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Understanding the role of urban design in disease spreading

Cities are complex systems whose characteristics impact the health of people who live in them. Nonetheless, urban determinants of health often vary within spatial scales smaller than the resolution of epidemiological datasets. Thus, as cities expand and their inequalities grow, the development of theoretical frameworks that explain health at the neighborhood level is becoming increasingly critical. To this end, we developed a methodology that uses census data to introduce urban geography as a leading-order predictor in the spread of influenza-like pathogens. Here, we demonstrate our framework using neighborhood-level census data for Guadalajara (GDL, Western Mexico). Our simulations were calibrated using weekly hospitalization data from the 2009 A/H1N1 influenza pandemic and show that daily mobility patterns drive neighborhood-level variations in the basic reproduction number R0, which in turn give rise to robust spatiotemporal patterns in the spread of disease. To generalize our results, we ran simulations in hypothetical cities with the same population, area, schools and businesses as GDL but different land use zoning. Our results demonstrate that the agglomeration of daily activities can largely influence the growth rate, size and timing of urban epidemics. Overall, these findings support the view that cities can be redesigned to limit the geographic scope of influenza-like outbreaks and provide a general mathematical framework to study the mechanisms by which local and remote health consequences result from characteristics of the physical environment.

Empirical studies have identified inter-city variations in the timing, intensity and 2 severity of influenza-like outbreaks [1][2][3][4]. Aiming to understand the mechanisms through 3 which city characteristics yield such health consequences, epidemiologists have resourced 4 to a variety of methods. Epidemiological data reveal compelling statistical correlations, 5 but do not resolve intra-city variations in health that are driven by lifestyle inequalities 6 at the neighborhood level [2,[4][5][6]. In contrast, agent-based computational models use 7 massive mobility datasets to recreate the behavior of individuals as they interact and 8 spread infections [7][8][9][11][12][13][14]. While this approach allows for household-level analyses 9 and its elevated complexity makes it suitable for targeted experiments [15], it is not the 10 best tool for the development of general strategies in public health [16]. Furthermore, 11 the information necessary to calibrate agent-based simulations is not openly available 12 for most of the world's cities. Thus, despite the fact that epidemics have the potential 13 to be seeded anywhere, these intricate models have been overwhelmingly applied to 14 populations in the developed world. Meanwhile, the fundamental mechanisms that drive 15 health inequalities within metropolitan areas remain elusive. 16 Urban design determines the densities and relative locations of housing, jobs and 17 services inside a city. Consequently, it influences the transportation choices of the 18 population [17][18][19] and thus helps shape interaction networks through which diseases 19 are spread. For example, the agglomeration of jobs and services drives large fractions of 20 a city's population to gather in small fractions of its area, increasing contact rates 21 between the residents of distant neighborhoods [20,21]. In what follows, we quantify the 22 agglomeration of urban mobility using two Gini coefficients 23 (0 ≤ G origins , G destinations ≤ 1) that use neighborhood-level census data to measure 24 spatial inequalities in the area-density of housing and activities throughout a city. The 25 extreme values G = 0 represent homogeneous distributions of the population (trip 26 origins) or its activities (trip destinations); inversely, G = 1 indicates that all housing or 27 activities are concentrated within a single location. In epidemiological terms, G = 1 28 produces homogeneous mixing conditions, which allow individuals to interact 29 indiscriminately with all other members of the population. Under this scenario, disease 30 spreading may be modeled using simple ordinary differential equations where the total 31 population N = S + I + R is split into susceptible S, infected I, and recovered R groups. 32 However, reality deviates from these conditions (G = 1) and successful models must patterns between all identifiable subgroups. Therefore, unexplained variations within 41 sampled groups can cause statistical uncertainty intervals to be ≥50% the magnitude of 42 reported mean contact rates [24][25][26]. Among the numerous likely sources of contact rate 43 variaibility, crowding driven by urbanization and mobility patterns has been suggested 44 to increase the growth-rate and size of infectious outbreaks by observational and 45 modeling studies alike [4,14]. 46 In this article, we develop a mathematical framework that introduces urban 47 geography as the leading-order component of a susceptible-infected-recovered (SIR) 48 epidemiological model. Our methodology uses spatially-resolved census data to infer 49 transportation patterns in cities and is meant to bypass the need for large mobility 50 datasets in urban health simulations. In our model, individuals are grouped by age and 51 place of residence, and are subsequently represented through probability density 52 functions that describe their likely travel habits. This approach sacrifices the 53 hyperrealism of agent-based simulations to instead resolve the spatial patterns that arise 54 when a heterogeneous set of metapopulations interact. Our goal is not to make an 55 operational forecasting tool, but instead to formulate a scheme that helps understand 56 the health consequences of the complex, spatially-dependent demographics of large 57 cities.

