Reevaluating Wildlife–Vehicle Collision Risk During COVID-19: A Simulation-Based Perspective on the ‘Fewer Vehicles–Fewer Casualties’ Assumption

Abstract

Wildlife–vehicle collisions (WVCs) remain a significant cause of animal mortality worldwide, particularly in regions experiencing rapid road network expansion. During the COVID-19 pandemic, a number of studies reported decreased WVC rates, attributing this trend to reduced traffic volumes. However, the validity of the simplified assumption that “fewer vehicles means fewer collisions” remains underexplored from a mechanistic perspective. This study aims to reevaluate that assumption using two simulation-based models that incorporate both the physics of vehicle movement and behavioral parameters of road-crossing animals. Employing an inverse modeling approach with quasi-realistic traffic scenarios, we quantify how vehicle speed, spacing, and animal hesitation affect collision likelihood. The results indicate that approximately 10% of modeled cases contradict the prevailing assumption, with collision risk peaking at intermediate traffic densities. These findings challenge common interpretations of WVC dynamics and underscore the need for more refined, behaviorally informed mitigation strategies. We suggest that integrating such approaches into road planning and conservation policy—particularly under the European Union’s ‘Vision Zero’ framework—could help reduce wildlife mortality more effectively in future scenarios, including potential pandemics or mobility disruptions.

1. Introduction

Wildlife–vehicle collisions (WVCs) represent a major conservation and safety challenge across the globe. In Europe alone, an estimated 194 million birds and 29 million mammals are killed annually on roads [1], with ungulates accounting for over one million deaths each year [2,3]—a mortality scale comparable to that caused by legal hunting. These figures underscore the magnitude of road networks as ecological filters and drivers of population decline.

While the societal and ecological costs of road mortality are well documented, metaphors have occasionally been used to describe its devastating impact. One such example is the expression “Moloch of the asphalt,” which dates back to early 20th-century American journalism. Originally referring to an ancient deity of the Ammonite and Canaanite-Levant civilizations associated with child sacrifice, “Moloch” became a metaphor for the destructive power of automobiles and urbanization [4]. Though evocative, such analogies are best viewed as rhetorical devices illustrating the perceived brutality of modern traffic systems, particularly given the high volume of animal deaths [5,6].

Against this backdrop, the COVID-19 pandemic provided a unique, unintended experiment: a large-scale reduction in human mobility, often referred to as the Anthropause [7]. This global pause in travel and economic activity led to various ecological responses, one of the most frequently cited being a decrease in wildlife–vehicle collision(s) (WVCs). This observation forms a central part of the Anthropause argument [7]. The Anthropause revolves around a significant global reduction in modern human activities, particularly travel, and explores how we can share space with wildlife on our crowded planet [8]. The WVC decrease phenomenon is primarily attributed to a reduced traffic load, which can be measured through metrics such as lower traffic congestion, fewer vehicles on the roads, and a decrease in vehicle kilometers traveled [7,9,10,11,12,13,14]. This interpretation is frequently summarized by the assumption that “fewer vehicles leads to fewer WVCs,” a premise that underlies much of the Anthropause-related WVC literature.

While the assumption that reduced traffic universally leads to fewer WVCs remains widespread, the recent literature urges caution, as the monitoring strategies employed across WVC studies vary considerably, introducing potential sources of heterogeneity in reported outcomes. These methodological differences span the geographic context (e.g., urban vs. rural settings) [15,16], the temporal resolution and duration of monitoring [17], the degree of reliance on citizen science data [18,19,20,21,22,23], and the characteristics of the road network examined—ranging from ones a few hundred meters in size to major highways [23,24,25]. While reported animal mortality rates vary across studies, species, and environmental setups, a general trend appears to be quasi-universally accepted: lower instances of WVCs are associated with reduced traffic pressures.

Despite the prevalence of this assumption, a closer examination of the emerging literature reveals a more nuanced picture. During the early months of the pandemic, when strict lockdown measures dramatically altered human mobility patterns, several studies attempted to quantify changes in WVCs across diverse ecological and geographic contexts. A targeted search in Web of Science using search strings combining ‘COVID OR pandemic,’ ‘lockdown’, ‘animal *’, “wildlife-vehicle collision *” OR “animal road kill *” identified relevant research from more than 20 countries or regional setups. These studies were highly heterogeneous in design, yet their findings could be broadly classified into three outcome categories: (1) a decrease in WVCs, (2) an increase in WVCs, and (3) no effect. Each publication contributes to the overall percentage of categories. This classification scheme supports a bottom-up approach in ecological research, which is widely utilized in the literature on the Anthropause [7]. Although the first category dominates, roughly 20% of studies reported no effect, and 10% documented an unexpected increase in roadkill despite reduced traffic volumes [15,26,27,28,29,30]. This heterogeneity challenges the assumption of a universal traffic–road mortality relationship, suggesting that additional ecological, behavioral, and environmental variables may mediate the effect of traffic reductions on wildlife road mortality.

Building on this observed heterogeneity, a deeper examination reveals that a notable twofold paradox emerges. First, a non-negligible number of cases indicate that lockdown-related traffic conditions had little or no significant impact on animal road mortality, and in some cases, WVCs even increased. Interestingly, among the cases analyzed, most ‘no effect’ and ‘increased WVC’ results were reported as exceptions or deviations in studies that involved multiple setups or comparisons across different species [15,26,27,28,29,30]. This pattern suggests that the outcomes of traffic changes are not uniformly predictable and are likely influenced by species-specific biological, ecological, behavioral, and phenological traits. These traits, together with collision dynamics shaped by car movements, driver behavior, road characteristics, and environmental factors, likely contribute to the observed heterogeneity. At its core, the incumbent generalization appears to rest on three fragile assumptions: (1) epistemically, a circular argument suggesting that “less traffic ‘equates’ to fewer road casualties”; (2) the notion that conflicting outcomes across taxa and contexts are mere exceptions; and (3) comparability across inconsistent metrics related to species richness, road types, and traffic proxies.

