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Emergent Dynamical Spatial Boundaries in Emergency Medical Services: A Navier-Stokes Framework from First Principles
Emergency medical services (EMS) response time is a critical determinant of patient survival. For cardiac arrest patients, each minute of delay reduces survival probability by 7-10% larsen1993predicting, vukmir2006survival. For stroke patients, rapid treatment within 60 minutes of symptom onset (the "golden hour") can prevent permanent neurological damage saver2006time, meretoja2014reducing. For trauma patients, the first hour after injury—the so-called "golden hour"—determines mortality risk dinh2013redefining, newgard2010national. Yet despite this clinical importance, the spatial analysis of EMS coverage remains methodologically underdeveloped.
Current approaches to EMS coverage analysis rely primarily on two methods: (1) discrete distance buffers, where facilities are assumed to serve all areas within a fixed radius (e.g., 8 km for urban areas, 12 km for rural areas), and (2) GIS-based isochrones, which calculate travel-time polygons based on road networks and traffic conditions mclafferty2012rural, mccoy2013geographic. While these methods have practical appeal, they suffer from fundamental limitations. Discrete buffers impose arbitrary thresholds with no theoretical justification, creating sharp discontinuities where response effectiveness supposedly drops from full coverage to zero. GIS isochrones, while more sophisticated in incorporating road networks, remain descriptive tools that do not provide a causal framework for understanding how response effectiveness decays with distance or time.
This paper develops a theoretically grounded framework for emergency response boundaries by deriving the spatial decay function from first principles using fluid dynamics. The key insight is that emergency response can be modeled as a mass transport problem governed by the Navier-Stokes equations—the fundamental partial differential equations (PDEs) describing fluid motion. Just as pollutants disperse from a source following advection-diffusion dynamics, emergency response "flows" from EMS stations to incident locations, with effectiveness decaying as a function of travel time due to physiological deterioration of patient conditions.
The theoretical framework yields a tractable parametric form for response effectiveness:
where $\tau(x,t)$ is response effectiveness at location $x$ and time $t$, $\tau_0$ is baseline effectiveness (response at the source), $\kappa$ is the temporal decay parameter governing how quickly effectiveness diminishes, and $d(x,t)$ is the distance (or equivalently, travel time) from the nearest EMS station. This exponential form arises naturally from the advection-diffusion equation under steady-state assumptions, providing both theoretical justification and empirical tractability.
The framework enables calculation of critical boundaries $d^*$—the maximum distance beyond which response effectiveness falls below a policy-relevant threshold $\varepsilon$:
These boundaries provide actionable guidance for EMS station placement: areas beyond $d^*$ require additional stations to achieve adequate coverage. Unlike arbitrary distance buffers, $d^*$ is derived from the estimated decay dynamics and can be tailored to different policy objectives by varying the threshold $\varepsilon$.
This paper extends my research program on continuous functional frameworks for spatial treatment effects. The approach builds on theoretical foundations kikuchi2024unified, kikuchi2024navier, kikuchi2024stochastic, nonparametric methodology kikuchi2024nonparametric1, kikuchi2024nonparametric2, and empirical applications kikuchi2024dynamical, kikuchi2024healthcare.
The motivation comes from recognizing that treatment propagation follows physical principles. Just as heat diffuses continuously from sources, economic treatments spread through space following diffusion-advection dynamics captured by the Navier-Stokes equations.
Pollution dispersion: kikuchi2024nonparametric1 analyzes 42 million TROPOMI satellite observations of NO$_2$ from coal plants, finding spatial boundaries of 50 km with exponential decay validated. Nonparametric kernel methods reduce prediction errors by 1.0 percentage point, demonstrating when flexible methods outperform parametric specifications.
Banking deserts: kikuchi2024nonparametric2 applies the framework to bank branch consolidation, finding negative decay parameters that correctly signal urban confounding---branches locate in high-quality areas rather than causing quality. This demonstrates diagnostic capability.
Healthcare access: kikuchi2024healthcare uses CDC PLACES data (32,520 ZCTAs) to analyze hospital access, finding logarithmic decay strongly outperforms exponential ($\Delta$AIC $>$ 15,000). Spatial boundary is 37.1 km with 5-13$\times$ heterogeneity across education levels.
Dynamical boundaries: kikuchi2024dynamical develops the continuous functional framework emphasizing dynamic evolution and model selection, demonstrating predictive capability for boundary evolution over time.
Emergency response (this paper): The current application focuses on time-critical EMS where patient survival depends directly on response speed. Unlike spatial applications above (boundaries in kilometers), emergency response exhibits primarily temporal decay with boundaries in minutes. This demonstrates framework flexibility across spatial and temporal dimensions.
Together, these applications span environmental economics (pollution), financial services (banking), healthcare (hospital and EMS access), validating the framework's generality while revealing context-dependent functional form selection.
Using 10,000 simulated emergency incidents matched with demographic data from the U.S. Census, this paper documents four main findings:
Finding 1: Exponential decay characterizes response effectiveness. Estimating the temporal decay parameter yields $\hat{\kappa} \approx 0.344$ per minute (standard error = 0.023) across urgency levels (critical, urgent, routine). The exponential functional form fits well ($R^2 = 0.93$), and non-parametric kernel regression confirms this specification is appropriate, with kernel methods achieving mean squared error (MSE) 8-12 times smaller than parametric alternatives, validating the exponential assumption.
Finding 2: Critical boundaries $d^*$ vary by urgency. For critical incidents (8-minute threshold: cardiac arrest), $\hat{d}^* = 5.95$ minutes. For urgent incidents (15-minute threshold: stroke), $\hat{d}^* = 5.88$ minutes. For routine incidents (30-minute threshold), $\hat{d}^* = 5.96$ minutes. Currently, 36.3% of incidents fall beyond the 8-minute threshold, 14.5% beyond 15 minutes, and 2.2% beyond 30 minutes, revealing substantial coverage gaps.
Finding 3: Demographic heterogeneity is substantial. Elderly populations (85+) experience significantly longer response times (8.40 minutes) compared to younger adults aged 18-44 (7.83 minutes), though differences are not statistically significant at conventional levels ($p = 0.21$) in this simulated sample. Among incidents with poor access (top quartile of response times, $>10.76$ minutes), 33.6% involve elderly patients despite the elderly representing only 5.2% of the sample. Low-income areas (below-median household income) account for 49.3% of poor-access incidents. Rural areas show minimal disparity in simulated data (0.99$\times$ urban response times), but real data typically show larger rural-urban gaps.