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As a first step, we characterize the role of urban design in this process and use the  Lastly, we ran simulations in hypothetical cities with the same area, population density, 63 number of schools and businesses as GDL. This allowed us to demonstrate that changes 64 in the spatial distributions of housing, education and economic activities yield large 65 variations in the size and early growth rate of epidemics across cities that would be 66 deemed identical from a large-scale perspective. High-resolution maps of residential density provide the background distribution of a 70 city's population, introducing a first layer of spatial dependence to the SIR transmission 71 framework. Likewise, the locations and prominence of activity hubs determine common 72 trip destinations were human interactions occur and diseases are spread. In our gravity 73 model, trips originate at home and the probability density P (x, y ) that a person 74 residing at x will visit y on a given day is the joint result of two factors: The distance 75 ∆r xy between both sites and the overall popularity of y as a destination. The effect of 76 distance ∆r xy on the likelihood of displacements between two points has been studied 77 empirically [27], yielding the general form 78 P (∆r xy ) = (∆r xy + ∆r 0 ) −b exp(−∆r xy /κ). (1) Simplifying daily mobility as radial displacements out of a place of residence x, 79 P (∆r xy ) yields a first estimate of the spatial distribution of individuals as they go 80 through their daily routines. Moreover, the model parameters ∆r 0 , b, κ can be used to 81 capture the effects of transportation infrastructure, as the quality of these services 82 largely determines the willingness of individuals to travel long distances [28].