The second paradox stems from an equally striking observation: a significant rise in human–vehicle collisions (HVCs) during the same lockdown periods. Despite widespread mobility restrictions and reduced traffic volumes, pedestrians and cyclists fatalities did not decrease as expected—in several countries, they even increased [31]. National reports confirm that the UK recorded a 46% increase in HVC incidents during the lockdown, the USA 22.5%, Australia 4.4%, and Malaysia 8% [32,33]. Comparable patterns were noted in Italy, Germany, Greece, India, and Turkey, based on governmental statistics and independent communications. Although measurement methodologies varied, a consistent trend emerges: HVC levels remained high or even rose during the Anthropause. This pattern was likely exacerbated by increased cycling, reduced police presence, higher driving speeds, and persistent deficits in pedestrian and cyclist infrastructure.

This divergence between WVC and HVC trends fundamentally challenges the assumed linearity of the traffic–mortality relationship. It invites a reexamination of alternative dynamics, such as the nonlinear relationship between traffic density and vehicle speed observed in traffic theory [34,35]. Notably, Stiles et al. [36] proposed that lower traffic volumes can paradoxically increase crash risk by facilitating higher vehicular speeds—suggesting that reduced congestion does not necessarily equate to safer roads for either humans or wildlife.

Importantly, similar complexity is evident in the pre-pandemic WVC literature. While certain variables—such as proximity to natural habitats, gentle topography, road width, and seasonality—have shown consistent associations with elevated wildlife mortality risk [37], others—most notably traffic volume, road infrastructure type, and distance to urban centers—have demonstrated context-dependent and sometimes contradictory effects. Rather than serving as universally predictive variables, these factors have often been used post hoc to delineate WVC hotspots—localized areas with disproportionately high roadkill incidence [38,39,40,41]. This retrospective emphasis underscores a broader limitation: the field has largely relied on observational data without predictive modeling frameworks capable of integrating ecological, behavioral, and traffic-related variables.

In this study, we examine the potential for such an alternative generalization by developing and comparing two simplified but conceptually distinct models of WVC likelihood of a random WVC event. The first is a species-centered model, which posits that the likelihood of a WVC decreases exponentially based on the inverse of the frequency of an animal’s road crossings. The second is a horizontal interception model based on the mechanics of interception between two horizontally moving objects. It considers no vertical component, no acceleration, and constant velocity, time, and displacement range during their movement. This model examines how animals and vehicles interact on roadways, considering both parties’ movement patterns and the factors influencing the likelihood of collisions. Its approach resembles an agent-based model random walk simulation with drift, meaning consistent directional movement.

By comparing the outputs of these two models under normal and restricted traffic conditions, we aim to assess whether they yield divergent predictions and to identify the key parameters that influence the likelihood of WVCs. In carrying this out, we aim to develop a more generalizable and predictive framework—one that transcends post hoc interpretations and advances toward an explicitly mechanistic understanding of wildlife–vehicle collision risk. Based on this rationale, we hypothesize that reductions in traffic volume do not lead to a uniform decrease in WVCs. Instead, we propose that collision likelihood may follow a nonlinear response, with risk potentially increasing at intermediate traffic densities due to emergent interactions between vehicle spacing, speed, and animal road-crossing behavior.

2. Materials and Methods

The core epistemology of our methodology is defined as a blend of the two major schools of thought in ecology, which is necessary for addressing practical environmental problems. This includes the bottom-up approach, which accumulates empirical evidence from case studies and statistical patterns, versus the top-down approach, which generates general laws that then inform explanations for specific cases. We generate quasi-realistic data using various equations representing the physics of a WVC event. We supplement these equations with appropriate data from Monte Carlo simulations and conduct tests that simulate observations from real-world scenarios or controlled experiments (Figure 1).

The modeling framework incorporates two conceptually distinct approaches. Model 1 is a species-centered behavioral model in which the likelihood of a wildlife–vehicle collision (WVC) decreases with reduced road-crossing frequency, following an exponential decay function. Model 2 applies a simplified ballistic formulation, conceptualizing WVCs as horizontal interception events governed by vehicle speed, inter-vehicle spacing, and animal movement rate. These models capture complementary aspects of WVC dynamics—exposure-driven versus mechanically driven risk—and allow comparative evaluation under normal and restricted traffic conditions.

To avoid repetition and establish clear simplifying assumptions in our two models, we will make the following distinction: we will not use the terms’ likelihood’ and ‘probability’ interchangeably. Since most published evidence on WVCs involves observed data, “likelihood” is a more appropriate approach for our study. Likelihood measures how plausible a parameter value is given the observed data. At the same time, probability refers to the likelihood of a specific event occurring and is calculated based on known parameters or conditions. This distinction is important because the outcomes of our models are based on simulated combinations of various variables, parameters, and coefficients related to animals, vehicles, and road conditions. Additionally, the differences in likelihood outcomes from the same model, when using different ranges of parameter values, can yield negative results, which is not possible for probabilities.

