
income and time, D) proportion in poverty and time, E) ICEraceinc and time, F)
political lean and time. In panels B-F the effect of age is marginalized out
using the median age in the US, 38.8 (US Census Bureau, 2022).

Likelihood ratio tests confirmed that models with covariates interacted with
time had significantly lower residual deviance ($p\leq 2.2e^{-16}$ for all
models) compared to models only including a spline term on median age and the
given covariate not interacted with time. The model interacting median age and
time was compared to a model with a spline for median age alone. Akaike
Information Criteria values were also lower for all models compared to models
that did not interact the covariates with time.

A more complex non-spatial application of GAMs to describe the distribution of
COVID-19 mortality in the US over time is to consider the associations of
mortality with three-way interactions of time and two covariates taken
together. In the following example, the formula used is deaths ~ te(date,
ICEraceinc, median_age, d=c(1,1,1)). Again, the d argument specifying the
marginal basis dimensions is used indicate that each of the date, ICEraceinc,
and median_age measures are in different units and should not be smoothed
assuming that a one unit difference in one variable is comparable to a one
unit difference in another variable. This approach is represented in Figure 4.

Figure 4: The changing interacted effect of the ICE for racialized economic
segregation and age over time. A) May 2020, B) November 2020, C) May 2021, D)
October 2021, E) February 2022, F) August 2022. An animated version is
available online at
https://github.com/ctesta01/covid.gradient.estimation/blob/main/analysis/05_two_variables_at_a_time/animation_ICEraceinc_age/readme.md

### Application 2: Spatiotemporal Smoothing

By fitting GAMs with a tensor-product smoothing term on latitude, longitude,
and time we can estimate a spatiotemporally smoothed trend in COVID-19
mortality. To do this, the GAM is constructed similarly as in Application 1
but with the smoothing term specified as te(latitude, longitude, time,
d=c(2,1)) where d=c(2,1) indicates that latitude and longitude share the same
dimensions (i.e., both are spatial and in units of degrees) while the time
data are in separate units. Note that results presented in Application 2 and 3
are based on county-level data from the contiguous US excluding Alaska and
Hawaii.

Using GAMs to present spatiotemporally smoothed estimates of COVID-19
mortality allows for the synthesis of trends over time. Whereas the raw rates
of COVID-19 mortality are noisy due to 72% of the land area in the US being
classified as rural and low population density (Health Resources & Services
Administration, 2022), the spatiotemporally smoothed GAM estimates use
population weighting via the offset specified to estimate a latent surface
that represents localized mortality rate averages in spatial coordinates and
in time.

Figure 5 shows the difference between crude mortality rates and spatially
smoothed mortality rates in January 2022 to illustrate the level of noise
present in raw rates and how spatially smoothing synthesizes local geographic
patterns into trends that can be meaningfully interpreted as local area
mortality risk levels with information information pooled across nearby county
rates.

Figure 6 shows the results from a spatiotemporally smoothed model in select
months to highlight how spatiotemporal smoothing can yield results that
synthesize trends in space and time.

Figure 5: Comparison of crude vs. smoothed mortality rates. A) Crude mortality
rates per 100,000 person-years in January 2022. B) Smoothed mortality rates
from a GAM applied to mortality rates from January 2022. Figure 6:
Spatiotemporally smoothed COVID-19 mortality estimates from a GAM fit to data
from March 2020-August 2022. Panels show months selected to highlight the
changing spatial patterns of COVID-19 mortality risk over time. An animated
version is available online at
https://github.com/ctesta01/covid.gradient.estimation/blob/main/analysis/09_spatiotemporal_models/animation/readme.md

### Application 3: Estimating Covariate Effects Adjusted for Spatiotemporal
Autocorrelation

The final application of GAMs presented here is to estimate the effects of
covariates after adjusting for a spatiotemporally smoothed latent risk
surface. This has the interpretation of asking what are the changes to
COVID-19 mortality rates associated with each covariate after taking into
account regional geographic patterns in COVID-19 mortality over time. In
particular, this is relevant for understanding what county-level measures are
associated with elevated COVID-19 mortality even after adjusting for where
COVID-19 rates were locally elevated or depressed due to variation in the
timing of local introduction, transmission, and diffusion.

Figure 7: The spatial autocorrelation of COVID-19 rates in January 2022. Panel
A shows the graph of US counties connected according to which are neighbors.
Panel B shows a scatterplot depicting each US counties’ COVID-19 mortality
rate (x-axis) compared to the average of its neighbors’ COVID-19 mortality
rates (y-axis). A regression line shows the association between counties’
COVID-19 rates and their neighboring counties’ COVID-19 mortality rates.
Dotted lines indicated the average for US county-level crude COVID-19
mortality rates (vertical) and for the average of each counties’ neighboring
counties’ COVID-19 mortality rates (horizontal). Figure 8: The median (black)
and quantiles (shaded bands) of temporal autocorrelation of counties’ COVID-19
mortality rates over time during April 2020 — August 2022.

To motivate the need for and application of spatiotemporal smoothing in
assessing the associated changes in COVID-19 mortality with county-level
covariates, Figure 7 presents the Moran’s I diagnostic plot for January 2022
which summarizes the amount of autocorrelation between counties’ COVID-19
mortality rates and the average of each counties’ neighboring counties’
COVID-19 mortality rates. The observed autocorrelation indicates that there is
an association between counties’ COVID-19 rates and their neighboring
counties’ COVID-19 mortality rates, implying that without taking this
patterning into account, regression models that do not explicitly model the
effect of spatial autocorrelation may be biased due to inappropriately
assuming that county-level data are independent of one another. Second, while
the Moran’s I plot demonstrates the spatial-correlation present in the data at
a single time-point, the following Figure 8 shows the tendency of county
COVID-19 observations to be autocorrelated between subsequent months.

Figure 9: Estimates of increases in COVID-19 mortality associated with county-
level covariates without (first) and with (second) adjustment for
spatiotemporal autocorrelation. Panels A and B show COVID-19 mortality
increases associated with varying levels of median age over time; C and D show
associations with log population density; E and F median income; G and H for
proportion in poverty; I and J the ICE for racialized economic segregation; K
and L political lean.

Figure 9 shows the additional COVID-19 mortality associated with each
covariate over time without and with adjustment for spatiotemporal trends. The
models for population density and median income show particularly marked
changes with the spike at high population density and high median incomes
disappearing after adjusting for a spatiotemporally smoothed term. This likely
reflects the early emergence and surge of COVID-19 in the greater New York
City area as being predominantly a feature of the combined effect of human
geography and infectious disease dynamics where the earliest introductions of
COVID-19 were made to dense social networks in the US. Other effects remain
qualitatively changed little, suggesting that the trends in COVID-19 mortality
associated with these county-level measures remains durable when taking into
account spatiotemporal autocorrelation.

