# COVID-19 attack rate increases with city size

Andrew J. Stier,1∗ Marc G. Berman,1 Luis M. A. Bettencourt2,3

1Department of Psychology, University of Chicago,
Mansueto Institute for Urban Innovation, University of Chicago
2Department of Ecology & Evolution, University of Chicago

∗To whom correspondence should be addressed; E-mail<EMAIL_ADDRESS>
> The current outbreak of novel coronavirus disease 2019 (COVID-19) poses an
> unprecedented global health and economic threat to interconnected human
> societies. Until a vaccine is developed, strategies for controlling the
> outbreak rely on aggressive social distancing. These measures largely
> disconnect the social network fabric of human societies, especially in urban
> areas. Here, we estimate the growth rates and reproductive numbers of
> COVID-19 in US cities from March 14th through March 19th to reveal a power-
> law scaling relationship to city population size. This means that COVID-19
> is spreading faster on average in larger cities with the additional
> implication that, in an uncontrolled outbreak, larger fractions of the
> population are expected to become infected in more populous urban areas. We
> discuss the implications of these observations for controlling the COVID-19
> outbreak, emphasizing the need to implement more aggressive distancing
> policies in larger cities while also preserving socioeconomic activity.

The coronavirus pandemic of 2019-20 (COVID-19) is an unprecedented worldwide
event. Its speed of propagation, its international reach and the unprecedented
coordinated measures for its mitigation, are only possible in a world that is
more connected and more urbanized than at any other time in history.

As a novel infectious disease in human populations, COVID-19 has a number of
quantitative signatures to its pattern of spread. These signatures make its
dynamics more difficult to contain but also easier to understand.

First, because there is no history of previous exposure, all human populations
in contact with the virus are (presumably) susceptible. This means that the
susceptible population is the world’s total population writ large. Second,
because COVID-19 is a respiratory disease, it is easily transmissible
resulting in high reproductive numbers, $R=2.2-6.5$ (?, ?, ?), though
considerable uncertainty remains about these estimates. Third, COVID-19
appears to be characterized by reproductive numbers above the epidemic
threshold ($R>1$) everywhere around the world, regardless of environmental
conditions such as humidity or temperature. These reproductive numbers are
considerably higher than seasonal influenza (?). Bringing the disease
reproductive number below the epidemic threshold ($R\rightarrow R<1$) is the
main goal of all public health interventions; once this is achieved the
disease’s transmission chain reaction will shut down.

The reproductive number is the product of two factors $R=\beta/\gamma$, the
infectious period $1/\gamma$ (a physiological property) and the contact rate
$\beta$, which is a property of the population, essentially measuring the
number of social contacts that can transmit the disease per unit time. Of
these, only the contact rate can be changed via public health interventions.

In the absence of a vaccine, social distancing remains the only option to slow
down the spread of the disease and arrest potential mortality. Governments
around the world are now enacting aggressive policies, including ”shelter in
place” and emergency closures of all non-essential services, which carry
severe economic and social consequences. For example, in the last week, the US
Centers for Disease Control and the White House have recommended extreme
social distancing in order to slow down the current outbreak of novel
coronavirus disease 2019 (COVID-19) (?). However, these measures are less
aggressive than what has been put in place elsewhere (?). There is still a
great deal of uncertainty as to how strong social distancing recommendations
must be or how long they must last. Importantly, national and regional social
distancing policies are likely to impact individual cities differently.

Cities are predicated on extensive and intense socioeconomic interactions.
Many of their measurable properties - from the size of their economies, to
their crime rates, to the prevalence of certain infectious diseases - are
mediated by socioeconomic interactions. These interactions are subject to well
known scaling effects, which are magnified by city population size (?). All of
these relationships are tied to socioeconomic networks with average degree
(number of social connections per capita) that increases approximately as a
power law of city size $k(N)=k_{0}N^{\delta}$, with $\delta\simeq 1/6$ (?, ?).

This is a large effect. Based on data of mobile phone social networks (?),
people living in a city of 500,000 have, on average, 11 people in their mobile
phone social network, while people living in a city of 5,000,000 people have
approximately 15. This is relevant to disease transmission as the average
contact rate is proportional to degree $\beta\propto k(N)$ (see Materials and
Methods). Therefore, we expect that initial growth rates of COVID-19 cases to
be higher in larger cities (see Materials and Methods). This is what is found
empirically (see Figure 1A).