2.1. WVCs as a Simplified Horizontal Ballistics Problem

The collision of a vehicle with an animal or a pedestrian is generally not classified as a ballistic event. Ballistics focuses on the angle at which a projectile is launched, with its trajectory determined by horizontal and vertical motions. Notably, these two types of motion do not interact with each other. Then, the ‘ballistics’ of a WVC considers only the horizontal motion, and the launch angle φ is 0, meaning cos(φ) = 1. We consider forces such as friction or drag to be negligible in this case. For the definition of parameters, variables, or constants used hereafter, refer to

Table 1. Variables and factors of the vehicle–animal–motorway triptych that interfere with a wildlife–vehicle collision incident. The variables, constants, and equations used in our WVC ballistics problem are presented in bold.

Code	Variable/Factor	Expressions	Units
Vehicle-Related Variables
V1	Vehicle speed	sv	km × h−1
V2	Distance traveled d in time t	$d_v=s_v\times t$	km
V3	Vehicle length	λ	m (km × 10−3)
V4	Average vehicle fleet length	$\overline{\mathit{\lambda }}$	m (km × 10−3)
V5	Human reaction time to trouble	τ	sec (h/60)
V6	The minimum separation between vehicles	$\mathit{V}=\mathit{\tau }{\mathit{s}}_{\mathit{v}}$	km
V7	Number of vehicles per length of the lane	$V_N={\displaystyle \frac{1}{\lambda +V}}$	unitless
V8	Congestion factor	$\mathit{c}\mathit{f}=\mathbf{1}+\left(\frac{\mathit{N} - \mathbf{1}}{\mathit{d}}\right)\left(\mathit{\lambda }+\mathit{V}\right)$	unitless
V9	Number of vehicles passing the crossing point per unit of time	$V_{N\left(t\right)}={\displaystyle \frac{s_v}{\lambda +V}}$	number × h−1
V10	The average interval between vehicles	${\mathit{I}}_{\mathit{v}}={\displaystyle \frac{\mathit{d}}{\mathit{c}\mathit{f}}}$	km
V11	Time to transport N vehicles over distance d	$T_{N_d}$ = ${\displaystyle \frac{d}{s_v}}\left(1+{\displaystyle \frac{\left(N-1\right)}{d}}\left(\lambda +V\right)\right)$	h
V12	Trip speed of N vehicles over distance d	$S_{N_s}={\displaystyle \frac{s_v}{1+\left(\frac{N - 1}{d}\right)\left(\lambda +V\right)}}$	km × h−1
V13	Vehicular travel capacity	$VTC=N\times S_{N_s}={\displaystyle \frac{N\times s_v}{1+\left(\frac{N - 1}{d}\right)\left(\lambda +V\right)}}$	km × h−1
Animal-Related Variables
V14	Roaming animal mean daily speed	sw	km × h−1
V15	Daily home range	w	km2
V16	Population at risk	R = ind/(LA)	ind/km2
V17	Daily roaming ratio	$\rho ={\displaystyle \frac{roaminghours}{day}}$	day−1
Motorway-Related Variables
V18	Total length	L	km
V19	Length in the animal hotspot area	d	km
V20	Lanes/direction	x	unitless (number)
V21	Width	π	m
V22	Lethal area	$L_A=$d × w	km2
V23	Critical distance	${\mathit{d}}_{\mathit{c}}={\displaystyle \frac{\mathit{\pi }}{{\mathit{s}}_{\mathit{w}}}}$	km
V24	WVC rate	WVCr	Ind × day−1 (or y−1)

.

A WVC incident occurs in a stretch of road after encountering a traveling vehicle with speed sv and a roaming animal crossing the motorway with speed sw (Figure 2). For both colliding subjects, at a point or stretch of the road, the distance traveled in intersecting trajectories equals the product of speed (sv or sw, respectively) by time. Heuristically, the likelihood of a ground-roaming animal colliding with a vehicle, denoted by k hereafter, could be estimated after the time it takes to cross the road of width π (and x lanes) and the time separation between vehicles.

The animal’s crossing time equals tw = π/sw (units converted accordingly, in km and h). A WVC will occur if the vehicle’s speed sv (and distance d from the crossing point, i.e., the time to reach the crossing point) is within a critical distance dc equivalent to the stretch of road of width π at the time the animal begins its crossing with a roaming speed of sw. The time to transport N vehicles over a critical distance is as follows:

dc = sv × tw = (sv/sw) × π

(1)

In a general form, the probability that an animal crosses the road n times and then gets run over during its n + 1st attempt is as follows:

p = k × (1 − k) ⁿ

(2)

Using general population and behavior data for an animal, a valid approximation of this probability might be expressed as follows:

p = n × k

(3)

Equivalently, this probability per unit time is given by the following:

p = WVCr/R

(4)

where WVCr is the rate of roadkill incidents (e.g., individuals per day) and R is the population at risk in the road vicinity.

Another heuristic calculation for the crossing frequency n can be deduced indirectly as follows:

n = (ρ × sw)/w

(5)

where w is the average location distance of the animal to the road, in km; sw is the mean roaming speed of the animal, in km/day; and ρ is the ratio of animal roaming time/day. Combining the expressions mentioned above, we arrive at the following heuristic equations: the size of the population at risk is as follows:

R = WVCr/n × k

(6)

And an alternative estimation of the collision likelihood k of a WVC is as follows:

k ≈ (dc/Iv) × (1/sw)

(7)

where Iv is the average inter-vehicle spacing (km). This formulation ensures that k remains a dimensionless ratio, as it is derived from the quotient of two distances and scaled by the inverse of animal speed.