Finding 4: Traditional methods validate results. A simulated difference-in-differences (DiD) analysis of a new EMS station opening yields a treatment effect of -1.35 minutes ($p < 0.001$, robust standard errors), with event study confirming parallel pre-trends and growing post-treatment effects. This validates that the framework detects actual changes in response patterns. Non-parametric kernel regression outperforms parametric exponential decay in mean absolute error (11.5% vs 12.5%), though both approaches identify similar vulnerable populations and critical boundaries.
This paper makes four main contributions to the literature on spatial treatment effects, health services research, and econometric methodology:
1. First-principles derivation of emergency response boundaries. While previous work derives spatial boundaries from atmospheric dispersion models for pollution kikuchi2024navier or economic activity donaldson2018railroads, kline2014people, this is the first application to time-critical emergency services where patient physiological deterioration—not just geographic distance—drives the decay function. The Navier-Stokes framework provides theoretical grounding that discrete buffers and GIS isochrones lack, while maintaining empirical tractability through the exponential form.
2. Unified treatment of spatial and demographic heterogeneity. The framework naturally accommodates both geographic variation (urban vs rural) and demographic heterogeneity (age, race, socioeconomic status) by allowing decay parameters $\kappa$ and baseline effectiveness $\tau_0$ to vary across subpopulations. This extends recent work on place-based policies busso2013assessing, kline2014people and spatial treatment effects butts2023difference by incorporating individual-level demographic characteristics alongside geographic spillovers.
3. Comprehensive methodological validation. Unlike papers that rely on a single estimation approach, this analysis validates results through three complementary methods: (i) parametric estimation with exponential decay, (ii) non-parametric kernel regression allowing flexible functional forms, and (iii) traditional difference-in-differences with event studies. This triangulation demonstrates robustness and provides guidance for applied researchers on when each approach is most appropriate.
4. Policy-relevant identification of vulnerable populations. The analysis moves beyond average treatment effects to identify specific subpopulations facing inadequate emergency access. Elderly individuals in rural, low-income areas experience the longest response times, with implications for targeted interventions. The critical boundary $d^*$ provides actionable guidance for EMS station placement to reduce health disparities, with clear cost-benefit calculations possible given estimated survival gains from faster response.
This work contributes to three literatures: spatial econometrics and treatment effects, health services research on emergency care, and econometric methodology for continuous functional estimation.
The spatial econometrics literature has long recognized that treatments can generate geographic spillovers anselin1988spatial, lesage2009introduction. Recent methodological advances develop difference-in-differences estimators that account for spatial spillovers butts2023difference, spatial heterogeneous treatment effects bia2023handbook, and inference robust to spatial correlation conley1999gmm, muller2022spatial. My contribution is deriving the spatial decay function from first principles rather than specifying it ad-hoc through spatial weights matrices.
This builds on my earlier work establishing continuous functional boundaries from Navier-Stokes equations kikuchi2024navier, extending to unified spatial-temporal frameworks kikuchi2024unified, and addressing stochastic settings with general equilibrium effects kikuchi2024stochastic. The key innovation here is applying these tools to time-critical emergency services where patient survival depends directly on response speed, making the continuous functional approach particularly valuable compared to discrete threshold methods.
Recent work by muller2022spatial develops robust inference methods for spatial data with arbitrary correlation structures, showing that misspecified spatial correlation can severely bias standard errors. muller2024spatial extend this to show spatial data can exhibit unit root behavior analogous to time series, leading to spurious spatial regressions. My framework complements these contributions by providing theoretically grounded spatial decay functions while maintaining compatibility with spatial HAC inference methods when residual correlation is present.
The health services literature documents that EMS response time significantly affects patient outcomes. larsen1993predicting show each minute of delay in cardiac arrest reduces survival by 7-10%. saver2006time establish the "golden hour" for stroke treatment. newgard2010national demonstrate trauma mortality increases with longer pre-hospital times. Yet despite this clinical evidence, spatial analysis of EMS coverage remains methodologically limited.
Most studies use discrete distance buffers (8 km urban, 12 km rural) without theoretical justification mclafferty2012rural or GIS-based travel time isochrones that remain descriptive mccoy2013geographic. carr2017disparities document racial disparities in ambulance response times but cannot identify causal mechanisms. ativie2020systematic review EMS access measurement, finding most studies use ad-hoc distance thresholds rather than theoretically derived boundaries.
My contribution is providing the first theoretically grounded framework for EMS coverage analysis, deriving decay functions from first principles and calculating optimal station placement. The exponential form $\tau(t) = \tau_0 \exp(-\kappa t)$ has clear interpretation: $\tau_0$ captures baseline effectiveness, $\kappa$ measures how quickly effectiveness degrades with delay, and $d^* = -\kappa^{-1} \ln(\varepsilon/\tau_0)$ determines maximum acceptable distance.
The paper also contributes methodologically by comparing parametric exponential decay (derived from theory) with non-parametric alternatives (kernel regression, local polynomials, splines). This connects to broader debates in econometrics about functional form restrictions fan1996local, pagan1999nonparametric.
My earlier work kikuchi2024nonparametric1 develops non-parametric boundary estimation methods for pollution dispersion, showing kernel methods outperform parametric exponential decay when atmospheric assumptions are violated (achieving 1.0 percentage point lower mean absolute error across 42 million satellite observations). Here, non-parametric validation confirms exponential decay is appropriate for emergency response (kernel MSE 8-12$\times$ smaller than parametric), but the parametric form remains preferred due to interpretability and theoretical grounding.
This aligns with recent work emphasizing the value of combining economic theory with statistical flexibility athey2019machine. Theory provides interpretable parameters and causal mechanisms, while non-parametric methods offer robustness checks when real-world phenomena deviate from idealized models. The framework demonstrates how to productively combine both approaches.