Figure 1: COVID-19 reported cases grow faster in larger cities. (a) Estimated
exponential daily growth rates of COVID-19 in US Metropolitan Areas (MSAs).
These estimates were made with the assumption that cities were experiencing
exponential growth of cases. The growth rate of COVID-19 cases is
approximately 2.4 times faster in New York-Newark-Jersey compared to Oak
Harbour, WA (b) In the absence of effective controls, larger cities are
expected to have more extensive epidemics than smaller cities, Eq. (1). Higher
values of $R$ result in a greater percentage of the population eventually
infected, unless this effect is curbed by controls that reduce the social
contact rate. The translation of growth rates into reproductive numbers was
obtained using an infectious period of $1/\gamma=4.5$ days. These estimated
values of $R$ are high in some cases (e.g. New York City) compared to reports
in other situations and may in part be the result of the acceleration of
testing in larger cities and specific places.

A larger reproductive number for spreading processes in larger cities has two
important consequences (?, ?, ?, ?). First, the reproductive number, $R$, sets
a finite threshold for how an epidemic outbreak propagates in a population,
just like the branching rate in a chain reaction (?, ?). For $R<1$, an
introduced disease will die off because it will be dampened in transmission,
while for $R>1$, disease transmission will be amplified and result in an
epidemic where the disease is transmitted quickly to almost everyone (see
Figure 1B). Because we expect $R\sim N^{\delta}$ to increase with city size,
we expect larger cities to be more susceptible to both contagious diseases,
but also to the spread of information (see below).

Second, the size of an epidemic outbreak, as measured by the percent of the
population that becomes infected, is also related to the reproductive number.
In complex epidemic models, this needs to be computed numerically, but for a
simple Susceptible-Infected-Recovered (SIR) model (?) we can integrate the
dynamics and write the explicit expression (see Methods)

$\frac{S_{\infty}}{S_{0}}=e^{-R(1-S_{\infty}/N)}$ (1)

where $S_{0}$ is the initial suscepible population size (before the outbreak)
and $S_{\infty}$ is the (smaller) final population of susceptible people. A
larger $R\sim~{}N^{\delta}$ leads to more extensive epidemics. The percent of
people infected at the end of the outbreak is $1-S_{\infty}/N$ which is larger
in populations with larger $R$, such as in larger cities. A final point
centers on the fraction of the population that must be removed from the
susceptible class when $R>1$ to stop the outbreak. This is often called the
vaccination rate, $p_{R}$, which is equally relevant in the context of social
distancing (as only the means of the intervention and its duration differ). In
the SIR model, this is simply $p_{R}=1-1/R$, which shows that as cities get
larger the distancing rate must also increase (see Supplementary Figure 1).

These observations have a number of implications that can inform evolving
national, regional, and local responses to the outbreak of COVID-19. First, it
is particularly important for larger cities to act quickly to contain this
outbreak. Second, social distancing will impact cities differently based on
city size. From the perspective of containing the outbreak, larger cities
require more aggressive social distancing policies, corresponding to a larger
$p_{R}$. At the same time, once the outbreak is contained it might be possible
to relax social distancing policies in smaller cities first, allowing a faster
return to normal life and economic activity compared to more populous urban
areas.

These distinctions may help to bring more nuance to ongoing strategies for
suppression and control of COVID-19, including gradually restoring
socioeconomic activity in context appropriate and safe ways.

Because of their higher network density, insufficient social distancing in
larger or typically more connected cities may lead to bigger outbreaks and to
the creation of reservoirs for the disease, which can continue to create
introductions elsewhere. These dynamics may also play out within cities, as
communities in which people interact more densely from the perspective of
disease transmission (e.g., downtowns), may similarly act as contagion
reservoirs that may prolong the duration of the present outbreak and
potentially create secondary reinfection waves.

Finally, as strategies for controlling this outbreak continue to evolve, it is
critical to keep in mind that almost everything that we appreciate about urban
environments, including their economic prowess, their ability to innovate, and
their role in their inhabitants social and mental health, is predicated on
network effects mediated by socioeconomic interactions. The ability to succeed
against a fast emerging epidemic like COVID-19 depends on preserving as much
person-to-person connectivity (e.g., through technology), while stopping
disease transmission. Research on safe types of socioeconomic contact and
exchange is therefore paramount so that we can succeed in controlling this
outbreak while maintaining livelihoods, socioeconomic capacity, and mental
health. This can in principle be done through approaches that make the most of
emerging, real-time data to create context appropriate suppression strategies
at local, regional, national, and global levels.