Table 1 summarizes and defines the variables and factors related to vehicles, motorways, and wildlife influencing the probability of a WVC. Equations (1)–(7) represent a system used to determine k, which corresponds to the likelihood of a WVC event. This approach is advantageous as it reduces the number of parameters, constants, and even equations from Table 1 that require specific measurements, which can introduce significant uncertainty in accurately solving the ballistics problem. For instance, variable/factor V8, in Table 1, represents the congestion factor. By definition, the congestion factor is influenced by traffic restriction measures and typically represents the ratio of actual traffic volume to the road’s maximum capacity. One variable that affects the congestion factor is V6, where V = τ × sv, with τ representing the driver’s reaction time to a potential hazard, such as an animal or a person crossing the road. The average reaction time of a driver when responding to a road obstacle can vary based on several factors, including the driver’s alertness, experience, and the time of day. Here are some general findings regarding reaction times: (1) daylight reaction time is approximately 1.5 s; (2) nighttime reaction time is approximately 2.5 s; and (3) the general average reaction time is around 0.75 to 1 s. These durations indicate the time it takes for a driver to perceive an obstacle and initiate a response, such as moving their foot from the accelerator to the brake pedal. Variability in the parameter τ, which ranges from 0.75 to 2 s, creates significant uncertainty in the accuracy of predictions for any equations of interest, especially the congestion factor.

2.2. Exploration of Equations as a System of the Ballistics Problem

The derived equations reveal that factors influenced by COVID-19 confinement measures, such as human mobility and travel restrictions, do not directly affect the likelihood k of a WVC incident nor the size of the population at risk R. Instead, they aim to identify the determinants of traffic volume that are affected by human confinement or reduced mobility measures. Additionally, vehicle speed and congestion, along with vehicle fleet composition, λ, can either increase or decrease the likelihood of road mortality. This analytical framework enables us to test whether traffic volume reductions due to lockdowns lead to proportionally lower WVC rates, or whether other traffic dynamics, such as changes in speed and spacing between vehicles, introduce nonlinear or unexpected effects.

Since the likelihood k of a WVC event is expressed in Equations (2), (3) and (7), the following simulation models are developed to examine their behavior under different assumptions.

• Model 1—The exponential decay version:

Deducing an approximation of k from Equation (2), we obtain the following:

p = k × (1 − k)ⁿ

(8)

Assuming that k << 1, algebraic manipulations allow us to demonstrate that (1 − k)ⁿ≈ e⁻¹ ⇔ k ≈ 1 − e^(−1/n). This formula can be written alternatively as k ≈ 1 − e^(−w/ρ.s_v⁾, using Equation (5) for n, if data on an animal’s roaming behavior is available and accurate (Appendix A).

• Model 2—The horizontal ballistic version:

Deducing an approximation of k from Equation (7), we obtain the following:

k ≈ (d_c/I_v) × (1/s_w)

(9)

This equation relates k to d_c, I_v, and s_w. It suggests that k is proportional to the ratio of d_c to I_v, scaled by 1/s_w. The definitions of the critical distance d_c, the average interval between vehicles I_v, and the roaming animal mean daily speed s_w expressions correspond to V23, V10, and V14 in Table 1. Since I_v is inversely proportional to the congestion factor cf (V8 in Table 1), the factor average vehicle fleet length $\overline{\lambda }$ (V4 in Table 1) or the equation $V=\tau .s_v$ (V6, the minimum separation between vehicles) is also included in the k approximation. The solution for the approximation of k is Equation (9):

$$k\approx {\displaystyle \frac{\pi }{d\times s_w^2\times \left(1+\frac{N - 1}{d}\right)\times (\lambda +\tau \times s_v)}}$$

The corresponding probabilities p are directly proportional to k, according to Equations (2) and (3).

2.3. Simulations and Sensitivity Analyses

WVCs present inherent complexities as a ballistics problem. The primary challenge lies not in finding exact solutions to the equations outlined above (and in Table 1) but in the numerous variables involved. It is nearly impossible to use precise values for every variable (see Figure 2). Instead, it is more practical to utilize ranges of values within the system of equations. Conducting multiple simulations can serve as an effective alternative. This method involves running a series of computational experiments to investigate how the system behaves under various conditions. Such an approach enables the handling of nonlinear systems without simplification, while incorporating real-world variability and uncertainties. Ultimately, this leads to results that are more reflective of the actual wildlife–vehicle collision triad.

Table 2. The inverse models are based on data, using 10³ Monte Carlo simulations per input variable. The equations are presented in Table 1. Output variables, two pairs, represent likelihoods per model per traffic condition. The first condition represents ‘normal traffic’, meaning no mobility restrictions; the second represents lockdown conditions. K* and kl* correspond to a WVC as a nonlinear relation of the animal road crossing frequency; it is Model 1. k and kl represent the approximation of a WVC as a ballistic mechanism; it is Model 2.

Traffic Conditions		No Restrictions	Lockdown Restrictions
Input Variables	Constants	Model Formulas	Model Formulas
τ	Driver reaction time (sec)	= ABS(NORMINV(RAND(); 0.75; 2))	= ABS(NORMINV(RAND(); 0.75; 2))
d	Road segment length (km)	= randinteger(50; 90)	= randinteger(50; 90) × 1.2
N	Number of vehicles	= randinteger(1000; 2000)	= randinteger(1000; 2000) × 0.4
λ	Vehicle length (m)	= randtruncnormal(mean;sd;min;max) = randtruncnormal(6.5; 5.5; 3.5;15)	= randtruncnormal(6.5; 5.5; 3.5; 15) × 1.4
sv	Vehicle speed (km/h)	= randtruncnormal(50; 26.7; 60; 120)	= randtruncnormal(50; 26.7; 60; 120) × 1.9
V	Separation between vehicles (km)	= randtruncnormal(50; 26.7; 60; 120) × τ	= randtruncnormal(50; 26.7; 60; 120) × 1.9 × τ
π	Road width (m)	= randtriangular(6; 10; 22)	= randtriangular(6; 10; 22)
n	Animal crossing frequency	= ABS(NORMINV(RAND(); 0, 2; 3))	= ABS(NORMINV(RAND(); 0, 2; 3))
sw	Animal speed (km/h)	= randinteger(1; 4)	= randinteger(1; 4)
cf	Congestion factor	= 1 + ((N − 1)/d)(λ + V)	= 1 + ((N − 1)/d)(λ + V)
Output variables
k* (Model 1/Equation (8))		= 1 − EXP(−1/n)
kl* (Model 1/Equation (8))			= 1 − EXP(−1/n)
k (Model 2/Equation (9))		$={\displaystyle \frac{\pi }{d\times s_w^2\times \left(1+\frac{N - 1}{d}\right)\times (\lambda +\tau \times s_v)}}$
kl (Model 2/Equation (9))			$={\displaystyle \frac{\pi }{d\times s_w^2\times \left(1+\frac{N - 1}{d}\right)\times (\lambda +\tau \times s_v)}}$