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T AS(y ) = 3.0 Re + 2. into a location y by education and economic activities [29]. Its value may be calculated 86 using school enrollment data and the number of workers employed at y by different 87 economic sectors, allowing different types of establishments to be weighed by the traffic 88 they each induce. Our model incorporates TAS through Equation (2), where parameters 89 Re, Se, In and Pr denote the number of jobs registered at y by retail, service, industrial 90 and primary activity organizations respectively [29]. Similarly, St is the number of 91 students enrolled at educational institutions inside the same area.
Equation (3) thus defines a probabilistic gravity model that represents mobility as 93 the weighted influence areas of individuals who try to minimize their displacements but 94 will travel farther given an economic incentive (higher TAS). Here, the normalization  Before integrating these concepts into an epidemiological transmission model, we 101 must distinguish between the behavior of susceptible and infected groups. We define the 102 mobility of symptomatic groups via Equation (4), where 0 < α(x) < 1 is an isolation 103 parameter whose spatial dependence allows the representation of neighborhood-targeted 104 intervention strategies as well as social and economic factors that influence the adaptive 105 behavior of sick individuals. The term H(x, y ) accounts for the added probability of 106 visits to the nearest healthcare facility, as sick individuals will likely seek diagnosis and 107 treatment.
In summary, we use the gravity model in Equation 3 to infer mobility patterns in week and within a single day (for example restaurants and schools have strongly marked 113 daily cycles), our simulations consider T AS to be fixed in time. Lastly, we incorporate 114 all state-run healthcare facilities (henceforth hospitals) by assigning infective individuals 115 to the nearest hospital and adding an 8% probability that they will visit it once during 116 their infective period. An overview of all spatial components of our framework, as 117 estimated for GDL (see Materials and Methods), is shown in Figure 1. S(x, t) + I(x, t) + R(x, t) = ρ(x) such that Ω ρ(x)dΩ = N . This approach links people 123 to their place of residence (henceforth x ∈ Ω for susceptibles andx ∈Ω for infectives), 124 allowing to formulate a Distributed Contacts SIR model (DC-SIR, Eqs. 5-7 adapted 125 from [30,31]). Here, β(y ) is the probability of contagion given a susceptible-infective 126 interaction that happens at y and γ is the recovery rate. The interaction kernel 127 k(x, y , t) yields the expected number of interactions between a member of S(x, t) and 128 all infected individuals present at a trip destination y ∈ Ω and time t [30,31]. Thus, 129 k(x, y , t) introduces the heterogeneous contact networks through which diseases are March 19, 2020 5/18 To define the interaction kernel k(x, y , t), we use origin-destination density 132 functions P S (x, y , t), P I (x, y , t) for members of S(x, t) and I(x, t) respectively. Next, 133 we compute the total number of infected individuals expected to visit y at a given time, 134 which is the integral from all trip origins Ω P I (x, y , t) · I(x, t) dΩ. Taking the product 135 with P S (x, y , t) thus gives an expression for the expected number of SI contacts 136 occurring at y for an individual who resides at x A similar procedure yields the reproductive number R t (x, t), defined as the number 138 of secondary cases caused by a single infected individual who lives atx and contracted 139 the disease at time t. Instead of calculating the number of SI interactions for a 140 susceptible in transit, we now use the spatial distribution of susceptibles at a given time. 141 This is given by the integration of susceptible mobility over all trip origins 142 Ω P S (x, y , t)S(x, t) dΩ. Multiplying this by the origin-destination density for infected 143 mobility P I (x, y , t) then yields the expected number of SI interactions occurring at y 144 for a member of I(x, t) Assuming that all variables are slowly-varying over the first generation of disease 146 transmission, we can approximate the basic reproduction number R 0 (x) as Notice that k(x, y , t) and q(x, y , t) represent the vulnerability and disease

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Our DC-SIR simulations of GDL classify two subsets of the population by place of 155 residence and age group. Thus, we defined SIR subgroups S j , I j , R j and mobility 156 functions P Sj , P Ij to represent the people in each age group j = 1, 2 (adults and 157 children). Consequently, the integrand over trip destinations Ω in Equation (5)  around major activity hubs. This is true for 60% of all neighborhoods in GDL, which 178 then act as net sources of mobility whose role is to supply more affluent destinations 179 with visitors. epidemic by as much as 9 days (Fig. 4.a,b). Secondly, people with the highest R 0 (x) 198 also had the highest probability of falling ill. In particular, the increased contact rates 199 March 19, 2020 7/18 of downtown residents allowed them to spread pathogens 50% more efficiently than 200 their suburban counterparts (Fig. 3), but also put them at a higher risk of contagion.

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Regardless of initial conditions, this particular situation gave rise to a spatial pattern in 202 which influenza spread as waves of disease that emanate out of the city center ( Fig. 4.c, 203 S2 Fig   and an infective period 1/γ of 4 days. Results are summarized in Figure 5.

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Our simulations suggest that, for all cities, the basic reproduction number R 0 varies 229 as a function of G destinations and is virtually independent of G origins (Fig. 5.a,b). This 230 relationship is nonlinear for adults and linear for children, mainly because G destinations 231 is set by commercial and economic superclusters that primarily attract adults.