## Discussion

This paper presents three applications of GAMs to county-level COVID-19
mortality data from the US during March 2020 to August 2022. This paper
addresses an important need, which is to assess what measures of the
socioeconomic and geopolitical landscape are associated with increases in
COVID-19 mortality in a way that takes into account how infectious disease
dynamics are locally correlated in space and time. However, the approaches
outlined in this paper are not without several limitations which include (at
least) the following: the potential for cross-level confounding, lack of
individual level data, an inability to include all possible metrics, further
improvements to the methods used may be warranted, and an inability to make
claims about the causal relationships between the variables investigated.

In analyses that use area-level aggregated data, the ecologic fallacy refers
to the possibility that associations between area-level measures and outcomes
may not reflect the associations that would be observed if analyzing
individual level data on the same measures and outcomes as a result of cross-
level confounding (Greenland, 2001). It’s important to note that some of the
variables in this analysis are included to contextualize the places in which
people live, reside, and work and area-based social metrics should be included
even in individual-level analyses so as to accurately capture the effects of
both individual-level risk factors and risk factors that stem from the
contexts in which people exist (Testa et al., 2022).

One particular improvement that could be made in future work is to address age
adjustment by using direct or indirect age standardization (Anderson et al.,
1998) or to create age-group stratified models. In the applications presented
here, county-level median age is adjusted for in order to control for age
effects as age-specific data were not publicly available without substantial
levels of suppression.

Additional variables like vaccination rates, mobility data, further variables
that relate to risk factors for COVID-19, and other COVID-19 related outcomes
like case trends and hospitalizations may be worthwhile to investigate under a
similar framework. The emphasis on trends in COVID-19 mortality in the
applications in this paper was done to prioritize presenting data that are
less subject to the reporting inconsistencies in COVID-19 cases and
hospitalizations (Galaitsi et al., 2021).

Further improvements to the methods employed here could include use of soap
film smoothing splines which are more appropriate in settings where non-convex
geographic boundaries are common (as in with peninsulas and bays) (S. N. Wood
et al., 2008). Taking a critical eye to the assumptions employed reveals yet
another area of potential improvement, which is the nature of how
smoothing splines are penalized in the GAM framework. It is conceivable that
geographic changes in COVID-19 mortality rates are not equally smooth across
all regions of the US, potentially warranting rapid changes in some areas and
more smooth changes in other areas in a risk surface model. The Bayesian
wombling literature provides the means to identify areas of rapid change in
latent surface models and may pose a fruitful direction of exploration for
future efforts to understand trends in COVID-19 mortality rates in the US
(Gelfand & Banerjee, 2015).

## Conclusions

This work introduces generalized additive models (GAMs) applied to COVID-19
mortality data to document the disparities and inequities in which US counties
suffered the worst mortality outcomes and at what times. The effort to
document inequities and explain the driving mechanisms behind them is crucial
in building up the evidence necessary to enact policy changes that can
mitigate unjust and preventable harm in the future. To this end, it is
critical that efforts to model infectious disease dynamics take into account
how disease transmission and outcomes are spatiotemporally patterned,
violating any misplaced assumptions that area-level disease rates can be
treated as independent observations. By presenting novel applications of GAMs
to trends in COVID-19 mortality, this paper illustrates new ways of
understanding and visualizing the disproportionate burden some communities
have suffered by incorporating complex patterns of interaction across
socioeconomic and geopolitical variables and spatiotemporal trends.

## References

reAnderson, R. N., Rosenberg, H. M., & National Center for Health Statistics.
(1998). Age standardization of death rates; implementation of the year 2000
standard. _National Vital Statistics Report_.
https://stacks.cdc.gov/view/cdc/13357

preBailey, Z. D., Feldman, J. M., & Bassett, M. T. (2021). How Structural
Racism Works — Racist Policies as a Root Cause of U.S. Racial Health
Inequities. _New England Journal of Medicine_ , _384_(8), 768–773.
https://doi.org/10.1056/NEJMms2025396

preBendix, A. (2022). Less than 4% of eligible people have gotten updated
Covid booster shots, one month into the rollout. In _NBC News_.
https://www.nbcnews.com/health/health-news/updated-covid-booster-shots-doses-
administered-cdc-rcna48960

preBlendon, R. J., Schoen, C., DesRoches, C. M., Osborn, R., Scoles, K. L., &
Zapert, K. (2002). Inequities In Health Care: A Five-Country Survey. _Health
Affairs_ , _21_(3), 182–191. https://doi.org/10.1377/hlthaff.21.3.182

preBrown, K. M., Lewis, J. Y., & Davis, S. K. (2021). An ecological study of
the association between neighborhood racial and economic residential
segregation with COVID-19 vulnerability in the United States’ capital city.
_Annals of Epidemiology_ , _59_ , 33–36.
https://doi.org/10.1016/j.annepidem.2021.04.003

preCarpenter, E. (2021). “The Health System Just Wasn’t Built for Us”: Queer
Cisgender Women and Gender Expansive Individuals’ Strategies for Navigating
Reproductive Health Care. _Women’s Health Issues_ , _31_(5), 478–484.
https://doi.org/10.1016/j.whi.2021.06.004

preCDC. (2021). Location, location, location – COVID Data Tracker Weekly
Review. In _Centers for Disease Control and Prevention_.
https://www.cdc.gov/coronavirus/2019-ncov/covid-data/covidview/past-
reports/07162021.html

preChen, J. T., & Krieger, N. (2021). Revealing the Unequal Burden of COVID-19
by Income, Race/Ethnicity, and Household Crowding: US County Versus Zip Code
Analyses. _Journal of Public Health Management and Practice_ , _27_(1),
S43–S56. https://doi.org/10.1097/PHH.0000000000001263

preChrisler, J. C., Barney, A., & Palatino, B. (2016). Ageism can be Hazardous
to Women’s Health: Ageism, Sexism, and Stereotypes of Older Women in the
Healthcare System. _Journal of Social Issues_ , _72_(1), 86–104.
https://doi.org/10.1111/josi.12157

preCoven, J., & Gupta, A. (2020). _Disparities in Mobility Responses to
COVID-19_. https://arpitgupta.info/s/DemographicCovid.pdf

preDel Rios, M., Chomilo, N. T., & Lewis, N. A. (2022). Covid-19 is an inverse
equity story, not a racial equity success story. In _STAT_.
https://www.statnews.com/2022/10/25/covid-19-inverse-equity-story-not-racial-
equity-success-story/

preDonovan, D. (2022). U.S. Officially Surpasses 1 Million COVID-19 Deaths. In
_Johns Hopkins Coronavirus Resource Center_. https://coronavirus.jhu.edu/from-
our-experts/u-s-officially-surpasses-1-million-covid-19-deaths