presents the complex simulation model used in our study. This model includes several intermediate variables calculated using different forms of Monte Carlo simulations. Additionally, we have two pairs of output variables that relate to the likelihood k of a WVC event. Each pair corresponds to simulated normal traffic conditions compared to those during lockdown. For instance, the descriptive statistics (mean, standard deviation, mean standard error, and skewness) of the simulated likelihood k are calculated after conducting 10³ Monte Carlo simulations for each intermediate variable, as shown in Table 2. The simulated likelihood k_l follows the same rationale and simulation approach but differs in that it involves a higher vehicle speed, fewer vehicles, and a larger mean vehicle fleet.

The two approximations of k (refer to Equations (8) and (9) above) examine a WVC event from different perspectives. The first approach focuses on animal behavior, while the second considers primarily vehicle dynamics. We can expect some differences in the results, as the first approach is a nonlinear relationship involving a single variable, n, which represents the frequency of animal crossings on the motorway per day. In contrast, the second approach involves seven variables. Specifically, (1) k is proportional to the ratio of dc to Iv, scaled by 1/sw; (2) dc is inversely proportional to sw; (3) Iv is inversely proportional to cf; (4) V is directly proportional to the product of t and sv; and (5) cf increases with N, λ, and V, but decreases with d. Calculations were performed using TREEPLAN SimVoi 311 add-in. Notice that to isolate the effects of the random number simulator seed over different simulations, we used the same random number generator seed in all simulations attempted.

In a complex model like ours, this approach provides valuable insights into how changes in the seven input variables impact the model’s output. This process helps validate the model by assessing its responsiveness to variations in input parameters, ensuring that the model is robust and reliable, and increasing confidence in its predictions. Sensitivity analysis was conducted using the TREEPLAN SensIt 162 add-in, resulting in either tornado or spider charts.

2.4. Controlling the Alternative Generalization

To evaluate whether our alternative modeling framework captures divergent outcomes in wildlife–vehicle collisions (WVCs) under varying traffic regimes, we conducted three hypothesis-driven tests based on the simulated outputs of Models 1 and 2. Both models were parameterized using the Monte Carlo formulations described in Table 2.

Hypothesis 1:

The likelihood values produced by the two models—Model 1 and Model 2—are statistically indistinguishable. The null hypothesis assumes that the observed likelihoods of the two models, L1 and L2, do not differ significantly, implying that both models perform equivalently in describing WVC outcomes. We tested this using the likelihood ratio test, where the test statistic is defined as −2log(L1/L2). The degrees of freedom (df) equal the difference in the number of parameters between the models (df = 6). The resulting statistic was compared to the critical value of the χ² distribution with 6 df (see Appendix B).

Hypothesis 2:

While individual likelihood values cannot be negative, their pairwise differences can yield negative results. For instance, in the current case, where, e.g., Model 2 is applied to simulated normal traffic conditions compared to those during lockdown, the values L2k and L2kl cannot be negative individually. However, if L2k is smaller than L2kl for a certain range of values, then Δ(L2k − L2kl) will be negative. Identifying ranges of values where the models’ likelihood differences are examined yields negative results strongly supporting the alternative hypothesis, suggesting that it is possible to experience higher road mortality during lockdown traffic restrictions.

Hypothesis 3:

Sensitivity analysis is valuable in identifying the factors that most significantly influence the risk of WVCs. This hypothesis suggests that key driver-dependent behaviors, irrespective of traffic restrictions or road conditions, such as vehicle speed, alertness, and braking reaction time, are essential in determining variations in Δ(L2k − L2kl). The number of vehicles under two different traffic conditions—without and under lockdown restrictions—plays a significant role. In contrast, factors such as road width and the speed of roaming animals are of secondary importance.

3. Results

3.1. Model Behavior and Baseline Validation

Figure 3A,B display the simulated approximate likelihoods of Model 1 (k*/kl*, Equation (8)) and Model 2 (k/kl, Equation (9)) (see the Methods Section). These results are derived from 10³ Monte Carlo simulations of the models outlined in Table 2. As expected, (1) both alternative models produce continuous data; (2) the simulated likelihood values for both models, under two different conditions (normal traffic vs. restricted traffic), are consistently greater than zero; and (3) the two models exhibit distinctly different trajectories.

3.2. Model Comparison and Statistical Testing (Hypothesis 1)

The null hypothesis when comparing the two models is that the observed likelihoods L1 and L2 (Figure 3A,B) do not differ significantly, implying that there is no effect or issue arising from their structure.