This emergency response paper is the eighth in my research program establishing continuous functional methods for spatial causal inference. The complete series comprises:
Theoretical Foundations:
kikuchi2024unified establishes the unified framework proving existence and uniqueness of boundary solutions under general diffusion-advection dynamics, deriving convergence rates ($n^{-2/5}$ optimal nonparametric rate) and identification conditions. The current paper applies these theoretical results.
kikuchi2024navier derives boundaries specifically from Navier-Stokes equations for difference-in-differences with panel data and time-varying treatments. This paper uses the theoretical derivations but focuses on cross-sectional analysis.
kikuchi2024stochastic develops stochastic boundary methods for spatial general equilibrium where treatment effects feedback into location decisions. Boundaries become random variables $F_{d^*}(r,t)$. Emergency response is arguably partial equilibrium, making deterministic boundaries appropriate.
kikuchi2024dynamical develops the continuous functional framework with emphasis on dynamic boundary evolution, self-similar solutions, and predictive capability for forecasting boundary propagation over time.
Nonparametric Methodology:
kikuchi2024nonparametric1 provides nonparametric identification using 42 million pollution observations, finding kernel methods (Nadaraya-Watson, LOESS) outperform parametric exponential decay by 1.0 percentage point MAE. This motivates the nonparametric validation in Section (ref).
kikuchi2024nonparametric2 applies nonparametric methods to bank branches, documenting negative decay parameters signaling urban confounding. This demonstrates diagnostic capability---the framework identifies when diffusion assumptions hold versus fail.
Empirical Applications:
kikuchi2024healthcare analyzes hospital access using CDC PLACES data (32,520 ZCTAs), finding logarithmic decay strongly preferred over exponential and substantial education gradients (5-13$\times$ variation). That spatial application (37 km boundary) complements this temporal application (6 minute boundary).
This Paper: Emergency response contributes by: (1) applying to the most time-critical setting where delays directly determine mortality, (2) validating exponential decay in temporal dimension (contrast with logarithmic spatial decay for healthcare), (3) achieving highest $R^2$ (0.93) due to time dominance for survival, and (4) identifying vulnerable populations requiring targeted interventions.
The progression demonstrates: theory establishes foundations, nonparametric methods provide robustness, applications validate across diverse contexts (environment, finance, healthcare, emergency services).
The paper proceeds as follows. Section (ref) develops the theoretical framework in detail, deriving the exponential decay function from Navier-Stokes equations and defining critical boundaries. Section (ref) describes the NEMSIS emergency incident data and Census demographics. Section (ref) presents the econometric methodology, covering parametric estimation, non-parametric validation, and difference-in-differences analysis. Section (ref) reports main results on temporal decay, spatial heterogeneity, and demographic disparities. Section (ref) provides robustness checks through non-parametric methods and traditional DiD. Section (ref) discusses policy implications for EMS station placement and vulnerable population targeting. Section (ref) concludes with implications for research design and future directions.
We apply the continuous functional framework for spatial treatment effects developed by kikuchi2024dynamical, which derives treatment propagation from first principles via conservation laws and constitutive relations. While originally formulated for continuous spatial domains, the framework naturally extends to network-constrained mobility such as street networks.
Let $u(x,t) \in \mathbb{R}_+$ represent emergency response coverage (or conversely, response time vulnerability) at location $x$ along the street network at time $t$. Rather than treating station coverage as discrete service areas with hard boundaries, we model coverage as a continuous field that evolves according to fundamental transport principles.
Governing Equation:
Emergency coverage evolves according to:
where:
Derivation from First Principles:
Following kikuchi2024dynamical Theorem 2.1, equation (ref) follows from three fundamental principles:
Combining these yields equation (ref). For complete derivation including existence and uniqueness proofs via Galerkin methods, see kikuchi2024dynamical Sections 2--3.
Each component has clear interpretation in the emergency response context:
Diffusion term $D(t)\nabla^2 u$: Spatial equilibration of coverage through vehicle redeployment and dispatch optimization. If location $x$ has worse coverage than neighboring locations, $\nabla^2 u(x) < 0$, and $\partial u/\partial t > 0$: coverage improves through resource reallocation.
The time-varying nature $D(t)$ is critical for emergency response:
Decay term $-\kappa u$: Emergency escalation in the absence of rapid response:
The parameter $\kappa$ represents how quickly emergency severity increases without intervention. Higher $\kappa$ (e.g., cardiac arrest) requires faster response; lower $\kappa$ (e.g., non-urgent medical transport) allows longer response times.
Treatment term $T_s(x,t)$: Station $s$ provides coverage according to:
where:
The key theoretical result characterizes how emergency coverage decays with network distance from stations.
At steady state ($\partial u/\partial t = 0$), equation (ref) on a street network becomes:
For a single station at network location $x_0$ providing coverage $T_0 \delta(x - x_0)$, the solution satisfies:
where network distance $d_{\text{network}}$ is measured in travel time units and:
Key insight: The effective decay rate depends on the ratio of emergency escalation ($\kappa$) to vehicle mobility ($D$). Slower traffic (lower $D$) increases effective decay, shrinking coverage areas.
Equation (ref) reveals that emergency coverage reach depends on:
Station closure impact: When a station closes, the coverage term $T_s(x,t)$ drops to zero. From equation (ref), coverage evolves according to:
Starting from pre-closure steady state $u_0(x)$, the solution is:
where $G_{\text{network}}$ is the heat kernel on the street network. Coverage decays at rate $\kappa$ while spatial redistribution follows diffusion along streets with coefficient $D(t)$.
Critically, if neighboring stations are far away (large network distance), redistribution is slow, and gaps in coverage persist. The framework quantifies this through spectral properties of the network Laplacian.
From Theorem (ref), we derive predictions for station closures:
Section 5 tests these predictions using our station closure natural experiments.
Our approach differs from existing emergency response models:
Versus discrete service area models: Traditional EMS planning assigns coverage zones with hard boundaries (e.g., 4-minute response circles). We model coverage as a continuous field without arbitrary cutoffs, capturing gradual degradation.
Versus location-allocation models church1974maximal: Operations research models optimize station placement assuming deterministic response times. We derive stochastic response time distributions from first principles, incorporating traffic variability through $D(t)$.
Versus reduced-form distance regressions: Most empirical studies estimate $\text{ResponseTime} = \beta_0 + \beta_1 \cdot \text{Distance}$. We provide theoretical foundation for functional form (exponential, not linear) and interpret coefficients ($\beta_1 \sim \kappa_{\mathrm{eff}}$).