preEichenbaum, A., & Tate, A. D. (2022). Health Inequity in Georgia During the
COVID-19 Pandemic: An Ecological Analysis Assessing the Relationship Between
County-Level Racial/Ethnic and Economic Polarization Using the ICE and SARS-
CoV-2 Cases, Hospitalizations, and Deaths in Georgia as of October 2020.
_Health Equity_ , _6_(1), 230–239. https://doi.org/10.1089/heq.2021.0118

preFeldman, J. L., Luhur, W. E., Herman, J. L., Poteat, T., & Meyer, I. H.
(2021). Health and health care access in the US transgender population health
(TransPop) survey. _Andrology_ , _9_(6), 1707–1718.
https://doi.org/10.1111/andr.13052

preGalaitsi, S. E., Cegan, J. C., Volk, K., Joyner, M., Trump, B. D., &
Linkov, I. (2021). The challenges of data usage for the United States’
COVID-19 response. _International Journal of Information Management_ , _59_ ,
102352. https://doi.org/10.1016/j.ijinfomgt.2021.102352

preGelfand, A. E., & Banerjee, S. (2015). Bayesian wombling: Finding rapid
change in spatial maps. _WIREs Computational Statistics_ , _7_(5), 307–315.
https://doi.org/10.1002/wics.1360

preGelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., &
Rubin, D. B. (2013). Bayesian Data Analysis. In _Routledge & CRC Press_.
https://www.routledge.com/Bayesian-Data-Analysis/Gelman-Carlin-Stern-Dunson-
Vehtari-Rubin/p/book/9781439840955

preGelman, A., Hill, J., & Vehtari, A. (2020). _Regression and Other Stories_
(1st ed.). Cambridge University Press. https://doi.org/10.1017/9781139161879

preGlenza, J. (2020). Covid cases increase across US as upper midwest sees
rapid rise. _The Guardian_.
https://www.theguardian.com/world/2020/oct/22/covid-cases-coronavirus-us-
midwest

preGoldstein, J. (2022). During the Omicron surge, Black New Yorkers were
hospitalized at a rate more than twice that of white residents. _The New York
Times_. https://www.nytimes.com/2022/03/03/nyregion/omicron-hospitalizations-
new-york-city.html

preGonzalez, K. E., James, R., Bjorklund, E. T., & Hill, T. D. (2021).
Conservatism and infrequent mask usage: A study of US counties during the
novel coronavirus (COVID-19) pandemic. _Social Science Quarterly_ , _102_(5),
2368–2382. https://doi.org/10.1111/ssqu.13025

preGreenland, S. (2001). Ecologic versus individual-level sources of bias in
ecologic estimates of contextual health effects. _International Journal of
Epidemiology_ , _30_(6), 1343–1350. https://doi.org/10.1093/ije/30.6.1343

preGrossman, G., Kim, S., Rexer, J. M., & Thirumurthy, H. (2020). Political
partisanship influences behavioral responses to governors’ recommendations for
COVID-19 prevention in the United States. _Proceedings of the National Academy
of Sciences_ , _117_(39), 24144–24153. https://doi.org/10.1073/pnas.2007835117

preHanage, W. P., Testa, C., Chen, J. T., Davis, L., Pechter, E., Seminario,
P., Santillana, M., & Krieger, N. (2020). COVID-19: US federal accountability
for entry, spread, and inequities—lessons for the future. _European Journal of
Epidemiology_ , _35_(11), 995–1006. https://doi.org/10.1007/s10654-020-00689-2

preHealth Resources & Services Administration. (2022). _Defining Rural
Population_. https://www.hrsa.gov/rural-health/about-us/what-is-rural

preHernandez, I., Dickson, S., Tang, S., Gabriel, N., Berenbrok, L. A., & Guo,
J. (2022). Disparities in distribution of COVID-19 vaccines across US
counties: A geographic information system–based cross-sectional study. _PLOS
Medicine_ , _19_(7), e1004069. https://doi.org/10.1371/journal.pmed.1004069

preHill, L., & Artiga, S. (2022). COVID-19 Cases and Deaths by Race/Ethnicity:
Current Data and Changes Over Time. In _KFF_. https://www.kff.org/coronavirus-
covid-19/issue-brief/covid-19-cases-and-deaths-by-race-ethnicity-current-data-
and-changes-over-time/

preJohnson, A., & Keating, D. (2022). Whites now more likely to die from covid
than Blacks: Why the pandemic shifted. In _Washington Post_.
https://www.washingtonpost.com/health/2022/10/19/covid-deaths-us-race/

preKaashoek, J., Testa, C., Chen, J., Stolerman, L., Krieger, N., Hanage, W.
P., & Santillana, M. (2021). _The Evolving Roles of US Political Partisanship
and Social Vulnerability in the COVID-19 Pandemic from February 2020 -
February 2021_ [{SSRN} {Scholarly} {Paper}].
https://doi.org/10.2139/ssrn.3933453

preKim, E. J., Marrast, L., & Conigliaro, J. (2020). COVID-19: Magnifying the
Effect of Health Disparities. _Journal of General Internal Medicine_ ,
_35_(8), 2441–2442. https://doi.org/10.1007/s11606-020-05881-4

preKrieger, N., Testa, C., Chen, J. T., Hanage, W. P., & McGregor, A. J.
(2022). Relationship of political ideology of US federal and state elected
officials and key COVID pandemic outcomes following vaccine rollout to adults:
April 2021–March 2022. _The Lancet Regional Health – Americas_ , _16_.
https://doi.org/10.1016/j.lana.2022.100384

preKrieger, N., Waterman, P. D., Spasojevic, J., Li, W., Maduro, G., & Van
Wye, G. (2016). Public Health Monitoring of Privilege and Deprivation With the
Index of Concentration at the Extremes. _American Journal of Public Health_ ,
_106_(2), 256–263. https://doi.org/10.2105/AJPH.2015.302955

preLambert, J. (2022). Most Americans Don’t Know About the Omicron Covid
Boosters. In _Grid News_. https://www.grid.news/story/science/2022/10/04/most-
americans-dont-know-about-the-omicron-covid-boosters-that-spells-trouble-for-
the-coming-winter/

preLeonhardt, D. (2021). Red Covid. _The New York Times_.
https://www.nytimes.com/2021/09/27/briefing/covid-red-states-vaccinations.html

preMayo Clinic. (2022). History of COVID-19: Outbreaks and Vaccine Timeline.
In _MayoClinic.org_. https://www.mayoclinic.org/coronavirus-covid-19/history-
disease-outbreaks-vaccine-timeline/covid-19

preMIT Election Data and Science Lab. (2022). _County Presidential Election
Returns 2000-2020_. Harvard Dataverse. https://doi.org/10.7910/DVN/VOQCHQ