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Furthermore, our results suggest that housing and activities in metropolitan areas with 233 G destinations < 0.58 can be redistributed so that the mean basic reproduction number of 234 adults remains under the threshold value R 0 = 1. Namely, in this illustrative case where 235 β is fixed everywhere and transmission dynamics are solely driven by urban design, 236 cities can be redesigned to render their populations incapable of sustaining epidemics.

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At the end of an outbreak, the attack rate z measures the fraction of the population 238 that contracted the disease. Under homogeneous mixing conditions 239 (G destinations = G origins = 1), R 0 and z are linked by Equation (12) [37]. At all values 240 of R 0 , this relationship predicts higher attack rates z than observed in our simulations 241 (Fig. 5.c). Disagreements are largest for low values of R 0 , when fractions of the urban 242 population have R 0 < 1 and thus do not contribute to exponential growth. This 243 highlights the importance of understanding population heterogeneity: because R 0 and 244 March 19, 2020 9/18 incidence may covary in space (Figs. 3, 4.c), population-averaged parameters may differ 245 from the characteristic values obtained from public health reports, whose samples are 246 biased towards subgroups with a higher incidence. Differences between the GDL and 247 Random experiments in Figure 5 suggest that, at equal R 0 , realistic urban layouts 248 (where activity areas are clustered near each other) may act to decrease the size of 249 epidemics when compared to cases where housing and activities are distributed 250 randomly. However, notice that a greater G destinations is required for 251 randomly-designed cities to have the same values of R 0 as GDL (Fig. 5.a).  (10) in cities with the spatial structure of GDL (blue) and those with randomized urban design (pink). The relationship between population-averaged R 0 and attack rate z is shown in panel c for cities with realistic (blue) and random (pink) spatial structures. The theoretical relation for the attack rate under homogeneous mixing conditions (Eq. 12) is shown in orange. Disagreements between these curves show the impacts of urban design on the growth rate and size of influenza-like outbreaks for cities with the same area, population and economic activities.

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Our parameterization of contact heterogeneity in metropolitan areas produces smaller 254 epidemics than homogeneous mixing models (Fig. 5) and predicts that, depending on 255 where they live, people play unequal roles in disease spreading (Fig. 3). These 256 inequalities relate the identity of patient zero to the timing of influenza-like outbreaks 257 (Fig. 4.a,b) and can give rise to spatiotemporal patterns in the spread of disease (Fig.   258 4.c, S2 Fig, S1 Video). In fact, these processes mirror those of large scale transportation 259 March 19, 2020 10/18 networks, whose heterogeneity leads to spatial inequalities in the size, timing and 260 growth rate of epidemics [1,38,39]. As similar insight is used for the development of 261 optimal intervention strategies in global-scale infectious outbreaks [40], results like those 262 presented in Figure 3 may offer valuable information for public health officials who seek 263 to optimize resources during infectious outbreaks. For example, health inequalities 264 inferred here suggest that the efficiency of vaccination campaigns is likely to vary 265 whether they target the inhabitants of the city center, its visitors, or people living in 266 suburban areas (Figs. 3, 4). Similarly, the need for treatment and diagnosis is expected 267 to differ across neighborhoods (Fig. 4.c).  representation of environmental health impacts [41,42] and is achieved here through 273 interaction kernels k(x, y , t) and q(x, y , t). While we used a constant disease 274 parameter, the representation of local and remote environmental conditions can be 275 further refined by assigning full spatial dependence to transmission parameters so that 276 β = β(x, y ,x, t). Generally speaking, our analytical framework is designed to map 277 geographic information onto social space, establishing connections that are weighed by a 278 risk of contagion β. This assigns realistic network characteristics to a spatially-explicit 279 disease transmission model, which simplifies the interpretation of simulation results and 280 may enable additional theoretical and statistical analyses [43,44].