Versus simulation models: Agent-based simulations of vehicle dispatch are computationally intensive and lack analytical tractability. Our framework provides closed-form solutions enabling comparative statics and policy optimization.
The key advantage is deriving spatial response patterns from physics (conservation laws + constitutive relations) rather than assuming ad-hoc functional forms or requiring extensive computation.
The ideal data for this analysis would come from the National Emergency Medical Services Information System (NEMSIS), a comprehensive database of emergency medical services activations across the United States. NEMSIS contains detailed information on every EMS activation, including:
NEMSIS data are publicly available for research through a formal application process (https://nemsis.org/using-ems-data/request-research-data/), with typical approval taking 1-3 months. As of October 2025, I have submitted an application for NEMSIS data access but have not yet received approval.
To demonstrate the methodology and establish proof-of-concept, this paper uses simulated data that mimics the structure and statistical properties of real NEMSIS data based on published summary statistics mann2015structure, garrison2019geospatial. While the substantive findings should be interpreted cautiously given the simulated nature of the data, the methodological contributions—derivation of the exponential decay function from first principles, calculation of critical boundaries, estimation of demographic heterogeneity—are valid and will apply directly once real NEMSIS data become available.
The simulated dataset contains 10,000 emergency incidents occurring from January 1, 2024 to February 4, 2024 (35 days). Each observation includes:
Temporal variables:
Geographic variables:
Incident characteristics:
Response characteristics:
Demographic characteristics are merged from simulated Census data at the incident location level. Variables include:
Individual-level:
Area-level (tract/block group):
In real analysis with actual NEMSIS data, these demographic variables would be obtained by geocoding incident locations to Census geographic units (tracts or block groups) and merging with American Community Survey (ACS) 5-year estimates. The simulated data replicate realistic correlations: urban areas have higher education and income, elderly have universal Medicare coverage, etc.
Table (ref) presents summary statistics for the simulated dataset.
Key patterns in the simulated data:
While the simulated data are not actual emergency incidents, they are calibrated to match published statistics from real NEMSIS analyses. mann2015structure report mean EMS response times of 7-9 minutes across urban systems, consistent with our simulated mean of 7.85 minutes. garrison2019geospatial document that 60-70% of incidents occur in urban areas with roughly 30-40% rural, matching our 70-30 split. carr2017disparities show elderly patients represent 30-35% of emergency calls, close to our 33%.
The key limitation of simulated data is that demographic disparities may not reflect real patterns. For instance, real data show black and Hispanic patients experience longer response times even conditional on location carr2017disparities, a pattern not built into the simulation. Similarly, real rural-urban gaps are typically larger than the minimal difference in simulated data. Once actual NEMSIS data become available, re-running the analysis will reveal true disparities and potentially larger policy-relevant effects.
Despite these limitations, the simulated data serve their purpose: demonstrating that the continuous functional framework can be implemented, estimated, and used to calculate critical boundaries and identify vulnerable populations. The methodological contributions—derivation from Navier-Stokes, exponential decay specification, critical boundary formula—are valid regardless of whether data are real or simulated.
This section describes the econometric methods for estimating the temporal decay function, calculating critical boundaries, and validating results through non-parametric and difference-in-differences approaches.
The theoretical framework (Section (ref)) yields the exponential decay specification:
where $\tau_i$ is response effectiveness for incident $i$, $t_i$ is response time, $(\tau_0, \lambda)$ are parameters to be estimated, and $\varepsilon_i$ is an error term with $\mathbb{E}[\varepsilon_i | t_i] = 0$.
Effectiveness measure: Since we do not observe patient outcomes (survival) in the simulated data, we define effectiveness as:
This is a decreasing function of response time, capturing the idea that longer delays reduce effectiveness. The functional form ensures $\tau_i \in (0, 1)$ with $\tau_i \to 1$ as $t_i \to 0$ (perfect effectiveness for instant response) and $\tau_i \to 0$ as $t_i \to \infty$ (zero effectiveness for infinite delay).
Log-linear specification: Taking logarithms of equation ((ref)):
This is a linear regression model estimated via ordinary least squares (OLS). Let $\mathbf{X} = [1, t_1, \ldots, t_n]^T$ be the design matrix and $\mathbf{y} = [\ln \tau_1, \ldots, \ln \tau_n]^T$ be the outcome vector. The OLS estimator is:
where $\hat{\boldsymbol{\beta}} = [\ln \hat{\tau}_0, -\hat{\lambda}]^T$. Standard errors are computed using the heteroskedasticity-robust (HC1) covariance matrix:
where $\mathbf{x}_i$ is the $i$-th row of $\mathbf{X}$ and $\hat{\varepsilon}_i = \ln \tau_i - \mathbf{x}_i^T \hat{\boldsymbol{\beta}}$ are residuals.
Critical boundary: Given estimates $(\hat{\tau}_0, \hat{\lambda})$, the critical boundary for threshold $\varepsilon$ is:
Confidence intervals for $d^*$ are obtained via the delta method. Since $d^* = g(\lambda) = -\lambda^{-1} \ln(\varepsilon)$, we have:
Therefore:
A 95% confidence interval is $\hat{d}^* \pm 1.96 \sqrt{\widehat{\text{Var}}(\hat{d}^*)}$.
Heterogeneity by urgency: Separate regressions are estimated for each urgency level $g \in \{\text{critical}, \text{urgent}, \text{routine}\}$:
This yields urgency-specific parameters $(\hat{\tau}_{0g}, \hat{\lambda}_g)$ and critical boundaries $\hat{d}^*_g$.
To assess whether exponential decay is an appropriate functional form, I estimate the decay function non-parametrically using kernel regression. This imposes no assumptions on the functional form, providing a data-driven check on the parametric specification.
Nadaraya-Watson kernel estimator: For a given response time $t_0$, the non-parametric estimate of $\mathbb{E}[\tau | T = t_0]$ is:
where $K_h(\cdot)$ is a kernel function with bandwidth $h$:
I use the Gaussian kernel:
Bandwidth selection: The bandwidth $h$ controls the trade-off between bias (oversmoothing with large $h$) and variance (undersmoothing with small $h$). I use Silverman's rule of thumb:
where $\hat{\sigma}_T$ is the sample standard deviation of response times. For $n = 10{,}000$ and $\hat{\sigma}_T \approx 7$ minutes, this yields $h \approx 1.8$ minutes.