preMollalo, A., Vahedi, B., & Rivera, K. M. (2020). GIS-based spatial modeling
of COVID-19 incidence rate in the continental United States. _Science of The
Total Environment_ , _728_ , 138884.
https://doi.org/10.1016/j.scitotenv.2020.138884

preMoreland, A., Herlihy, C., Tynan, M. A., Sunshine, G., McCord, R. F.,
Hilton, C., Poovey, J., Werner, A. K., Jones, C. D., Fulmer, E. B.,
Gundlapalli, A. V., Strosnider, H., Potvien, A., García, M. C., Honeycutt, S.,
Baldwin, G., CDC Public Health Law Program, CDC COVID-19 Response Team, &
Mitigation Policy Analysis Unit. (2020). Timing of State and Territorial
COVID-19 Stay-at-Home Orders and Changes in Population Movement — United
States, March 1–May 31, 2020. _MMWR. Morbidity and Mortality Weekly Report_ ,
_69_. https://doi.org/10.15585/mmwr.mm6935a2

preMurthy, B. P. (2021). Disparities in COVID-19 Vaccination Coverage Between
Urban and Rural Counties — United States, December 14, 2020–April 10, 2021.
_MMWR. Morbidity and Mortality Weekly Report_ , _70_.
https://doi.org/10.15585/mmwr.mm7020e3

preNational Bureau of Economic Research. (2021). Measuring the Virus Risk of
Essential Workers and Dependents. In _NBER_.
https://www.nber.org/digest/202103/measuring-virus-risk-essential-workers-and-
dependents

preNational Conference of State Legislatures. (2021). COVID-19: Essential
Workers in the States. In _COVID-19: Essential Workers in the States_.
https://www.ncsl.org/research/labor-and-employment/covid-19-essential-workers-
in-the-states.aspx

preOffice of the Commissioner. (2022). Coronavirus (COVID-19) Update: FDA
Authorizes Moderna, Pfizer-BioNTech Bivalent COVID-19 Vaccines for Use as a
Booster Dose. In _FDA_. https://www.fda.gov/news-events/press-
announcements/coronavirus-covid-19-update-fda-authorizes-moderna-pfizer-
biontech-bivalent-covid-19-vaccines-use

preOkonkwo, N. E., Aguwa, U. T., Jang, M., Barré, I. A., Page, K. R.,
Sullivan, P. S., Beyrer, C., & Baral, S. (2021). COVID-19 and the US response:
Accelerating health inequities. _BMJ Evidence-Based Medicine_ , _26_(4),
176–179. https://doi.org/10.1136/bmjebm-2020-111426

preOrtega, A. N., & Roby, D. H. (2021). Ending Structural Racism in the US
Health Care System to Eliminate Health Care Inequities. _JAMA_ , _326_(7),
613–615. https://doi.org/10.1001/jama.2021.11160

prePark, Y. M., Kearney, G. D., Wall, B., Jones, K., Howard, R. J., & Hylock,
R. H. (2021). COVID-19 Deaths in the United States: Shifts in Hot Spots over
the Three Phases of the Pandemic and the Spatiotemporally Varying Impact of
Pandemic Vulnerability. _International Journal of Environmental Research and
Public Health_ , _18_(17), 8987. https://doi.org/10.3390/ijerph18178987

prePress, V. G., Huisingh-Scheetz, M., & Arora, V. M. (2021). Inequities in
Technology Contribute to Disparities in COVID-19 Vaccine Distribution. _JAMA
Health Forum_ , _2_(3), e210264.
https://doi.org/10.1001/jamahealthforum.2021.0264

preRader, B., Astley, C. M., Sewalk, K., Delamater, P. L., Cordiano, K.,
Wronski, L., Rivera, J. M., Hallberg, K., Pera, M. F., Cantor, J., Whaley, C.
M., Bravata, D. M., & Brownstein, J. S. (2021). _Spatial Accessibility
Modeling of Vaccine Deserts as Barriers to Controlling SARS-CoV-2_. medRxiv.
https://doi.org/10.1101/2021.06.09.21252858

preRapp, K. S., Volpe, V. V., Hale, T. L., & Quartararo, D. F. (2022).
State–Level Sexism and Gender Disparities in Health Care Access and Quality in
the United States. _Journal of Health and Social Behavior_ , _63_(1), 2–18.
https://doi.org/10.1177/00221465211058153

preSaha, S., & Feldman, J. M. (2020). _Neighborhood-level Racial/Ethnic and
Economic Inequities in COVID-19 Burden Within Urban Areas in the US and
Canada_. medRxiv. https://doi.org/10.1101/2020.12.07.20241018

preScott, D. (2020). These 4 Midwestern states are seeing worrying Covid-19
spikes. In _Vox_. https://www.vox.com/2020/9/2/21418812/covid-19-coronavirus-
us-cases-midwest-surge

preShumaker, L., & Wu, T. (2020). COVID-19 cases surge in U.S. Midwest and
Northeast. _Reuters_. https://www.reuters.com/article/us-health-coronavirus-
usa-trends-idUSKBN26Q30B

preSigler, T., Mahmuda, S., Kimpton, A., Loginova, J., Wohland, P., Charles-
Edwards, E., & Corcoran, J. (2021). The socio-spatial determinants of COVID-19
diffusion: The impact of globalisation, settlement characteristics and
population. _Globalization and Health_ , _17_(1), 56.
https://doi.org/10.1186/s12992-021-00707-2

preSonderlund, A. L., Charifson, M., Schoenthaler, A., Carson, T., & Williams,
N. J. (2022). Racialized economic segregation and health outcomes: A
systematic review of studies that use the Index of Concentration at the
Extremes for race, income, and their interaction. _PLOS ONE_ , _17_(1),
e0262962. https://doi.org/10.1371/journal.pone.0262962

preSugg, M. M., Spaulding, T. J., Lane, S. J., Runkle, J. D., Harden, S. R.,
Hege, A., & Iyer, L. S. (2021). Mapping community-level determinants of
COVID-19 transmission in nursing homes: A multi-scale approach. _Science of
The Total Environment_ , _752_ , 141946.
https://doi.org/10.1016/j.scitotenv.2020.141946

preTesta, C., Chen, J., Hall, E., Javadi, D., Morgan, J., Rushovich, T., Saha,
S., Waterman, P., & Krieger, N. (2022). _Public Health Disparities Geocoding
Project 2.0 Training Manual_. https://phdgp.github.io/PHDGP2.0/index.html

preThe Lancet. (2020). The plight of essential workers during the COVID-19
pandemic. _The Lancet_ , _395_(10237), 1587.
https://doi.org/10.1016/S0140-6736(20)31200-9

preThe New York Times. (2021). _Coronavirus (Covid-19) Data in the United
States_. The New York Times. https://github.com/nytimes/covid-19-data