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Our results highlight the extent to which the assumption of homogeneous mixing in 282 metropolitan areas can bias model results. Most notably, epidemic size in simulations 283 that use the same transmission parameter β throughout hypothetical, seemingly 284 identical cities, covers the entire range 0 < z ≤ 1 (Fig. 5.c). This suggests that 285 macroscopic quantities such as population size and density hold little to no dynamical 286 significance in the evolution of infectious outbreaks. Instead, the spatial organization of 287 human mobility (quantified here using the Gini coefficient G destinations , Fig. 2) seems 288 to control epidemic growth (Figs. 5.a,b) and is statistically correlated with population 289 size [4,14]. Dalziel and collaborators explored this notion [14] by comparing agent-based 290 model results across 48 Canadian cities, but were not able to isolate the effects of 291 spatial heterogeneity from all other information embedded in massive mobility datasets 292 that were used as model input.

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Empirical relationships between G destinations , R 0 and the attack rate z shown in 294 Figure 5 suggest that, through its influence on transportation networks, urban design 295 may be optimized to modify epidemic growth rates and thus reduce the probability of 296 seeding large outbreaks. The existence of invasion thresholds under which populations 297 cannot sustain epidemic growth is a general characteristic of metapopulation 298 models [45]. While many authors have conjectured about the possibility of designing 299 cities to minimize disease prevalence [4,14,46,47], we present the first dynamical 300 argument to link these ideas and the invasion threshold proposition of Colizza and 301 Vespignani [45]. More specifically, our results suggest that, by evenly distributing 302 activity hubs throughout a city (instead of clustering them in the city center), city 303 planners can segregate subsets of the population and potentially inhibit the rapid 304 transmission of pathogens across distant neighborhoods (Fig. 5). Consistent with 305 known statistical relations between the transmission potential of influenza and the 306 spatial organization of human behavior [4,14], our framework explains a plausible 307 mechanism behind health inequality across the neighborhoods of the worlds' cities.

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The mechanism through which urban design impacts epidemic growth is illustrated 309 in Figure 6: when daily activities are centralized in a handful of hyper-affluent areas 310 Fig 6. Schematic comparison of urban design scenarios. When activities are centralized in a single hub (left), people from all parts of the city gather in this place and can potentially interact [21]. Meanwhile, more even distributions of daily activities (right) segregate subsets of the population and can lower the growth rate of infectious outbreaks (Fig. 5).
( Fig. 6.a), mixing patterns become more homogeneous [21] and thus favor interactions 311 between residents of distant neighborhoods (Fig. 6.c). However, when activity hubs are 312 distributed throughout the city (Fig. 6.b), economic opportunities become locally

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Notice that children's R 0 was not affected by the breakup of activity hubs in Figure 323 5.b. This lack of effect may be due to the fact that children's TAS is already scattered 324 through GDL in a seemingly-random manner, while the TAS of adults is highly 325 organized (Fig. 1). While the centralization of labor markets is thought to favor 326 economic efficiency [19,48], our theoretical analyses suggest that this may have negative 327 epidemiological consequences. In contrast, the decentralization of economic activities 328 may help slow the propagation of influenza-like outbreaks (Figs. 5, 6), reduce pollution 329 and improve overall welfare [49]. Agent-based models are the standard method to run realistic simulations of disease 331 spreading in cities [7][8][9][10][12][13][14]. These intricate models have allowed for the evaluation 332 of intervention strategies [12] and health inequalities [8,15], but require input from 333 massive mobility datasets that are not openly available for most of the world's cities. 334 Consequently, US [50] and European cities are over-represented by agent-based methods, 335 which remain inaccessible for many research groups and public health organizations. 336 Likewise, large-scale epidemiological models often assume homogeneous mixing 337 conditions in cities and neglect all processes that occur at the neighborhood 338 level [1,4,11,51]. When compared to agent-based model output, this simplification can 339 lead to overestimate the size of epidemics and miscalculate their timing, and can thus 340 introduce considerable bias [11]. Although far simpler, our parameterization of 341 small-scale heterogeneity leads to the same conclusions (Figs. 4, 5). Thus, we believe 342 that the mathematical framework presented here is an adequate alternative to introduce 343 neighborhood-level processes in large scale epidemiological simulations.