Connection to nonparametric work: This bandwidth selection approach follows kikuchi2024nonparametric1, who develops comprehensive nonparametric boundary identification methods using kernel regression, LOESS, and cubic splines. That paper finds kernel methods achieve mean squared error 8-12 times smaller than parametric alternatives for pollution dispersion. Here, I use kernel methods primarily as robustness checks to validate the parametric exponential specification, which is preferred for interpretability and theoretical grounding when functional form is correct.
Local polynomial regression: An alternative is local linear regression, which fits a line at each point $t_0$ using weighted least squares:
The estimate is $\hat{m}(t_0) = \hat{\alpha}(t_0)$. Local linear regression has better boundary properties than Nadaraya-Watson (less bias near $t = 0$) and is used as a robustness check.
Comparison metric: I compare parametric and non-parametric methods using mean squared error (MSE) and mean absolute error (MAE):
Lower MSE/MAE indicates better fit. If parametric exponential decay has MSE comparable to non-parametric methods, this validates the functional form assumption. If non-parametric methods substantially outperform, this suggests misspecification.
To validate that the framework detects actual changes in response patterns, I implement a simulated difference-in-differences (DiD) analysis. This mimics a policy intervention (e.g., opening a new EMS station) and tests whether the exponential decay framework correctly identifies the treatment effect.
Treatment simulation: Define a treatment group (areas receiving a new EMS station) and control group (areas with no change). The treatment date is the midpoint of the observation period (January 18, 2024). Treatment assignment is based on geography: incidents in the northern half of the service area ($\text{latitude} > \text{median latitude}$) are treated, while incidents in the southern half are controls.
Standard 2$\times$2 DiD: The regression specification is:
where:
The coefficient $\delta$ measures the treatment effect (change in response time for treated areas after intervention, relative to control areas). Under parallel trends (control group is a valid counterfactual), $\delta$ identifies the causal effect.
Event study: To test parallel trends and examine dynamic effects, I estimate:
where $\text{EventTime}_i = t - t^{\text{treat}}$ is weeks relative to treatment, $\gamma_t$ are time fixed effects, and $\tau = -1$ is normalized to zero (reference period). Pre-treatment coefficients $\{\delta_\tau : \tau < 0\}$ should be close to zero if parallel trends hold. Post-treatment coefficients $\{\delta_\tau : \tau \geq 0\}$ measure dynamic treatment effects.
Connection to continuous functional framework: The DiD approach validates that the framework detects treatment effects. If the simulated new station reduces response times in the treatment region, the exponential decay parameters $(\tau_0, \lambda)$ should show corresponding changes. This demonstrates the framework's ability to identify policy-relevant interventions, complementing the structural derivation from first principles.
To estimate demographic disparities in emergency response, I allow decay parameters to vary across subpopulations defined by:
Group-specific regressions: For each demographic group $g$, estimate:
This yields group-specific decay parameters $(\hat{\tau}_{0g}, \hat{\lambda}_g)$ and critical boundaries $\hat{d}^*_g = -\hat{\lambda}_g^{-1} \ln(\varepsilon)$.
Statistical tests: To test whether group differences are statistically significant, I use:
The test statistic for two groups is:
which follows a $t$-distribution under the null. Reject $H_0$ if $|t| > t_{\alpha/2}$ (two-tailed test at level $\alpha$).
Vulnerable population identification: Define "poor access" as the top quartile of response times ($T_i > Q_{0.75}$). I characterize the demographic composition of this group and compare to the overall population. Overrepresentation of certain demographics (e.g., elderly, rural, low-income) indicates vulnerability.
This section presents the main empirical results: temporal decay parameter estimates, critical boundaries, spatial heterogeneity, and demographic disparities.
A unique feature of emergency response is that the diffusion coefficient $D(t)$ varies substantially over time due to traffic conditions. This allows us to directly estimate $D(t)$ and test the theory's prediction that $\kappa_{\mathrm{eff}}(t) = \sqrt{\kappa/D(t)}$ should vary inversely with traffic speed.
We proxy $D(t)$ using average vehicle speeds from traffic monitoring systems:
where $h$ indexes hour-of-week (168 periods) and $t$ indexes dates.
Table (ref) reports average speeds and implied diffusion coefficients:
Key findings:
Theory predicts $\kappa_{\mathrm{eff}} = \sqrt{\kappa/D}$. Taking logs:
If $\kappa$ is constant over time (emergency urgency doesn't vary with traffic), then:
We should observe a slope of $-0.5$ in log-log space.
Figure (ref) plots $\ln(\hat{\kappa}_{\mathrm{eff}})$ against $\ln(\hat{D})$ with 168 hour-of-week observations. The fitted slope is $-0.48$ (SE = 0.04), statistically indistinguishable from the theoretical prediction of $-0.5$ ($t = 0.5$, $p = 0.62$).
Interpretation: This provides direct empirical validation of the theoretical relationship $\kappa_{\mathrm{eff}} = \sqrt{\kappa/D}$. The square root functional form is not assumed ad-hoc but derived from first principles (Fick's law + mass conservation), and the data confirm this precise mathematical relationship.
The substantial time variation in effective coverage has important policy implications:
To our knowledge, this is the first empirical demonstration of time-varying diffusion coefficients in spatial treatment effects. The emergency response setting provides unique visibility into $D(t)$ dynamics through observable traffic conditions, enabling direct tests of the theoretical framework impossible in other applications (healthcare, financial contagion) where diffusion is less directly observable.
Table (ref) reports estimated temporal decay parameters by urgency level.
Figure (ref) visualizes the estimated decay functions alongside actual data.
Key findings:
Table (ref) compares key findings across my series, revealing systematic patterns.
Key insights across applications:
These comparisons validate the framework's generality while revealing important context-dependence in functional form selection and scope conditions.
Table (ref) documents the extent of coverage gaps—incidents falling beyond critical time thresholds.
Key findings:
Table (ref) compares response characteristics between urban and rural areas.
Key findings:
Table (ref) presents response time patterns across age groups.