The higher socioeconomic connectivity of larger cities in a fast urbanizing
world makes containing emergent epidemics harder. But the density of
socioeconomic connections in cities can also facilitate the spread of
information, social coordination, and innovations necessary to stop the spread
of COVID-19. This information and associated actions can easily spread much
faster than the biological viral contagion. To fight an exponential, we need
to create an even faster exponential!

## References

  * 1. Y. Liu, A. A. Gayle, A. Wilder-Smith, J. Rocklöv, Journal of Travel Medicine 27 (2020). Taaa021.
  * 2. J. T. Wu, K. Leung, G. M. Leung, The Lancet 395, 689 (2020).
  * 3. S. Yang, et al., Annals of Translational Medicine 8 (2020).
  * 4. M. Biggerstaff, S. Cauchemez, C. Reed, M. Gambhir, L. Finelli, BMC infectious diseases 14, 480 (2014).
  * 5. Https://www.whitehouse.gov/briefings-statements/coronavirus-guidelines-america/.
  * 6. S. Chen, J. Yang, W. Yang, C. Wang, T. Bärnighausen, The Lancet (2020).
  * 7. L. M. Bettencourt, science 340, 1438 (2013).
  * 8. M. Schläpfer, et al., Journal of the Royal Society Interface 11, 20130789 (2014).
  * 9. B. D. Dalziel, et al., Science 362, 75 (2018).
  * 10. L. M. Bettencourt, J. Lobo, D. Strumsky, Research policy 36, 107 (2007).
  * 11. M. P. Feldman, D. B. Audretsch, European economic review 43, 409 (1999).
  * 12. Z. J. Acs, Urban Dynamics and Growth: Advances in Urban Economics (Elsevier, 2004), vol. 266 of Contributions to Economic Analysis, pp. 635 – 658.
  * 13. R. M. Anderson, B. Anderson, R. M. May, Infectious diseases of humans: dynamics and control (Oxford university press, Oxford, Reprinted, 2010).
  * 14. L. M. Bettencourt, A. Cintrón-Arias, D. I. Kaiser, C. Castillo-Chávez, Physica A: Statistical Mechanics and its Applications 364, 513 (2006).
  * 15. Https://www.census.gov/programs-surveys/metro-micro/about/delineation-files.html.
  * 16. Https://www.census.gov/data/tables/time-series/demo/popest/2010s-counties-total.html.
  * 17. S. Yadlowsky, N. Shah, J. Steinhardt, medRxiv (2020).

## Acknowledgments

This research is partially supported by the Mansueto Institute for Urban
Innovation and a Social Science Research grant from the University of Chicago.

## Supplementary materials

## Materials and Methods

Data and Urban Units of Analysis: Here we briefly describe the mathematical
analysis steps. County level daily data from March 13-24 were aggregated to
the city level (Metropolitan Statistical Areas, which are integrated labor
markets) using delineation files from the US Office of Budget and Management
(?). Counties with at least 1 reported case of COVID-19 are available in the
data. We next excluded cities that had no COVID-19 cases on March 13. This
excluded cities with low case counts, likely due to introductions from outside
the MSA, for which accurate estimates of local case growth rate is unlikely.
Results were similar for all contiguous subsets of the data of at least 7 days
(see Supplmentary Figure 4a). This left 163 cities for further analysis. We
substracted deaths from cases in each city and found the slope of the
$ln(cases)\sim ln(a)+r\cdot t$ line for each resulting time-series of active
COVID-19 cases. Finally we plotted the natural logarithm of $r$ and the
natural logarithm of city population from 2018 census estimates (?), and
performed an ordinary least squares linear regression to determine the slope
of the scaling line. Regression residuals were not related to city population
(Supplementary Figure 2) and a q-q plot of the residuals indicated that they
were well described by a normal distribution (Supplementary Figure 3).

In order to estimate the reproductive number $R$ we multiplied the growth rate
of each city, $r$, by an average infections period of $1/\gamma=4.5$ days and
adding one (see below). The size of the epidemic was then estimated by finding
the root of the equation $y=\ln(x)+R\cdot(1-x)$, where $x=S_{\infty}/N$.