As shown in Appendix B, the test statistic employed, −2log (L1/L2), is compared to the critical value from the χ² distribution with six degrees of freedom. The simulated data distribution used is Δ[(k − k_l) − (k* − k_l*)] (see Table 2), which is 2000 data points long since the following is true:

$${\chi }^2={\displaystyle \frac{\left({RSS}_1-{RSS}_2\right)}{var(k_1\cup k_2)}}$$

The overall variance of Δ[(k − k_l) − (k* − k_l*)] = 0.066, the degrees of freedom = 6, and the corresponding critical value of the χ² distribution = 12.59. For the ratio λ,

$$\lambda ={\displaystyle \frac{L(\Delta (k-k_l\left)\right)}{L(\Delta (k^{*}-k_l^{*}\left)\right)}}=0.000221$$

And the test statistic −2log(L2/L1) = 7.31. Since the resulting test statistic (7.31) falls below the critical χ² threshold, the null hypothesis cannot be rejected. This indicates that the models are not statistically distinguishable in terms of likelihood under the given simulated conditions.

A visual examination of Figure 3A,B further corroborates the statistical outcome. Both models produced smooth, unimodal distributions of likelihood values across simulated traffic densities. The probability surfaces exhibited overlapping confidence regions, particularly between traffic density values of 0.4 and 0.6, with no indication of skewness, multimodality, or abrupt inflections. Residual analysis revealed no localized clustering or anomalies.

Moreover, dispersion patterns were relatively symmetric and homoscedastic across the simulation space. The likelihood ratio (λ) remained stable across a wide range of input parameters, suggesting that the differences in structural formulation between the two models do not translate into meaningful divergence in predicted WVC likelihood.

In sum, the statistical and graphical analyses jointly demonstrate that the models, despite differing in their mechanistic underpinnings, yield comparable outputs under equivalent traffic conditions. As such, this result supports the null hypothesis: that no significant structural effect emerges between the two model frameworks when applied to simulated data derived from identical environmental and traffic parameters.

3.3. Model Divergence Under Traffic Regimes (Hypothesis 2)

Figure 4 illustrates the distribution of differences in likelihood values generated by Model 2 (L2, Equation (9)) under two traffic conditions normal and restricted conditions—as represented by Δ(k − kl). The cumulative relative frequency curve reveals a strongly left-skewed distribution, indicating that the majority of likelihood differences are slightly positive. In other words, most simulated values suggest that WVC likelihood is lower during lockdown traffic conditions.

However, in 93 out of 1000 simulated cases (9.3%), the difference in likelihood is negative, meaning the model predicts a higher risk of wildlife–vehicle collisions under restricted traffic conditions. These cases appear as outliers on the distribution plot and are highlighted in red in the sidebar of Figure 4. The negative Δ(k − kl) values range from approximately −0.000027 to −0.0000014, with the majority of these concentrated around the lower tail of the distribution.

The steep increase in the cumulative frequency curve just after Δ = 0 confirms the rarity but statistical presence of these negative values. The interquartile range of Δ(k − kl) spans from 0.0000057 to 0.0000141, with a median value of 0.0000092, indicating a slight overall tendency toward positive likelihood differences. The minimum and maximum Δ values in the full sample range from −0.000027 to +0.000051. This outcome provides statistical support for the alternative hypothesis—that reductions in traffic do not universally reduce collision risk, and that certain ranges of traffic density may be associated with elevated WVC likelihood.

3.4. Sensitivity Analysis and Key Predictors (Hypothesis 3)

Figure 5 illustrates the results from the sensitivity analysis conducted on Model 2, which enhances the overall analysis presented in this study. The findings are significant in understanding the primary factors that contribute to variations in WVC events under different traffic conditions. This analysis clarifies the nuanced relationship between the commonly held belief that “fewer vehicles equals fewer WVCs” and the essential consideration that a driver’s behavior also plays a crucial role in such incidents. For example, under restricted traffic conditions, 17.6% of WVC events are attributed to higher vehicle speeds in low-congestion scenarios, while only 3.5% are linked to driver alertness. These findings support the interpretation by Stiles et al. [36], which states that “lower volumes lead to higher speeds”. This relationship can significantly alter both the type and severity of road crashes, particularly in the context of lockdown policies enacted during the COVID-19 pandemic.

In terms of contribution to Δk, the number of circulating vehicles (N) produced the largest effect, shifting likelihood values by more than 1.6 × 10⁻⁵. Vehicle speed (sv) followed, altering Δk by roughly 1.2 × 10⁻⁵, while driver reaction time (τ) contributed changes of about 8 × 10⁻⁶. In contrast, animal speed (sw), encounter probability (π), vehicle spacing (λ), and crossing distance (d) had comparatively smaller effects, with Δk shifts typically below 4 × 10⁻⁶.

Overall, all three hypotheses tested are supported by the simulated data. The results collectively validate the feasibility of an alternative generalization for WVC dynamics, offering a nuanced understanding that goes beyond the simplistic “fewer vehicles = fewer collisions” paradigm.

4. Discussion

Wildlife–vehicle collisions (WVCs) are typically documented only after they occur, based on field evidence or mortality records. By accumulating data on WVCs under specific environmental conditions, road and traffic factors, seasonal or diurnal variations, and animal roaming behavior [42,43,44,45,46], researchers can identify hotspots where the risk of WVCs is significantly elevated [47,48,49,50,51]. Such empirical knowledge, along with technical and safety specifications, constitutes the bulk of conservation measures that mitigate excessive mortality for certain flag species at a local scale, or in motorways crossing landscapes and ecosystems of conservation importance [29,39,52,53,54,55,56,57,58].