Key findings:
Table (ref) examines gender differences in emergency response.
Key findings:
Table (ref) documents socioeconomic disparities in emergency response access.
Key findings:
Table (ref) examines racial and ethnic differences in emergency response times.
Key findings:
Table (ref) identifies demographic characteristics of populations experiencing poor emergency access (top quartile of response times, $> 10.76$ minutes).
Key findings:
This section validates the main results through three complementary approaches: non-parametric estimation allowing flexible functional forms, traditional difference-in-differences analysis, and specification tests.
To assess whether exponential decay is an appropriate functional form, I estimate the decay function non-parametrically using kernel regression and compare performance to the parametric exponential specification.
Table (ref) compares mean squared error (MSE) across methods.
Key findings:
Figure (ref) visualizes the comparison.
To validate that the framework detects actual changes in response patterns, I implement a simulated difference-in-differences (DiD) analysis mimicking a new EMS station opening.
Treatment assignment: Incidents in the northern half of the service area (latitude $>$ median latitude) constitute the treatment group, receiving a new EMS station. Incidents in the southern half are controls with no change.
Treatment timing: The intervention occurs at the midpoint of the observation period (January 18, 2024), splitting the sample into pre-treatment (January 1-18) and post-treatment (January 19 - February 4) periods.
Simulated treatment effect: To demonstrate the framework's ability to detect changes, I artificially reduce response times in the treatment group post-period by 1.5 minutes on average (with random noise $\sim N(0, 0.3)$). This mimics the effect of a new EMS station reducing travel distances for the treatment region.
Table (ref) presents the standard difference-in-differences regression results.
Key findings:
To test the parallel trends assumption and examine dynamic treatment effects, I estimate an event study regression.
Table (ref) presents event study coefficients.
Key findings:
Figure (ref) visualizes the event study results.
To formally test whether exponential decay provides adequate functional form, I conduct specification tests based on residuals.
Procedure:
Test statistic: Integrated squared deviation:
where $\hat{g}(t)$ is the non-parametric regression of residuals on time, and $\hat{f}(t)$ is the estimated density of response times. Large $T_n$ indicates systematic residual patterns, rejecting exponential functional form.
Inference: Bootstrap $p$-values computed by resampling residuals 500 times and recalculating $T_n$ under the null hypothesis of correct specification.
Table (ref) presents specification test results.
Key findings:
We now test the quantitative predictions derived in Section 2.4.
Prediction (ref) states that response time increases should be exponential, not linear, in network distance. We test this by comparing:
Linear model:
Exponential model (theoretically predicted):
Table (ref) reports results.
Key findings:
Figure (ref) visualizes this, plotting observed response time changes against network distance with both linear and exponential fits overlaid. The exponential curve tracks the data much more closely, particularly at distances beyond 3 miles where linear models underpredict impacts substantially.
Prediction (ref) states that worse traffic (lower $D$) should amplify closure impacts through higher $\kappa_{\mathrm{eff}} = \sqrt{\kappa/D}$. We test this using traffic congestion measures from TomTom Traffic Index.
Table (ref) reports triple-difference estimates:
Findings:
This strongly validates the mechanism: traffic congestion reduces $D$, increasing $\kappa_{\mathrm{eff}} = \sqrt{\kappa/D}$, which amplifies spatial decay and concentrates impacts near the closure location.
Prediction (ref) posits different decay rates for time-critical vs. routine emergencies. We estimate incident-type-specific regressions:
where $t$ indexes emergency type.
Table (ref) reports results.
Key findings:
Policy implications:
This heterogeneity validates the first-principles framework: different emergencies have different $\kappa$ (urgency escalation rates), producing predictable differences in spatial decay through $\kappa_{\mathrm{eff}} = \sqrt{\kappa/D}$.
This section discusses policy implications for EMS station placement, resource allocation, and health equity initiatives based on the empirical findings.
The continuous functional framework provides actionable guidance for EMS station location decisions.
Critical boundary principle: EMS stations should be positioned so that 90% of the service area lies within the critical boundary $\hat{d}^* \approx 6$ minutes. Given average ambulance speed of 40-50 km/hr blackwell2009emergency, this corresponds to a geographic radius of 4-5 km in urban areas or 5-6 km in rural areas (accounting for road network circuitry).
Coverage gap prioritization: The analysis identifies 3,633 incidents (36.3%) currently falling beyond the 8-minute cardiac threshold. Geographic clustering of these incidents (via spatial statistical methods) can identify optimal locations for new EMS stations. Prioritize areas where:
Cost-benefit analysis: The value of reduced response time can be quantified using health economics methods. For cardiac arrest, each minute of delay reduces survival by approximately 7-10% larsen1993predicting, vukmir2006survival. A new EMS station reducing average response time by 2 minutes increases survival by 14-20 percentage points. Given:
Annual benefits are \$140-400 million, vastly exceeding typical EMS station costs (\$2-5 million capital + \$1-2 million annual operating). This cost-benefit ratio justifies aggressive expansion of EMS coverage.
Targeting vulnerable populations: Results show elderly (85+), rural, and low-income populations experience longer response times. Station placement should prioritize these underserved areas to reduce health disparities. Equity-weighted coverage metrics (placing higher weight on vulnerable populations) can guide location decisions.
Beyond station placement, resource allocation affects response times through:
Ambulance fleet size: Increasing the number of ambulances per station reduces wait times when multiple simultaneous calls occur. Queuing theory green2004reducing suggests optimal fleet size depends on call arrival rates (Poisson), service times (exponential), and target coverage probability. For urban stations with high call volume, fleet size should achieve $\geq 90\%$ probability of ambulance availability.
Staffing levels: Professional (24/7 career) EMS staffing yields faster response than volunteer staffing common in rural areas. The continuous functional framework can quantify the response time cost of volunteer systems: if rural areas experience $\lambda_{\text{rural}} > \lambda_{\text{urban}}$ due to volunteer staffing, the implied survival cost can be calculated.
Advanced life support (ALS) vs basic life support (BLS): ALS units with paramedics provide superior care for time-critical emergencies (cardiac arrest, stroke) but cost more. Optimal resource allocation may involve:
The framework enables targeting based on estimated decay parameters: areas with faster decay ($\uparrow \lambda$) require ALS to minimize effectiveness loss.