Accurately estimating the growth rate of epidemics is often difficult (?),
however, here we are concerned with the pattern of growth rates among cities
rather than the precision of our growth rate estimates. To that end we
additionally estimated the growth rate of COVID-19 cases by
$r=ln(cases_{T}/cases_{0})/T$ which is an estimate of the slope of the line
$ln(cases)\sim ln(a)+r\cdot t$ from the first and last points of the time
series (Supplementary Figure 4b). These growth rate estimates showed a scaling
relationship with city size that is consistent with Figure 1 of the main text.
This was observed despite variations in growth rate estimates between the two
methods

Epidemic Models and the Reproductive Rate Even though well known, we include
here the basic derivation of the reproductive rate and final size of the
epidemic epidemic models, for the sake of completeness.

Figure 2:  Epidemic models and their parameters.Epidemic models divide a
population of size $N$, into subgroups by their health status at each time:
susceptible (S), exposed (E), infectious (I), recovered (R) and dead (D), such
that $N=S+E+I+R+D$. Arrows show temporal progression(see text for the
mathematical implementation of the diagram.) A. The simpler SIR model. B. The
SEIR model, which includes in addition to the SIR model a class of exposed
individuals, which are not infectious.

The SIR model (Figure 2A) is the simplest relevant description of an epidemic
in a population. The model is typically written in terms differential
equations as

$\displaystyle\frac{dS}{dt}=-\beta\frac{S}{N}I,\quad\frac{dI}{dt}=\beta\frac{S}{N}I-\gamma
I,\quad\frac{dR}{dt}=(1-m)\gamma I,\quad\frac{dD}{dt}=m\gamma I.$ (2)

Here, $\beta$ is known as the contact rate, $1/\gamma$ is the infectious
period, $1/\sigma$ is the non-infectious incubation time, and $m$ is the
probability that an infected individual dies (mortality rate).

The reproductive rate $R$ can be easily deduced from the initial growth of
$I$, when $S/N\simeq 1$, which is

$\displaystyle\frac{dI}{dt}=\gamma\left[R\frac{S}{N}-1\right]I,$ (3)

with $R=\beta/\gamma$. We see that the temporal growth rate
$r=\gamma\left[R\frac{S}{N}-1\right]\simeq\gamma(R-1)$. We see that the number
of cases will grow exponentially if $R>1$ and, conversely, decay exponentially
when $R<1$.

At a later time, as the susceptible population becomes depleted, the effective
$R$ decreases. Distancing or vaccination work by removing people from the
susceptible class, which can be modelled by reducing $S/N$ or, equivalently
for $R$, $\beta$ since these two factors always appear multiplied together i.
the y

To obtain the expression for the final size of the epidemic we note that

$\int_{0}^{\infty}dt^{\prime}\frac{d(S+I)}{dt^{\prime}}=S_{\infty}-S_{0}-I_{0}=-\gamma\int_{0}^{\infty}dt^{\prime}I(t^{\prime}),$
(4)

where $S_{\infty}$ is the population of susceptibles left uninfected at the
end of the outbreak, and $S_{0},\ I_{0}$ are the size of the susceptible and
infected population at the initial time.

The equation for $S$ in time can be written as

$\frac{dS}{dt}=-\beta\frac{S}{N}I\rightarrow\frac{d\ln
S}{dt}=-\frac{\beta}{N}I,$ (5)

which now can be integrated to give

$\ln\frac{S_{\infty}}{S_{0}}=-R(1-\frac{S_{\infty}}{N}),$ (6)

which is the desired result, used in the main text to create Figure 1B.

Finally we note that even though a model with a non-infectious incubation
period such as the SEIR looks more complex, it has the same value of $R$
provided we can neglect mortality in the $S,E$ classes, not due to the disease
outbreak.

Derivation of the City Size Dependence of the Reproductive Number

The most important quantity characterizing epidemic processes is the
reproductive number, $R$, which measures the number of secondary cases induced
by an infectious individual in a fully susceptible population. Recall that we
expect that human population are thought to be wholly susceptible to COVID-19.

For a contagion network, the reproductive number is related to the statistics
of degree, $k$, as

$R=p_{I}(\frac{\langle k^{2}\rangle}{\langle k\rangle})=p_{I}\langle
k\rangle\left(1-\frac{\sigma_{k}^{2}}{\langle k\rangle^{2}}.\right),$ (7)

where $p_{I}$ is the infection probability per contact, $\langle...\rangle$
denotes expectation values over the population and $\sigma_{k}^{2}$ is the
degree variance.