preThe White House. (2021). Executive Order On Advancing Racial Equity and
Support for Underserved Communities Through the Federal Government. In _The
White House_. https://www.whitehouse.gov/briefing-room/presidential-
actions/2021/01/20/executive-order-advancing-racial-equity-and-support-for-
underserved-communities-through-the-federal-government/

preThe White House. (2022). Advancing Equity and Racial Justice Through the
Federal Government. In _The White House_. https://www.whitehouse.gov/equity/

preThompson, C. N. (2020). COVID-19 Outbreak — New York City, February 29–June
1, 2020. _MMWR. Morbidity and Mortality Weekly Report_ , _69_.
https://doi.org/10.15585/mmwr.mm6946a2

preUnited States COVID - Coronavirus Statistics - Worldometer. (2022). In
_Worldometers_. https://www.worldometers.info/coronavirus/country/us/

preU.S. Department of Commerce Economics and Statistics Administration, & U.S.
Census Bureau. (2000). _Census Regions and Divisions of the United States_.
https://www2.census.gov/geo/pdfs/maps-data/maps/reference/us_regdiv.pdf

preUS Census Bureau. (2022). Nation Continues to Age as It Becomes More
Diverse. In _Census.gov_. https://www.census.gov/newsroom/press-
releases/2022/population-estimates-characteristics.html

preUS Census Bureau. (2021). 2020 Census: Redistricting File (Public Law
94-171) Dataset. In _Census.gov_.
https://www.census.gov/data/datasets/2020/dec/2020-census-redistricting-
summary-file-dataset.html

preUS Census Bureau. (2020). American Community Survey 2015-2019 5-Year Data
Release. In _Census.gov_. https://www.census.gov/newsroom/press-
kits/2020/acs-5-year.html

preWalker, K., & Herman, M. (2022). _Tidycensus: Load US Census Boundary and
Attribute Data as ’tidyverse’ and ’sf’-Ready Data Frames_.

preWei, C.-F., Lan, F.-Y., Hsu, Y.-T., Lowery, N., Dibona, L., Akkeh, R.,
Kales, S. N., & Yang, J. (2022). Risk of SARS-CoV-2 Infection Among Essential
Workers in a Community-Based Cohort in the United States. _Frontiers in Public
Health_ , _10_. https://www.frontiersin.org/articles/10.3389/fpubh.2022.878208

preWhitehead, J., Shaver, J., & Stephenson, R. (2016). Outness, Stigma, and
Primary Health Care Utilization among Rural LGBT Populations. _PLOS ONE_ ,
_11_(1), e0146139. https://doi.org/10.1371/journal.pone.0146139

preWhittle, R. S., & Diaz-Artiles, A. (2020). An ecological study of
socioeconomic predictors in detection of COVID-19 cases across neighborhoods
in New York City. _BMC Medicine_ , _18_(1), 271.
https://doi.org/10.1186/s12916-020-01731-6

preWood, S. (2022). _Mgcv: Mixed GAM Computation Vehicle with Automatic
Smoothness Estimation_. https://CRAN.R-project.org/package=mgcv

preWood, S. N. (2017). _Generalized Additive Models: An Introduction with R_
(2nd ed.). Chapman & Hall/CRC Texts in Statistical Science.
https://www.routledge.com/Generalized-Additive-Models-An-Introduction-with-R-
Second-Edition/Wood/p/book/9781498728331

preWood, S. N., Bravington, M. V., & Hedley, S. L. (2008). Soap film
smoothing. _Journal of the Royal Statistical Society: Series B (Statistical
Methodology)_ , _70_(5), 931–955.
https://doi.org/10.1111/j.1467-9868.2008.00665.x

p


such that $\frac{1}{\rho}+\frac{2}{\sigma}=1$. Then we estimate

$\displaystyle\left\lvert\int_{0}^{\infty}\int_{\Omega}gh_{1}h_{2}\varphi\
dxdt\right\rvert$
$\displaystyle\leq\int^{\infty}_{0}\left(\int_{\Omega}\left\lvert
gh_{1}h_{2}\right\rvert^{r^{\prime}}dx\right)^{{}^{1}\\!/_{r^{\prime}}}\left\lVert\varphi(t)\right\rVert_{L^{r}(\Omega)}\,dt$
$\displaystyle\leq\left(\int^{\infty}_{0}\left(\int_{\Omega}\left\lvert
gh_{1}h_{2}\right\rvert^{r^{\prime}}dx\right)^{{}^{2}\\!/_{r^{\prime}}}dt\right)^{{}^{1}\\!/_{2}}\left(\int^{\infty}_{0}\left\lVert\varphi\right\rVert^{2}_{L^{r}(\Omega)}dt\right)^{{}^{1}\\!/_{2}}.$

This further implies that

$\left\lvert\int_{0}^{\infty}\int_{\Omega}gh_{1}h_{2}\varphi\
dxdt\right\rvert\leq C_{0}\left[\int_{0}^{\infty}\left\lVert
g(t)\right\rVert^{2}_{L^{r^{\prime}\rho}(\Omega)}\left\lVert
h_{1}(t)\right\rVert^{2}_{L^{r^{\prime}\sigma}(\Omega)}\left\lVert
h_{2}(t)\right\rVert^{2}_{L^{r^{\prime}\sigma}(\Omega)}dt\right]^{{}^{1}\\!/_{2}}\left\lVert\varphi\right\rVert_{L^{2}(I;V)}\longleftarrow(a).$
(6.3)

Here and below $C_{i},i=0,1,2,\dots$ denote constant which are independent of
$y,\varphi,h_{1},h_{2}$. We next recall Gagliardo’s inequality [BF, p 173]:

$\left\lVert u\right\rVert_{L^{q}(\Omega)}\leq\left\lVert
u\right\rVert^{1-{}^{d}\\!/_{q}-{}^{d}\\!/_{2}}_{L^{2}(\Omega)}\left\lVert
u\right\rVert^{{}^{d}\\!/_{2}+{}^{2}\\!/_{q}}_{W^{1,2}(\Omega)},\text{ for all
}q>2,\text{ and }u\in W^{1,2}(\Omega)\equiv V,$

where $q\in[2,2^{*}]$ and

$q^{*}\begin{cases}\in[2,\infty]\text{ for }d=1,\\\ \in[2,\infty)\text{ for
}d=2,\\\ \in[2,\frac{2d}{d-2}],\text{ for }d>2.\end{cases}$

In the above estimate we take, $q=r^{\prime}\sigma$. We obtain

$1+\frac{d}{q}-\frac{d}{2}=\frac{(2-d)r^{\prime}\sigma+2d}{2r^{\prime}\sigma},\
\frac{d}{2}-\frac{d}{q}=\frac{d(r^{\prime}\sigma-2)}{2r^{\prime}\sigma},\text{
also }r^{\prime}\sigma>2,\ (2-d)r^{\prime}\sigma+2d>0.$