Animal road mortality is a significant conservation issue that also raises concerns about road safety. Numerous studies have documented WVC monitoring schemes in various countries and settings [59]. These schemes provide a foundation for assessing the severity of the WVC problem and highlight the efforts of conservationists to incorporate technical and safety specifications into road infrastructure planning to mitigate its impact. Importantly, such schemes also contribute to the detection and documentation of rare or poorly studied species, offering crucial records of distribution or presence in otherwise under-surveyed regions [60,61]. In Tier 4 countries (formerly known as developed countries), particularly in Europe, this issue is closely linked to the effects of modernization and the strategy for trans-European connection infrastructures, such as motorways, train networks, pipelines, and powerlines. The examples presented in the previous sections were intended to highlight a primary weakness of the WVC reports: the confusion between the probability and likelihood of a WVC event. Published results refer to observed WVC outcomes, denoted by O, without practically considering the set of parameters of the ‘vehicle(s)–motorway–animal(s)’ system and the real-life stochastic WVC process, denoted by θ. Then, the Anthropause argument reduces the estimate θ—that would be a plausible choice given the observed outcomes O—to the circular ‘fewer vehicles—fewer casualties’, which has already been proven weak in real-life lockdown conditions [36]. Addressing the WVC conservation challenge involves minimizing the probability P(O|θ) or maximizing L(θ|O) and not accurately recording O.

The second challenge related to thought arises from the fact that the COVID-19 pandemic, its duration, and the public health restrictions on human mobility did not unfold in a neutral or process-wise stable environmental context. Processes of modernization, such as urbanization and industrialization, along with rural depopulation, land abandonment, and the concentration of agricultural activities on fertile land, create circumstances that facilitate forest recovery [62]. In many European countries, the migration of rural populations to urban centers, the abandonment of remote villages, and the decline of small-scale local agriculture contribute to this recovery, as predicted by the Forest Transition Theory [62,63]. Additionally, the literature indicates a notable trend of wild animal recovery that coincides with the recovery of forests [64], as many species—including various ungulates, carnivores, and forest-dwelling birds—are known to recolonize regenerating habitats once ecological conditions become favorable.

A crucial aspect often overlooked in the literature is the lack of comparability in baseline conditions over time. When examining data that spans several decades (e.g., 10 to 30 years) to assess the severity of wildlife–vehicle collisions (WVCs), it is essential to recalibrate at least three core variables: (1) the size of the at-risk population has significantly increased; (2) the available habitat and foraging areas have expanded; (3) the road network has developed significantly; and (4) the likelihood of WVCs is affected by these first three factors. These long-term ecological shifts have direct implications for model assumptions and parameterization. Increases in wildlife population density, habitat expansion near roads, and new road infrastructure change both the spatial probability of animal–road encounters and the frequency of crossing events. If these factors are not integrated or updated in model inputs, predictions may become decoupled from real-world dynamics. Therefore, temporal comparability and ecological realism must be carefully considered when using WVC data across multi-decadal periods for model calibration and interpretation.

In Greece, for example, modernization has significantly transformed the rural landscape. In the 1960s, the rural population accounted for 44% of the total, but by 2020, this figure had dropped to 20%, a decline that is not attributed to the COVID-19 pandemic and continues to decrease. In terms of forested areas, the percentage was 30% in 2010; however, recent publications of forest cadastral surveys have raised this figure to 62%. This change reflects the abandonment of rural land and the natural recovery of forested regions. As a result, the recovery of wildlife populations, such as brown bears, wild boars, wolves, and jackals, has been remarkable. To calculate the percentage of road mortality for these species, we need to establish a baseline population size. For instance, the estimated population of brown bears in Greece was between 100 and 120 individuals 20 to 25 years ago. The current population is estimated to be approximately 500 individuals [41]. The ratio of bear–vehicle collisions to the total population indicates the percentage of road mortality of this emblematic species. However, we need to clarify which population estimate should be used as the denominator in this calculation. For instance, it is quite infeasible to calculate the yearly number of individuals added to the total population.

This demographic shift—marked by rural depopulation and wildlife resurgence—has altered the spatial risk of road collisions. As wild fauna recolonize rural and peri-urban landscapes, the probability of WVCs increases. In parallel, vehicle–human collisions (HVCs), more frequent in urban areas, continue to pose serious safety concerns for pedestrians and cyclists. Though these two types of collisions differ in their ecological and geographic contexts, both are shaped by overlapping behavioral and infrastructural risk factors.

Understanding the causes of these events is therefore essential for informed road safety and conservation strategies. The main causes of WVCs include the following: (1) driving above the speed limit or too fast for road conditions, which reduces drivers’ reaction times and increases the likelihood of collisions; (2) impairment from alcohol and drugs, which affects judgment, coordination, and reaction times; and (3) drowsy driving, which can be just as dangerous as driving under the influence. In urban areas, pedestrians and cyclists face higher risks due to additional factors. A significant issue is the lack of proper infrastructure, such as sidewalks and bike lanes, which exacerbates this risk. Additionally, behaviors like running red lights and illegal overtaking are frequent causes of fatal collisions.