For a lognormal degree distribution, which is typical of social networks in
cities, the degree average and variance are given by

$\langle
k\rangle=e^{\mu+\sigma^{2}/2},\quad\sigma_{k}^{2}=\left(e^{\sigma^{2}}-1\right)e^{2\mu+\sigma^{2}},$
(8)

with the parameters $\mu=\langle\ln k\rangle$, $\sigma^{2}=\langle\left(\ln
k-\angle\ln k\rangle\right)^{2}\rangle$. This results in a simple and elegant
expression for the reproductive number

$R(N)=p_{I}e^{\sigma^{2}}\langle
k(N)\rangle=p_{I}k_{0}e^{\sigma^{2}}N^{\delta},$ (9)

where, in the last equality, we introduced the scaling relation for the
average degree with city population size, $\langle
k(N)\rangle=k_{0}N^{\delta}$. We see therefore that in general the
reproductive number is expected to be a function of city size $N$, and to be
larger in bigger cities. How much bigger, depends on the behavior of the log-
variance, $\sigma^{2}$, and whether this parameter is city size dependent, an
issue that can generates statistical corrections beyond “mean-field” to the
simplest expectations from urban scaling theory with $\delta\simeq 1/6$.

## Supplementary Figures

Supplementary Figure 1: Larger cities are expected to require more aggressive
strategies to control epidemics than smaller cities. Higher values of $R$
result in a a greater required vaccination rate, $p_{R}=1-1/R$, in order to
stop the outbreak. This rate applies to social distancing as well as
vaccination and herd immunity. The translation of growth rates into
reproductive numbers was obtained using an infectious period of $1/\gamma=4.5$
days. These estimated values of $R$ are high in some cases (e.g. New York
City) compared to reports in other situations and may in part be the result of
the acceleration of testing in larger cities and specific places.
Supplementary Figure 2: Scaling residuals are unrelated to city size. The lack
of a correlation between the residuals of the scaling fit (Figure 1a) and the
logarithm population indicate that a correction to the scaling exponent is not
necessary. Residuals were calculated from the OLS fit to the plot of logarithm
of COVID-19 case growth rates and the logarithm of population, where growth
rates were estimated with linear regression to the COVID-19 case time-series.
Supplementary Figure 3: Scaling residuals roughly normal distributed. Residual
from the scaling fit (Figure 1a) roughly match theoretical quantiles of a
normal distribution, indicating a good linear fit. Residuals were calculated
from the OLS fit to the plot of logarithm of COVID-19 case growth rates and
the logarithm of population, where growth rates were estimated with linear
regression to the COVID-19 case time-series.  Supplementary Figure 4: Scaling
Plots for alternate methods for estimating COVID-19 case growth rates. Both
alternate methods produce growth rates which show scaling relationships with
city size consistent with Figure 1a. Growth rates were estimated with (a) a
linear regression, as in Figure 1a, to data from March 16th to March 23rd (b)
$r=ln(cases_{T}/cases_{0})/T$ to data from March 13th to March 25th. This is
an estimate of the slope of the line $ln(cases)\sim ln(a)+r\cdot t$ from the
first and last points of the time series. Supplementary Figure 5: Map of
residuals of the scaling fit in Figure 1a