We estimate (6.3), (and check the conditions on the ranges of the parameters
below)

$\displaystyle\sup_{\left\lVert\varphi\right\rVert_{L^{2}(I;V)}\leq 1}(a)$
$\displaystyle=C_{1}\left(\int_{0}^{\infty}\left\lVert
g(t)\right\rVert^{2}_{L^{r^{\prime}\rho}(\Omega)}\left(\left\lVert
h_{1}(t)\right\rVert_{L^{2}(\Omega)}\left\lVert
h_{2}(t)\right\rVert_{L^{2}(\Omega)}\right)^{\frac{(2-d)r^{\prime}\sigma+2d}{r^{\prime}\sigma}}\left(\left\lVert
h_{1}(t)\right\rVert_{V}\left\lVert
h_{2}(t)\right\rVert_{V}\right)^{\frac{d(r^{\prime}\sigma-2)}{r^{\prime}\sigma}}\right)^{{}^{1}\\!/_{2}}dt$
$\displaystyle\leq C_{2}\left(\left\lVert
h_{1}\right\rVert_{W_{\infty}}\left\lVert
h_{2}\right\rVert_{W_{\infty}}\right)^{\frac{(2-d)r^{\prime}\sigma+2d}{2r^{\prime}\sigma}}\left(\int_{0}^{\infty}\left\lVert
g(t)\right\rVert^{2}_{L^{r^{\prime}\rho}(\Omega)}\left(\left\lVert
h_{1}(t)\right\rVert_{V}\left\lVert
h_{2}(t)\right\rVert_{V}\right)^{\frac{d(r^{\prime}\sigma-2)}{r^{\prime}\sigma}}\right)^{{}^{1}\\!/_{2}}dt\longleftarrow(b)$

We set $\frac{d(r^{\prime}\sigma-2)}{r^{\prime}\sigma}=\frac{2}{3}$. This
yields,

$\displaystyle(b)$ $\displaystyle\leq C_{3}\left(\left\lVert
h_{1}\right\rVert_{W_{\infty}}\left\lVert
h_{2}\right\rVert_{W_{\infty}}\right)^{{}^{2}\\!/_{3}}\left(\left\lVert
h_{1}\right\rVert_{W_{\infty}}\left\lVert
h_{2}\right\rVert_{W_{\infty}}\right)^{{}^{1}\\!/_{3}}\left(\int_{0}^{\infty}\left\lVert
g(t)\right\rVert^{6}_{L^{r^{\prime}\rho}(\Omega)}\right)^{{}^{1}\\!/_{6}}dt$
$\displaystyle=C_{4}\left\lVert h_{1}\right\rVert_{W_{\infty}}\left\lVert
h_{2}\right\rVert_{W_{\infty}}\left(\int_{0}^{\infty}\left(\int_{\Omega}\left\lvert
g(t)\right\rvert^{r^{\prime}\rho}dx\right)^{{}^{6}\\!/_{r^{\prime}\rho}}dt\right)^{{}^{1}\\!/_{6}}.$
(6.4)

Now we check the conditions on the parameter $r,\sigma,d$ , and
$r^{\prime},\sigma$. Since
$\frac{d(r^{\prime}\sigma-2)}{r^{\prime}\sigma}=\frac{2}{3}$, together with
the conditions on $r^{\prime}$ and $\sigma$ these parameters need to satisfy

$r^{\prime}\sigma=\frac{6d}{3d-2},\quad
r^{\prime}\in\left[\frac{2d}{d+2},\infty\right),\quad\sigma\in(2,\infty)\quad
r^{\prime}\sigma\in[2,2^{*}],$ (6.5)

and $r^{\prime}>1$ if $d=2$. The last condition above holds without
restricting the dimension $d$. From the first three relations we infer that
necessarily $\frac{6d}{3d-2}=r^{\prime}\sigma>\frac{4d}{d+2}$ which is only
possible for $d\leq 3$.

Let us focus on $d=2$. Then the choice of parameters
$r=6,r^{\prime}={}^{6}\\!/_{5},\sigma={}^{5}\\!/_{2},\rho=5$ satisfies all the
above requirements and it is convenient to further estimate (6.4). In fact we
obtain

$\left\lVert{\mathcal{F}}^{\prime}(y+h_{2})h_{1}-{\mathcal{F}}^{\prime}(y)h_{1}-{\mathcal{F}}^{\prime\prime}(y)(h_{1},h_{2})\right\rVert_{L^{2}(I;V^{*})}\\\
\leq C_{5}\left\lVert h_{1}\right\rVert_{W_{\infty}}\left\lVert
h_{2}\right\rVert_{W_{\infty}}\left(\int_{0}^{\infty}\left(\int_{\Omega}\left\lvert
g(t,x)\right\rvert^{6}dx\right)dt\right)^{{}^{1}\\!/_{6}}\leq C_{6}\left\lVert
h_{1}\right\rVert_{W_{\infty}}\left\lVert
h_{2}\right\rVert_{W_{\infty}}\left(\int_{0}^{\infty}\left(\int_{\Omega}\left\lvert
g(t,x)\right\rvert^{2}dx\right)dt\right)^{{}^{1}\\!/_{6}}$

for all $y,h_{1},h_{2}\in W_{\infty}$. Here we use the boundedness of $g$. By
Lebesgue’s bounded convergence theorem the last factor converges to $0$ for
$\left\lVert h_{2}\right\rVert_{W_{\infty}}\to 0$ and hence the fact that
${\mathcal{F}}$ is twice differentiable is verified. The continuity of the
second derivative follows with the above estimates and again by the Lebesgue
theorem.

Next we consider $d=3$. In this case an analogous procedure is not possible,
since the relations (6.5) and $r^{\prime}\rho\leq 6$ cannot be fulfilled
simultaneously. In fact,
$r^{\prime}\sigma={}^{18}\\!/_{7},r^{\prime}\geq{}^{6}\\!/_{5}$, and thus
necessarily $\sigma\in(2,{}^{15}\\!/_{7}]$. The condition $r^{\prime}\rho\leq
6$ is equivalent to
$12\leq\sigma(6-r^{\prime})={}^{18}\\!/_{7r^{\prime}}(6-r^{\prime})$, which in
turn is equivalent to $17r^{\prime}\leq 18$, which contradicts
$r^{\prime}\geq{}^{6}\\!/_{5}$.