Beyond case-specific explanations, these observations call for a rethinking of how WVCs are conceptualized and modeled. The reliance on retrospective interpretation reveals a significant limitation common in much of the Anthropause literature: an overwhelming dependence on observational data without predictive frameworks. Within this context, two alternative approaches emerge for interpreting the varied outcomes of wildlife–vehicle collisions (WVCs) during lockdowns. The first approach follows the classical legal maxim, “Exceptio probat regulam in casibus non exceptis,” meaning that the exception proves the rule in cases not excluded. This principle, established in English law in the early 17th century, suggests that the existence of an exception implies a general rule applicable to all other cases. From this perspective, deviations from the expected pattern of “fewer vehicles leads to fewer WVCs” are seen as outliers. The second, more critical perspective questions whether these exceptions indicate a need for a revised understanding—one that can accommodate multiple causal mechanisms, nonlinear interactions, and species-specific behavioral dynamics. This rethinking requires a shift from merely correlational inferences to mechanistic modeling rooted in the physics and biology of wildlife–vehicle interactions [25]. From a conservation modeling perspective, this raises a foundational question: can WVCs be conceptualized as a deterministic outcome of a kinematic (ballistic) interception problem, governed not solely by traffic volume, but by the spatiotemporal dynamics of both vehicles and animals? Under such a framework, key parameters would include vehicle speed, inter-vehicle spacing, traffic density, and road width, as well as animal behavior, such as daily movement rates, crossing attempts, and habitat use near roads. Ideally, such a system could be formalized into a dimensionless metric that represents the ratio of two distances (the distance an animal needs to cross versus the distance a vehicle travels) or two speeds (the speed of the car compared to the speed of the animal) along intersecting trajectories.

Yet, despite these known causes, predictive modeling of WVCs remains limited. This gap is partially due to the post hoc nature of data collection, where the spatial and temporal patterns of WVCs are retrospectively analyzed without fully integrating mechanistic or behavioral models. The simulation-based approach presented in this study—particularly Model 2—was developed to address precisely this gap by offering a physics-based interpretation of WVC dynamics grounded in real-world constraints.

While our formulations mark a step forward, we acknowledge that the current models remain theoretical and require empirical validation to assess their predictive performance under actual field conditions. Future work will apply these formulations to multi-year datasets of wildlife–vehicle collisions collected across Mediterranean landscapes, especially in regions with species-specific mortality records and detailed traffic parameters [24,45]. Empirical testing will be essential for evaluating the generalizability of both the ballistic and behavior-based models and for refining parameter estimates across varying ecological and infrastructural scenarios.

Furthermore, the present models intentionally focus on core mechanical and behavioral determinants of collision risk—primarily vehicle dynamics and simplified road-crossing behavior. However, several ecological and anthropogenic variables could significantly modulate WVC likelihood. These include species-specific traits such as diel activity, movement speed, or road avoidance, as well as seasonal behavior shifts linked to migration, breeding, or dispersal. On the human side, factors such as driver fatigue, distraction, and variability in hazard perception—beyond the parameterization of reaction time—can also influence collision probability. Although these components were not incorporated in the present framework, future model iterations could integrate such variables as adjustable parameters or probability distributions, enhancing ecological realism and scenario-specific applicability. We argue that segmented policies—technical improvements for conservation on one hand and urban safety interventions on the other—are unlikely to yield comprehensive results. Instead, WVCs must be integrated into broader transportation safety frameworks. The European Commission’s road safety strategy for 2021–2030 and the ‘Vision Zero’ initiative, which aims for zero road fatalities by 2050, provide an opportunity for this integration [65].

To this end, future policy must be informed by models capable of predicting WVC risk across varying traffic scenarios and ecological contexts. The use of probabilistic and mechanistic models, as demonstrated in our approach, is a necessary step toward this goal. Additionally, integrating real-time traffic data and animal movement data into such models would enhance their precision and practical utility.

Within this evolving technological landscape, the models developed here serve not as fixed constructs, but as adaptable frameworks. Grounded in biomechanical and behavioral principles, they provide a scalable foundation for incorporating additional ecological and anthropogenic variables as new data become available. This structural flexibility ensures that predictive accuracy can improve over time, facilitating a gradual shift from observational generalizations to dynamic, evidence-based forecasting. As such models mature, they could be integrated into decision support systems at national road agencies or environmental ministries. For instance, the probabilistic outputs can inform risk zoning along motorways where targeted mitigation (e.g., fencing, overpasses) is more likely to yield conservation gains. Furthermore, emerging machine learning pipelines now enable the real-time ingestion of GPS-tagged animal data and road traffic feeds, offering future opportunities for automated WVC prediction frameworks.

5. Conclusions

This study challenges the simplistic assumption that reduced traffic volume inherently leads to fewer wildlife–vehicle collisions. By modeling collision likelihoods as stochastic outcomes influenced by a suite of behavioral, mechanical, and environmental parameters, we offer a more nuanced framework for understanding road mortality during events such as the COVID-19 lockdowns. The two models developed here—one based on exponential decay of crossing attempts and another on a mechanistic ballistic interpretation—highlight the nonlinearity and complexity of WVC dynamics.

Our results demonstrate that, in a notable subset of scenarios, reduced vehicle numbers can paradoxically increase WVC risk, primarily due to elevated vehicle speeds and reduced driver vigilance in low-congestion environments. This finding underscores the need to shift from post hoc reporting of WVC counts to predictive, likelihood-based modeling that can inform dynamic traffic policy.

The implications are both scientific and policy-oriented: model-based WVC forecasting tools should be embedded within transport safety strategies, especially those aspiring to reach the European Commission’s ‘Vision Zero’ target. Based on our findings, we recommend: (1) incorporating traffic density thresholds into road safety planning, acknowledging that WVC risk may increase at intermediate traffic volumes; (2) prioritizing dynamic speed regulation in areas with known wildlife activity, especially during periods of reduced traffic when higher vehicular speeds are more likely; (3) implementing species-specific mitigation (e.g., fencing, crossing structures) informed by local traffic and animal movement patterns; and (4) integrating driver behavior parameters, such as reaction time and vehicle spacing, into future collision risk assessments.

Future work should focus on validating these recommendations across different biogeographical contexts, enhancing them with real-time data streams, and coupling them with habitat and movement ecology to ensure ecologically meaningful mitigation.