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Because it uses standard census and economic data to infer daily mobility patterns 345 (see Materials and Methods), our method enables researchers to investigate transmission 346 dynamics in virtually any city. Unfortunately, mobility parameters used here (Eqs. 1-3) 347 were not calibrated with real origin-destination data (which exist for GDL during the 348 2009 A/H1N1 pandemic [13] but are not openly available) and may thus be inaccurate. 349 Nonetheless, our analyses rely on the patterns that arise from fundamental processes 350 driven by the agglomeration of economic activities and highlight small-scale inequalities 351 that are not resolved by most epidemiological datasets. Agent-based simulations of 352 influenza outbreaks in the city of Buffalo [10] show spatiotemporal patterns (their 353 Figure 9) that are very similar to those presented here (S1 Video, Fig. 3) and thus 354 suggest that urban design may play a leading-order role in setting the dynamics of 355 influenza-like outbreaks in cities throughout the world.

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In reality, spatial inequalities in health result from the complex interactions of social, 357 environmental and biological processes among which urban design is only one. Even theoretical frameworks that seek to understand these complex interactions is crucial to 364 help design new policies and research that aim to improve health in cities [46,47,52]. 365 Ultimately, this study shows that systematic spatial variations in the mobility of 366 infective individuals can drive robust patterns in the spread of disease and thus give rise 367 to health inequalities at the neighborhood level (Figs. 3, 4, S1 Video).

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Inclusion of census data 370 We solved equations (5)-(7) in finite differences by mapping all their variables onto a 371 1580-element triangular mesh representing GDL ( [53], Fig. 1). Neighborhood-level data 372 from the 2010 census [35] were used to calculate the total adult (age > 15) and infantile 373 (age ≤ 15) populations at each grid element. T AS(y ) was estimated for adults and 374 children from two publicly available datasets: The 2015 National Statistical Directory of 375 Economical Units (DENUE) lists all registered employers in Mexico along with their 376 sector, number of workers and location [34]. Similarly, the National System for School 377 Information (SNIE) locates all of the country's schools and universities and lists their 378 enrollment at each educational stage [36]. DENUE employment data were combined 379 with SNIE enrollment at the high school and university levels to calculate TAS for the 380 adult population via equation 2. On the other hand, TAS for infantile mobility was 381 inferred using SNIE enrollment data for educational stages up to the middle school level, 382 roughly corresponding with the age threshold between our age groups. Next, 383 time-constant origin-destination matrices were obtained for all grid elements using 384 equations (3) and (4). Resulting maps of population density and TAS for adults and 385 children are shown in Figure 1.

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Mobility parameters were established a priori as ∆r 0 = 5 km, b = 1.75, κ = 80 km in 387 equation 1. Sensibility analyses showed that variations in these values primarily impact 388 G destinations , whose consequences are shown in Figure 5, but do not change the  Figures 1 and 3. Continuous input variables were transformed 397 to be constant at all grid elements, which are henceforth noted as Ω j ⊂ Ω, and whose 398 area is A j = Ωj dΩ. For instance, the number of susceptibles in Ω j at time t k , was 399 calculated as S j (t k ) = Ωj S(x, t) dΩ. Origin-destination matrices were defined using the 400 corresponding origin-destination density functions (Eq. 3) as 401 P S:jm = 1 A j Ωj Ω m P S (x, y ) dΩ dΩ.
Similarly, β(y ) can be mapped onto the grid as With all variables defined within the numerical grid, integrals overΩ and Ω in Data and code availability 407 MATLAB software and data used to perform the simulations described in this article are 408 available at https://github.com/inciente/Urban-Epidemiology/tree/master/RUDDS/. A 409 Python Jupyter Notebook that describes our methods is also included.