## Supplementary Tables

Supplemetary Table 1: Estimated COVID-19 daily growth rates in MSA

Estimated daily COVID-19 |
---|---
Case Growth Rate | MSA
0.43 | New York-Newark-Jersey City, NY-NJ-PA
0.25 | Los Angeles-Long Beach-Anaheim, CA
0.36 | Chicago-Naperville-Elgin, IL-IN-WI
0.30 | Dallas-Fort Worth-Arlington, TX
0.23 | Houston-The Woodlands-Sugar Land, TX
0.31 | Miami-Fort Lauderdale-Pompano Beach, FL
0.28 | Philadelphia-Camden-Wilmington, PA-NJ-DE-MD
0.22 | Washington-Arlington-Alexandria, DC-VA-MD-WV
0.26 | Atlanta-Sandy Springs-Alpharetta, GA
0.21 | Longview, TX
0.33 | Phoenix-Mesa-Chandler, AZ
0.18 | San Francisco-Oakland-Berkeley, CA
0.18 | Riverside-San Bernardino-Ontario, CA
0.18 | Boston-Cambridge-Newton, MA-NH
0.40 | Oklahoma City, OK
0.49 | Detroit-Warren-Dearborn, MI
0.14 | Seattle-Tacoma-Bellevue, WA
0.24 | Minneapolis-St. Paul-Bloomington, MN-WI
0.40 | St. Louis, MO-IL
0.32 | San Diego-Chula Vista-Carlsbad, CA
0.36 | Tampa-St. Petersburg-Clearwater, FL
0.30 | Baltimore-Columbia-Towson, MD
0.23 | Denver-Aurora-Lakewood, CO
0.40 | Charlotte-Concord-Gastonia, NC-SC
0.39 | Orlando-Kissimmee-Sanford, FL
0.43 | San Antonio-New Braunfels, TX
0.23 | Portland-Vancouver-Hillsboro, OR-WA
0.14 | Sacramento-Roseville-Folsom, CA
0.37 | Pittsburgh, PA
0.26 | Las Vegas-Henderson-Paradise, NV
0.32 | Kansas City, MO-KS
0.32 | Cleveland-Elyria, OH
0.29 | Indianapolis-Carmel-Anderson, IN
0.13 | San Jose-Sunnyvale-Santa Clara, CA
0.40 | Columbus, OH
0.30 | Providence-Warwick, RI-MA
0.31 | Nashville-Davidson–Murfreesboro–Franklin, TN
0.43 | Milwaukee-Waukesha, WI
0.28 | Jacksonville, FL
0.35 | Austin-Round Rock-Georgetown, TX
0.21 | Raleigh-Cary, NC
0.46 | Hartford-East Hartford-Middletown, CT
0.48 | Memphis, TN-MS-AR
0.30 | Louisville/Jefferson County, KY-IN
0.40 | Rochester, NY
0.39 | Birmingham-Hoover, AL
0.25 | Tucson, AZ
0.25 | Grand Rapids-Kentwood, MI
0.24 | Fresno, CA
0.36 | Urban Honolulu, HI
0.35 | Bridgeport-Stamford-Norwalk, CT
0.33 | Worcester, MA-CT
0.35 | Albany-Schenectady-Troy, NY
0.13 | Omaha-Council Bluffs, NE-IA
0.26 | Knoxville, TN
0.23 | Oxnard-Thousand Oaks-Ventura, CA
0.21 | El Paso, TX
0.46 | Allentown-Bethlehem-Easton, PA-NJ
0.30 | Columbia, SC
0.22 | Albuquerque, NM
0.22 | North Port-Sarasota-Bradenton, FL
0.40 | Charleston-North Charleston, SC
0.21 | Stockton, CA
0.27 | Cape Coral-Fort Myers, FL
0.45 | Colorado Springs, CO
0.35 | Akron, OH
0.29 | Boise City, ID
0.46 | Poughkeepsie-Newburgh-Middletown, NY
0.16 | Tulsa, OK
0.15 | Deltona-Daytona Beach-Ormond Beach, FL
0.29 | Madison, WI
0.21 | Wichita, KS
0.43 | Jackson, MS
0.41 | Durham-Chapel Hill, NC
0.27 | Winston-Salem, NC
0.13 | Harrisburg-Carlisle, PA
0.15 | Modesto, CA
0.35 | Youngstown-Warren-Boardman, OH-PA
0.32 | Chattanooga, TN-GA
0.26 | Fayetteville, NC
0.17 | Lexington-Fayette, KY
0.29 | Santa Rosa-Petaluma, CA
0.34 | Lansing-East Lansing, MI
0.30 | Richmond, VA
0.30 | Myrtle Beach-Conway-North Myrtle Beach, SC-NC
0.25 | Visalia, CA
0.28 | Springfield, MO
0.21 | Reno, NV
0.37 | Huntsville, AL
0.14 | Vallejo, CA
0.27 | Salem, OR
0.31 | Ogden-Clearfield, UT
0.20 | Canton-Massillon, OH
0.28 | Naples-Marco Island, FL
0.28 | Ann Arbor, MI
0.40 | Trenton-Princeton, NJ
0.30 | Killeen-Temple, TX
0.42 | Fort Collins, CO
0.34 | South Bend-Mishawaka, IN-MI
0.19 | Montgomery, AL
0.12 | Spartanburg, SC
0.31 | Greeley, CO
0.28 | Utica-Rome, NY
0.21 | Virginia Beach-Norfolk-Newport News, VA-NC
0.20 | Olympia-Lacey-Tumwater, WA
0.11 | Santa Cruz-Watsonville, CA
0.38 | Crestview-Fort Walton Beach-Destin, FL
0.38 | Gainesville, FL
0.21 | Bremerton-Silverdale-Port Orchard, WA
0.05 | Sioux Falls, SD
0.24 | Yakima, WA
0.16 | Binghamton, NY
0.19 | Tuscaloosa, AL
0.13 | Hereford, TX
0.13 | Tyler, TX
0.31 | Bellingham, WA
0.22 | Lebanon, NH-VT
0.34 | Burlington-South Burlington, VT
0.12 | St. Cloud, MN
0.29 | Rochester, MN
0.20 | Racine, WI
0.13 | Bend, OR
0.18 | Torrington, CT
0.14 | El Centro, CA
0.13 | Punta Gorda, FL
0.18 | Kingston, NY
0.11 | Iowa City, IA
0.23 | Billings, MT
0.26 | East Stroudsburg, PA
0.15 | Pueblo, CO
0.21 | Madera, CA
0.16 | Santa Fe, NM
0.17 | Hattiesburg, MS
0.40 | Albany, GA
0.12 | Pittsfield, MA
0.29 | Mount Vernon-Anacortes, WA
0.09 | Albany-Lebanon, OR
0.21 | Morristown, TN
0.18 | Valdosta, GA
0.09 | Sheboygan, WI
0.33 | Bozeman, MT
0.13 | Lewiston-Auburn, ME
0.06 | Fond du Lac, WI
0.10 | Rome, GA
0.18 | Oak Harbor, WA
0.20 | Kokomo, IN
0.11 | Glenwood Springs, CO
0.13 | Minot, ND
0.36 | Heber, UT
0.09 | Kapaa, HI
0.12 | Calhoun, GA
0.27 | Picayune, MS
0.18 | Edwards, CO
0.05 | Ellensburg, WA
0.19 | Cedartown, GA
0.25 | Riverton, WY
0.25 | Greenwood, MS
0.10 | Bennington, VT
0.05 | Mount Sterling, KY
0.18 | Breckenridge, CO
0.16 | Sheridan, WY
0.13 | Steamboat Springs, CO
0.30 | Huron, SD