Thus we fix parameters $r$ and $\sigma$ such that (6.5) are satisfied for
$d=3$, as for instance $r^{\prime}={}^{6}\\!/_{5},\sigma={}^{15}\\!/_{7}$,
which implies that $\rho=15$ and $r^{\prime}\rho=18$. Then for the finite
horizon problem we can estimate by Hölder’s inequality with
$\eta={}^{r^{\prime}\rho}\\!/_{6}$:

$\displaystyle(b)$ $\displaystyle\leq C_{7}\left\lVert
h_{1}\right\rVert_{W_{\infty}}\left\lVert
h_{2}\right\rVert_{W_{\infty}}\left(\int_{0}^{T}\left(\int_{\Omega}\left\lvert
g(t,x)\right\rvert^{2}dx\right)^{{}^{6}\\!/_{r^{\prime}\rho}}dt\right)^{{}^{1}\\!/_{6}}$
$\displaystyle\leq C_{8}\left\lVert h_{1}\right\rVert_{W_{\infty}}\left\lVert
h_{2}\right\rVert_{W_{\infty}}\left(\int_{0}^{T}\left(\int_{\Omega}\left\lvert
g(t,x)\right\rvert^{2}dx\right)dt\right)^{{}^{1}\\!/_{6\eta}}T^{{}^{1}\\!/_{6\eta^{\prime}}}.$

From here we can proceed as in the case $d=2$ to assert the continuous second
Fréchet differentiability of ${\mathcal{F}}$ in $d=3$ for the finite horizon
case.

####  6.2.2 Assumptions (A4) and (A5).

In order to verify (A4), we show that ${\mathcal{F}}(y):W(0,T)\to
L^{1}(0,T;V^{*})$ is continuous for every $T>0$. We consider the sequence
$y_{n}\rightharpoonup\hat{y}$ in $W(0,T)$ and let $z\in L^{\infty}(0,T;V)$ be
given. Then we estimate

$\int_{0}^{T}\langle{\mathcal{F}}(y_{n})-{\mathcal{F}}(\hat{y}),z\rangle_{V,V^{*}}dt=\int_{0}^{T}\int_{\Omega}(\mathfrak{f}(y_{n})-\mathfrak{f}(\hat{y}))z\
dxdt\leq C\left\lVert y_{n}-\hat{y}\right\rVert_{L^{2}(0,T;Y)}\left\lVert
z\right\rVert_{L^{2}(0,T;Y)}.$

Then by the compactness of $V$ in $Y$, we obtain (A4).

Now we verify (A5). We recall Remark 3.1, and proceed as in (6.2) for $y\in
W_{\infty},\varphi\in L^{2}(I;V)$,

$\displaystyle\left\lVert{\mathcal{F}}^{\prime}(y)^{*}p\right\rVert_{L^{2}(I;V^{*})}=\left\lVert({\mathcal{F}}^{\prime}(y)^{*}-{\mathcal{F}}^{\prime}(0)^{*})p\right\rVert_{L^{2}(I;V^{*})}=\sup_{\left\lVert\varphi\right\rVert_{L^{2}(I;V)}\leq
1}\int_{0}^{\infty}\int_{\Omega}\langle({\mathcal{F}}^{\prime}(y)^{*}-{\mathcal{F}}^{\prime}(0)^{*})p,\varphi\rangle_{V^{*},V}$
$\displaystyle=\sup_{\left\lVert\varphi\right\rVert_{L^{2}(I;V)}\leq
1}\int_{0}^{\infty}\int_{\Omega}(\mathfrak{f}^{\prime}(y)-\mathfrak{f}^{\prime}(0))p\varphi\
dxdt\leq C\left\lVert y\right\rVert_{W_{\infty}}\left\lVert
p\right\rVert_{L^{2}(I;V)}.$

This shows ${\mathcal{F}}^{\prime}(y)^{*}$ satisfies (A5).

###  6.3 Cubic nonlinearity $y^{3}$ in one dimension
($\Omega\subset\mathbb{R}$).

We can also consider the optimal stabilization problem with cubic
nonlinearity, i.e. ${\mathcal{F}}(y)=y^{3}$ in one dimension. This is a
special case of Schlögl model.

$\displaystyle(\mathcal{P}_{Sch})\qquad\mathcal{V}(y_{0})=\min_{\begin{matrix}(y,u)\in
W_{\infty}\times U_{ad}\end{matrix}}\ \frac{1}{2}\int_{0}^{\infty}\left\lVert
y\right\rVert^{2}_{Y}dt+\frac{\alpha}{2}\int_{0}^{\infty}\left\lVert
u\right\rVert^{2}_{{\mathcal{U}}}dt,$ (6.6a) subject to $\displaystyle y_{t}$
$\displaystyle=\Delta y+y^{3}+Bu\quad$ $\displaystyle\text{ in
}Q=(0,\infty)\times\Omega$ (6.6b) $\displaystyle y$ $\displaystyle=0\quad$
$\displaystyle\text{ on }\Sigma=(0,\infty)\times\Gamma$ (6.6c) $\displaystyle
y(0)$ $\displaystyle=y_{0}\quad$ $\displaystyle\text{ in }\Omega.$ (6.6d)

In this model, one can easily verify assumption (A1) is satisfied by taking
${\mathcal{A}}y=\Delta y,\ y|_{\Gamma}=0,$ and
${\mathcal{D}}({\mathcal{A}})=H^{2}(\Omega)\cap V$. Clearly ${\mathcal{A}}$
can be extended to ${\mathcal{A}}\in{\mathcal{L}}(V,V^{*})$. Moreover
${\mathcal{A}}$ generates an analytic semigroup on $Y$ which is uniformly
stable. Assumption (A2) is satisified under the same argumentation as in
Fisher’s equation. Differentiability assumption (A3), and continuity
assumption (A4) are satisfied along similar computations as in subsections
6.2.1, 6.2.2. For (A5) we require that $y_{0}\in V$. Indeed in this case for
$\bar{y}\in W_{\infty}$ by Gagliardo’s inequality

$\int_{0}^{\infty}\int_{\Omega}|\bar{y}^{3}|^{2}dxdt=\int_{0}^{\infty}\left\lVert\bar{y}\right\rVert^{6}_{L^{6}(\Omega)}dt\leq\int^{\infty}_{0}\left\lVert\bar{y}\right\rVert^{4}_{L^{2}(\Omega)}\left\lVert\bar{y}\right\rVert^{2}_{V}dt\leq
C\left\lVert\bar{y}\right\rVert^{4}_{W_{\infty}}\int^{\infty}_{0}\left\lVert\bar{y}\right\rVert^{2}_{V}dt\leq
C\left\lVert\bar{y}\right\rVert^{6}_{W_{\infty}}.$

Thus $\bar{y}^{3}\in L^{2}(I;Y)$ and parabolic regularity theory implies that
$\bar{y}\in C(I;V)$ if $y_{0}\in V$. We estimate for $h,\varphi\in
L^{2}(I;V)$, suppressing the arguments $(t,x)$,

