Evolving spatiotemporal patterns and urban scaling of deaths from external causes

Abstract

Urban scaling theory posits that urban indicators follow power-law relations with population, yet the evolution of these patterns - and the role of regional differences in settings marked by social inequalities and unplanned urbanization - remains poorly understood. Here, we analyze nearly three decades of mortality data from Brazilian cities to investigate the scaling of external causes of death: homicides, suicides, and accidents. Using a hierarchical Bayesian framework and spatial correlation analysis, we find that these mortality indicators exhibit distinct, regionally heterogeneous scaling trajectories. Homicide mortality has significantly attenuated its typical superlinear scaling with increased spatial clustering, suggesting a redistribution of violence to smaller cities and intensified intercity interactions, possibly linked to the consolidation of organized crime. Suicide mortality, usually sublinear, has trended upward, implying a weakening of urban agglomerations' protective effect. Accident mortality remains superlinear, with transport fatalities scaling nearly proportionally, and non-transport accidents becoming superlinear. The scaling changes for suicides and accidents coincide with less correlated and stable spatial patterns, suggesting that the underlying processes predominantly operate within city boundaries. Finally, while scaling exponents have evolved more homogeneously across Brazilian states, scale-adjusted mortality remains highly heterogeneous, indicating that fundamental processes govern scaling laws, whereas state-specific factors drive scale-adjusted metrics.

Figures (8)

• Figure 1: Evolution of mortality rates from external causes in Brazil between 1996 and 2023. (A) Nationwide mortality rates due to homicides (red circles), suicides (green downward triangles), and accidents (blue upward triangles). Homicide rates showed an increasing trend from 1996 to 2006, followed by a temporary dip in 2004 and 2005 that started a prolonged period of increase until 2017, after which rates declined sharply -- from $31$ to $21$ deaths per $100{,}000$ individuals between 2018 and 2019 -- stabilizing around this value ever since. Suicide rates exhibited a monotonically increasing trend, rising from $4.1$ to $8.4$ deaths per $100{,}000$ people over the study period. Accident rates initially decreased to a minimum of $30.3$ deaths per $100{,}000$ people in 2001, followed by an increasing trend that resulted in a peak of $37.7$ deaths per $100{,}000$ people in 2012, after which rates decreased to $31$ deaths per $100{,}000$ people in 2019, initiating a new upward trend. Panels (B), (C), and (D) present mortality rates for each external cause of death stratified by city population size, grouped into four categories as indicated in the legends. (B) Homicide rates exhibited a size-dependent hierarchy until 2017, with larger cities consistently having higher rates. After 2017, rates across all size categories converged, suggesting city size ceased to be a determinant of homicide mortality. Before this convergence, smaller cities displayed a more pronounced rise, peaking in 2017, while larger cities showed a steeper decline thereafter. (C) Suicide rates maintained a stable, size-dependent hierarchy, with smaller cities experiencing consistently higher rates than larger ones. All city size categories displayed a similar increasing trend, mirroring the national pattern, although larger cities experienced a faster rise in suicide rates, particularly after 2016. (D) Accident rates vary less across city size categories, with smaller cities typically experiencing slightly lower rates and an evolution closely following the overall national trend.
• Figure 2: Evolution of mortality rates from transport and non‐transport accidents in Brazil between 1996 and 2023. (A) Nationwide mortality rates for transport (light blue rightward triangles) and non‐transport (dark blue leftward triangles) accidents. Transport-related mortality exceeded that from non‐transport accidents until 2018, with a more pronounced gap during the 2000s and early 2010s. Transport accident rates began declining in 2012, whereas non‐transport rates exhibited a small but steady upward trend, leading to convergence in 2018. After that year, both rates followed a similarly slight upward trajectory. Panels (B) and (C) present mortality rates for transport and non‐transport accidents stratified by city population size, grouped into four categories as indicated in the legends. (B) Transport accident rates were less dependent on city size and followed trends similar to those observed nationwide. (C) Non‐transport accident rates showed a clear dependence on city size, with larger cities exhibiting significantly higher rates, particularly after 2010.
• Figure 3: Urban scaling of external causes of death in Brazil. Population scaling relations for (A) homicides, (B) suicides, and (C) accidents in 2000, 2010, and 2020. Markers represent the number of deaths ($Y$) due to each external cause versus the population ($N$) of Brazilian cities on a base-10 logarithmic scale ($\log Y$ versus $\log N$). Continuous lines indicate the nationwide scaling laws estimated using our Bayesian hierarchical approach, while dashed lines represent scaling laws obtained via the ordinary least-squares (OLS) method applied to log-transformed data. The urban scaling exponents and their standard errors are provided in the plot legends. The Bayesian estimated scaling laws closely match the OLS estimates for suicides and accidents, whereas for homicides, the Bayesian approach consistently yields a higher scaling exponent across all years. These findings indicate that homicides scale superlinearly, suicides sublinearly, and accidents linearly with population size, with the scaling exponent for homicides exhibiting a significant decline over time.
• Figure 4: Evolution of the urban scaling exponent for external causes of death in Brazil between 1996 and 2023. (A) Homicide exponents initially increased from $\beta=1.17$ in 1996 to a peak of $\beta=1.33$ in 2012, then decreased sharply to a minimum of $\beta=1.12$ in 2022. (B) Suicide exponents exhibited an overall upward trend, rising from a minimum of $\beta=0.70$ in 1999 to $\beta=0.92$ in 2022. (C) Accident exponents remained relatively stable over the years, oscillating around $\beta\approx1.1$. Disaggregating accidents into (D) transport-related and (E) non-transport-related categories reveals distinct trends. Transport-related accidents display an almost constant exponent near $1$, whereas the exponents for non-transport-related accidents, which were close to $1$ until 2010, increased thereafter, reaching a plateau at approximately $\beta\approx1.2$ in recent years. In all panels, markers denote the mean of the posterior distribution of $\beta$, and the shaded regions represent the 96% highest density (credible) intervals, with dashed lines indicating the isometric scaling regime.
• Figure 5: Individual evolution of mortality scaling exponents for each Brazilian state from 1996 to 2023. Colored curves represent the posterior probability distribution of $\beta_j$ for each state $j$, estimated for (A) homicides, (B) suicides, and (C) accidents. Each row of plots corresponds to a Brazilian state, with colors indicating different years (as shown in the legend). Vertical lines in all three panels denote the isometric regime. The evolution of each state generally aligns with nationwide trends for homicides and suicides, although some states exhibit distinct scaling exponents and rates of change. Consistent with national patterns, the posterior distributions of $\beta_j$ remain approximately stationary and show little variation across states.