## References

  * 1. Y. Liu, A. A. Gayle, A. Wilder-Smith, J. Rocklöv, Journal of Travel Medicine 27 (2020). Taaa021.
  * 2. J. T. Wu, K. Leung, G. M. Leung, The Lancet 395, 689 (2020).
  * 3. S. Yang, et al., Annals of Translational Medicine 8 (2020).
  * 4. M. Biggerstaff, S. Cauchemez, C. Reed, M. Gambhir, L. Finelli, BMC infectious diseases 14, 480 (2014).
  * 5. Https://www.whitehouse.gov/briefings-statements/coronavirus-guidelines-america/.
  * 6. S. Chen, J. Yang, W. Yang, C. Wang, T. Bärnighausen, The Lancet (2020).
  * 7. L. M. Bettencourt, science 340, 1438 (2013).
  * 8. M. Schläpfer, et al., Journal of the Royal Society Interface 11, 20130789 (2014).
  * 9. B. D. Dalziel, et al., Science 362, 75 (2018).
  * 10. L. M. Bettencourt, J. Lobo, D. Strumsky, Research policy 36, 107 (2007).
  * 11. M. P. Feldman, D. B. Audretsch, European economic review 43, 409 (1999).
  * 12. Z. J. Acs, Urban Dynamics and Growth: Advances in Urban Economics (Elsevier, 2004), vol. 266 of Contributions to Economic Analysis, pp. 635 – 658.
  * 13. R. M. Anderson, B. Anderson, R. M. May, Infectious diseases of humans: dynamics and control (Oxford university press, Oxford, Reprinted, 2010).
  * 14. L. M. Bettencourt, A. Cintrón-Arias, D. I. Kaiser, C. Castillo-Chávez, Physica A: Statistical Mechanics and its Applications 364, 513 (2006).
  * 15. Https://www.census.gov/programs-surveys/metro-micro/about/delineation-files.html.
  * 16. Https://www.census.gov/data/tables/time-series/demo/popest/2010s-counties-total.html.
  * 17. S. Yadlowsky, N. Shah, J. Steinhardt, medRxiv (2020).