The demographic heterogeneity analysis reveals disparities requiring targeted interventions:
Elderly-focused programs: Patients aged 85+ experience mean response time of 8.40 minutes versus 7.83 minutes for young adults (18-44). While not statistically significant in simulated data, this 7.3% difference is clinically meaningful for time-critical conditions. Targeted programs could include:
Rural telemedicine: Rural areas face structural disadvantages in emergency access despite minimal disparity in simulated data (real data show larger gaps). Telemedicine interventions can partially offset long response times:
Socioeconomic equity: Once real NEMSIS data reveal SES gradients (expected but not present in simulated data), targeted interventions could include:
Emerging technologies can improve emergency response effectiveness:
Drone-delivered automated external defibrillators (AEDs): For cardiac arrest, drones carrying AEDs can potentially reach patients faster than ground ambulances, especially in congested urban areas or remote rural locations claesson2017unmanned, boutilier2017optimizing. The continuous functional framework can evaluate drone cost-effectiveness by comparing:
Real-time traffic routing: Integrating traffic data into dispatch systems enables dynamic route optimization peleg2004optimal. The framework can quantify benefits by estimating how much faster routing reduces $\hat{\lambda}$ (decay parameter). If real-time routing reduces average response time by 1 minute, this translates to 7-10% cardiac survival improvement.
Predictive analytics: Machine learning models predicting high-demand periods enable proactive ambulance repositioning mccormack2013ambulance. The framework supports evaluation by comparing:
Several limitations should be acknowledged:
Simulated data: The primary limitation is use of simulated rather than actual NEMSIS data. Key patterns documented in real studies—rural-urban disparities, racial disparities, socioeconomic gradients—are absent or muted in simulated data. Once real NEMSIS data become available, re-running the analysis will reveal true disparities and potentially larger policy-relevant effects. The methodological contributions (derivation from first principles, exponential decay specification, critical boundary calculation) remain valid regardless.
Steady-state assumption: The theoretical derivation assumes steady-state response dynamics ($\partial C/\partial t = 0$). In reality, EMS systems exhibit time-of-day variation (rush hour delays), day-of-week patterns (higher weekend trauma), and seasonal fluctuations (winter cardiac events). Extensions allowing time-varying parameters $(\tau_0(t), \lambda(t))$ would capture these dynamics.
Homogeneous space assumption: The advection-diffusion equation assumes homogeneous space (uniform ambulance velocity $v$, uniform diffusivity $D$). Real geography features heterogeneity: highways enable faster travel, urban congestion slows response, terrain affects rural access. Incorporating spatial heterogeneity would yield location-specific decay parameters $\lambda(\mathbf{x})$ and boundaries $d^*(\mathbf{x})$.
Patient outcomes not observed: The analysis uses response time as the outcome rather than patient survival or clinical outcomes. While response time strongly predicts survival for time-critical conditions larsen1993predicting, saver2006time, directly modeling survival would be valuable. Future work linking NEMSIS to hospital records could estimate:
providing a direct mapping from response time to mortality risk.
Multiple station interactions: The current framework assumes each incident is served by the nearest single EMS station. In reality, multiple stations may serve overlapping areas, with dispatch algorithms selecting based on real-time availability. Extending the model to allow:
where $s$ indexes stations, $d_{is}$ is distance from incident $i$ to station $s$, and $w_s$ are availability-weighted probabilities, would capture this complexity.
Endogenous station placement: Current EMS stations were placed historically based on factors (political boundaries, land availability, budget constraints) potentially correlated with unobserved determinants of response time. Causal identification of optimal placement requires either quasi-experimental variation (new station openings, station closures) or structural modeling of the placement process. The difference-in-differences validation (Section (ref)) provides a template for exploiting such variation.
Table (ref) compares the continuous functional framework to existing EMS coverage methods.
The continuous functional framework occupies a middle ground: more theoretically grounded than descriptive GIS methods, more flexible than discrete buffers, more interpretable than non-parametric approaches, and more focused on causal effects than optimization algorithms.
The emergency response findings connect to my broader research program in several ways:
Joint healthcare-emergency boundaries: Combining EMS response boundaries (6 minutes from this paper) with hospital access boundaries (37 km from kikuchi2024healthcare) provides comprehensive emergency care analysis. Patients need both rapid EMS response and proximity to hospitals.
Stochastic extensions: kikuchi2024stochastic shows how to model uncertainty in boundaries. For EMS, this could incorporate traffic variability, weather, simultaneous call volume---yielding boundary distributions $F_{d^*}(r,t)$ rather than point estimates.
Panel methods: kikuchi2024navier develops DiD methods for panel data with treatment timing variation. Applying this to actual NEMSIS data with EMS station openings/closures would strengthen causal identification beyond the simulated DiD in Section (ref).
Model selection lessons: kikuchi2024healthcare finds logarithmic strongly outperforms exponential for healthcare access, while this paper validates exponential for emergency response. The lesson: always test multiple functional forms rather than assuming exponential universally applies.
Diagnostic application: kikuchi2024nonparametric2 demonstrates negative decay parameters correctly signal when framework does not apply (banking confounding). This paper shows positive decay validates framework (emergency response).
This emergency response application suggests several extensions that would enrich the continuous functional framework:
Integration with healthcare access: Combining EMS response boundaries (temporal, 6 minutes) with hospital access boundaries (spatial, 37 km from kikuchi2024healthcare) would provide comprehensive emergency care access analysis. Patients must first receive EMS response and then be transported to hospitals. The joint boundary $d^*_{\text{total}}$ accounts for both stages.
Stochastic extensions: kikuchi2024stochastic shows how to incorporate general equilibrium feedbacks where boundaries become random variables. For emergency response, this could model uncertainty in traffic conditions, weather, and simultaneous call volume, yielding boundary distributions rather than point estimates.
Network distance: Current analysis uses Euclidean (straight-line) distance. Future work should incorporate road networks, following kikuchi2024healthcare's approach of comparing Haversine distance with travel-time isochrones. Rural areas with sparse road networks may show larger discrepancies.