$\displaystyle\left\lvert\int_{0}^{\infty}\int_{\Omega}{\mathcal{F}}^{\prime}(\bar{y})h\varphi\,dxdt\right\rvert\leq\left\lvert\int_{0}^{\infty}\int_{\Omega}\bar{y}^{2}h\varphi\,dxdt\right\rvert$
$\displaystyle\leq\int_{0}^{\infty}\left\lVert\bar{y}\right\rVert^{2}_{L^{4}(\Omega)}\left\lVert
h\right\rVert_{L^{4}(\Omega)}\left\lVert\varphi\right\rVert_{L^{4}(\Omega)}\,dt,$
$\displaystyle\leq C\left\lVert\bar{y}\right\rVert^{2}_{C(I;V)}\left\lVert
h\right\rVert_{L^{2}(I;V)}\left\lVert\varphi\right\rVert_{L^{2}(I;V)}$

which implies (A5). Moreover we have ${\mathcal{F}}(\bar{y})\in
C([0,T_{y_{0}});Y)$, since $V\subset C(\bar{\Omega})$ in dimension 1, and thus
the extra regularity demanded in Theorem 3.2 is satisfied.

## References

  * [BPDM] A. Bensoussan, G. Da Prato, M. Delfour, S. Mitter  Representation and Control of Infinite Dimensional Systems,  Birkhäuser Boston Basel Berlin, 2007.
  * [BM] M. Bergounioux, N. Merabet,  Sensitivity analysis for optimal control problems governed by semilinear parabolic equations,  Control and Cybernetics, vol: 28(1999), No 3.
  * [BF] F. Boyer, P. Fabrie,  Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models,  Springer-Verlag New York, 2013, pp 526.
  * [BKP1] T. Breiten, K. Kunisch, L. Pfeiffer,  Infinite-horizon bilinear optimal control problems: Sensitivity analysis and polynomial feedback laws,  SIAM Journal on Control and Optimization, 56 (2018), pp. 3184-3214.
  * [BKP2] T. Breiten, K. Kunisch, L. Pfeiffer,  Taylor Expansion of the value Function associated with a bilinear optimal control problem,  Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire, (2019).
  * [BKP3] T. Breiten, K. Kunisch, L. Pfeiffer,  Feedback Stabilization of the Two-Dimensional Navier-Stokes Equations by Value Function Approximation,  Appl. Math. Optim. 80, 599–641 (2019).
  * [CF] P. Cannarsa, H. Frankowska,  Local regularity of the value function in optimal control,  Systems & Control Letters, Vol 62, Issue 9, 2013, pp 791-794, ISSN 0167-6911, https://doi.org/10.1016/j.sysconle.2013.06.001.
  * [Cas] E. Casas,  Pontryagin’s Principle for State-Constrained Boundary Control Problems of Semilinear Parabolic Equations,  SIAM J. Control Optim., 35(4), 1297-1327. (31 pages), 1997. https://doi.org/10.1137/S0363012995283637
  * [CK] E. Casas, K. Kunisch,  Stabilization by Sparse Controls for a Class of Semilinear Parabolic Equations,  SIAM J. Control Optim., 55(1): 512-532, 2017.
  * [Don] A. L. Dontchev,  Implicit function theorems for generalized equations,  Mathematical Programming 70, 91–106 (1995). https://doi.org/10.1007/BF01585930.
  * [Emm] E. Emmrich,  Gewöhnliche und Operator Differentialgleichungen,  Vieweg, Wiesbaden, 2004.
  * [Goe] R. Goebel,  Convex Optimal Control Problems with Smooth Hamiltonians,  SIAM J. Control Optim., 43(5), 1787-1811 (2005). (25 pages). https://doi.org/10.1137/S0363012902411581.
  * [Gri] R. Griesse,  Parametric sensitivity analysis in optimal control of reaction diffusion system - Part I: Solution Differentiability,  Numerical Functional Analysis and Optimization, 25(1-2):93-117, 2004.
  * [GHH] R. Griesse, M. Hintermüller, Michael Hinze,  Differential stability of control-constrained optimal control problems for the Navier-Stokes equations,  Numerical Functional Analysis and Optimization, 26:7-8, 829-850, 2005.
  * [GV] R. Griesse, S. Volkwein,  Parametric Sensitivity Analysis for Optimal Boundary Control of a 3D Reaction-Diffusion System,  Large-Scale Nonlinear Optimization. Nonconvex Optimization and Its Applications, vol 83. (2006) Springer, Boston, MA.
  * [IK] K. Ito, K. Kunisch,  Lagrange Multiplier Approach to Variational Problems and Applications,  Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2008.
  * [LM] J.-L. Lions, E. Magenes,  Non-Homogeneous Boundary Value Problems and Applications: Vol. 1,  Die Grundlehren der mathematischen Wissenschaften 181, Springer-Verlag, Berlin, 1972.
  * [Mal] K. Malanowski,  Sensitivity analysis for parametric optimal control of semilinear parabolic equations,  Journal or Convex Analysis, Vol 9(2002), No. 2, 543-569.
  * [MT] K. Malanowski, F. Tröltzsch,  Lipschitz Stability of Solutions to Parametric Optimal Control for Parabolic Equations,  Journal of Analysis and its Applications, 18(2):469-489, 1999.
  * [MS] C. Meyer, L. M. Susu,  Optimal control of nonsmooth, semilinear parabolic equations,  SIAM J. Control Optim. Vol. 55(2017), No. 4, pp. 2206-2234.
  * [MZ] H. Maurer, J. Zowe,  First and second-order necessary and sufficient optimality conditions for infinite-dimensional programing problems,  Math. Programming 16(1979), 98-110.
  * [Paz] A. Pazy,  Semigroups of Linear Operators and Applications to Partial Differential Equations,  Springer-Verlag New York, 1983.
  * [Rob] S. M. Robinson,  Strongly regular generalized equations,  Math. of Oper. Res. 5(1980), 43-62.
  * [Tri] R. Triggiani,  On the Stabilizability Problem of Banach Spaces,  J. Math. Anal. Appl. 55,1975, pp 303-403.
  * [Tro1] F. Tröltzsch,  Lipschitz stability of solutions to linear-quadratic parabolic control problems with respect to perturbations,  Dynamics of Continuous, Discrete and Impulsive Systems Series A Mathematical Analysis, 7(2):289-306, 2000.
  * [Tro2] F. Tröltzsch,  Optimal Control of Partial Differential Equations: Theory, Methods and Applications  AMS, ISBN 978-0-8218-4904-0, 2010, pp 399.
  * [Wac] D. Wachsmuth,  Regularity and stability of optimal controls of nonstationary Navier-Stokes equations,  Control and Cybernetics, vol 34(2005), No 2.

Received xxxx 20xx; revised xxxx 20xx.