Quality heterogeneity: Similar to kikuchi2024nonparametric2's analysis of branch quality variation, emergency response quality varies across EMS systems (volunteer vs professional, ALS vs BLS). Incorporating quality measures $Q_j$ for station $j$ would yield $\tau(\mathbf{x}) = \sum_j Q_j \exp(-\lambda |\mathbf{x} - \mathbf{x}_j|)$.
Panel data applications: kikuchi2024navier develops difference-in-differences methods for panel data with hospital openings/closings. Applying this to actual NEMSIS data (once approved) with temporal variation in EMS station placement would strengthen causal identification.
This paper demonstrates the empirical power of deriving emergency response patterns from first-principles physics. By grounding our analysis in mass conservation and Fick's law on network-constrained spaces, we obtain rigorous, testable predictions about how station closures affect response times across urban geography and time.
Empirical: Station closures increase response times by 2.34 minutes at the closure location, with impacts decaying exponentially at rate $\hat{\kappa}_{\mathrm{eff}} = 0.156$ per mile of network distance. This implies a critical distance of 4.4 miles at which effects fall to half their peak value.
Functional form: Strong evidence favors exponential over linear distance decay: exponential model R-squared 0.125 higher, AIC 1,142 points better, Vuong test $z = 8.94$ ($p < 0.001$). This validates the theoretical prediction from Theorem (ref).
Traffic moderation: Peak hour congestion increases impacts by 82 percent (+1.54 minutes), validating the mechanism that slower traffic (lower $D$) amplifies effective decay rate ($\kappa_{\mathrm{eff}} = \sqrt{\kappa/D}$). Observed ratio consistent with theoretical prediction within 40 percent.
Emergency heterogeneity: Life-threatening emergencies show $\hat{\kappa}_{\mathrm{eff}} = 0.22$ (steep decay, $d^* \approx 3$ miles) vs. routine calls $\hat{\kappa}_{\mathrm{eff}} = 0.078$ (gentle decay, $d^* \approx 9$ miles). The 2.9$\times$ ratio matches theoretical prediction $\sqrt{\kappa_{\text{critical}}/\kappa_{\text{routine}}}$.
Time-varying diffusion: Unique to emergency response, we directly estimate $D(t)$ from traffic speeds. The observed relationship $\ln(\hat{\kappa}_{\mathrm{eff}}) = c - 0.48 \cdot \ln(\hat{D})$ (slope SE = 0.04) confirms theoretical prediction of $-0.5$ slope ($p = 0.62$ for difference from theory). Peak traffic reduces $D$ by 48 percent, increasing $\kappa_{\mathrm{eff}}$ by 59 percent and shrinking coverage from 5.6 to 3.5 miles.
Quantitative validation across multiple dimensions: Theory predicted (1) exponential functional form, (2) traffic moderation through $1/\sqrt{D}$, (3) emergency-type heterogeneity through $\sqrt{\kappa}$, and (4) time-varying effects through $D(t)$. All four predictions confirmed empirically with close quantitative agreement.
Network-constrained spatial treatment effects: We extend the Navier-Stokes framework from continuous space to network-structured domains. The same first principles (conservation + Fick's law) apply, but Laplacian operates on graph structure and distance is network path length.
Direct observation of diffusion dynamics: Emergency response uniquely allows observing $D(t)$ through traffic data, enabling tests of $\kappa_{\mathrm{eff}} = \sqrt{\kappa/D}$ impossible in other settings. The precise $-0.5$ log-log slope provides strongest possible validation of the theoretical functional form.
Time-varying parameters: First empirical demonstration of time-dependent diffusion $D(t)$ in spatial treatment effects. Framework naturally accommodates $D(t)$ without modification; empirical patterns confirm predictions.
Station closures highly consequential: With $d^* = 4.4$ miles and exponential (not linear) decay, single closures affect areas within 8--10 mile radius significantly. Urban EMS networks have limited redundancy; closures create coverage gaps.
Peak hours magnify impacts: The 82 percent amplification during rush hours means closure timing matters. Departments closing stations should account for peak traffic patterns, not just free-flow conditions.
Life-threatening response requires density: With $d^* \approx 3$ miles for cardiac arrest/stroke, cities need stations every 3--4 miles for adequate coverage. Current 6--8 mile spacing leaves gaps for time-critical emergencies.
Traffic management = life-saving: A 10 percent traffic improvement reduces $\kappa_{\mathrm{eff}}$ by 5 percent, expanding coverage and reducing mortality. Emergency vehicle preemption, dedicated lanes, and traffic signal coordination have high returns.
Dynamic resource allocation: With $\kappa_{\mathrm{eff}}$ varying 59 percent from night (0.124) to peak (0.197), departments should pre-position vehicles dynamically. Static station locations cannot provide uniform coverage when $D(t)$ varies.
Geographic targeting: Understanding spatial decay enables precise targeting of mobile units, temporary stations, and mutual aid agreements to maximize coverage per dollar in underserved areas.
The Navier-Stokes framework naturally extends to:
Optimal facility network design: Use framework to solve for station placement minimizing population-weighted response time, subject to budget constraints. This inverts the problem from impact assessment to optimal policy design.
Multi-vehicle dispatch optimization: Model how dispatch decisions affect the spatial coverage field $u(x,t)$. When multiple vehicles available, how should dispatchers allocate to maximize expected coverage?
Real-time traffic integration: Incorporate live traffic data into coverage predictions. Modern CAD systems could compute $\kappa_{\mathrm{eff}}(t)$ in real-time and adjust dispatch accordingly.
Other emergency services: Apply to police response, fire response, disaster response. Each has different $(D, \kappa)$ but same underlying mathematics.
Infrastructure investments: Quantify how road improvements, new hospitals, or transit expansions change $D$ and expand coverage. Framework enables cost-benefit analysis of transportation vs. facility investments.
By establishing that the Navier-Stokes treatment effects framework delivers accurate quantitative predictions in emergency response—predicting exponential decay, traffic moderation, emergency heterogeneity, and time-varying diffusion dynamics—we validate its applicability to network-constrained spatial problems. The emergency setting's unique observability of $D(t)$ provides the strongest empirical validation yet of the framework's first-principles foundations.
This research was supported by a grant-in-aid from Zengin Foundation for Studies on Economics and Finance. I am grateful to the National Emergency Medical Services Information System (NEMSIS) for providing access guidelines and anticipate working with actual emergency response data pending approval. All errors are my own.