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# A simple contagion process describes spreading of traffic jams in urban networks

## Abstract

The spread of traffic jams in urban networks has long been viewed as a complex spatio-temporal phenomenon that often requires computationally intensive microscopic models for analysis purposes. In this study, we present a framework to describe the dynamics of congestion propagation and dissipation of traffic in cities using a simple contagion process, inspired by those used to model infectious disease spread in a population. We introduce two macroscopic characteristics for network traffic dynamics, namely congestion propagation rate β and congestion dissipation rate μ. We describe the dynamics of congestion spread using these new parameters embedded within a system of ordinary differential equations, similar to the well-known susceptible-infected-recovered (SIR) model. The proposed contagion-based dynamics are verified through an empirical multi-city analysis, and can be used to monitor, predict and control the fraction of congested links in the network over time.

## Introduction

Traffic congestion in cities, also known as traffic jams, propagate over time and space. Existing approaches to model city traffic often rely on microscopic models with high computational burden as well as excessive parameterization required for calibration1,2,3. Further, the lack of available transportation network data in many countries, especially those that are less economically developed, poses a challenge for traffic modelers. However, the fast-paced development and deployment of mobile sensors offers the opportunity to generate continuous spatial data, which further enables the estimation of road traffic conditions in real time and at the macroscopic level.

Numerous studies have explored different macroscopic approaches to model the spread of traffic jams in cities4,5,6,7,8,9, including through the lens of percolation theory10,11, machine-learning methods12, and queuing theory13,14. However, machine-learning models12 and models based on queuing theory13,14 are unable to capture and quantitatively describe the propagation and dissipation patterns of congestion over time and space. They are instead formulated based on input (arrival) and output (departure) rates to estimate the number of vehicles (i.e., accumulation) in an area, similar to conservation-based models using Macroscopic or Network Fundamental Diagram (MFD or NFD)4. In this work, we show that traffic systems exhibit underlying spreading dynamics similar to those observed and applied in other network systems, for example, social networks, population contact networks, or technological networks. Specifically, we propose and empirically demonstrate that traffic congestion in urban networks can be characterized using a simple contagion process, similar to the well-known susceptible-infected-recovered (SIR) model used to describe spread of infectious diseases in a population, wherein traffic spreads and recovers throughout the network over time. A previous study by Wu et al.15 conjectured that traffic congestion spread can be described with a SIR model, but provided no empirical evidence. To the best of the authors’ knowledge, characterizing and modeling congestion propagation and dissipation as a network spreading phenomenon has never been empirically validated.

Urban traffic often exhibits high spatial correlation in which links adjacent to a congested link are more likely to become congested. Additionally, there is a strong temporal correlation in urban congestion, which is known to be driven by the time-dependent profile of travel demand. The spread of congestion at the link level is well theorized and understood with queuing and kinematic wave theories16,17,18. However, our understanding of congestion propagation dynamics at the network level is still incomplete. Queue spillbacks in networks are shown to be sensitive to link capacities19, which could remain stable in both under- and over-saturated conditions. Congestion also exhibits fragmentation during recovery20 leading to greater spatial heterogeneity and thus results in a drop of network production21,22,23. Propagation and dissipation of gridlocks can be characterized by the number of congested links or the length of congestion in the network23 with propagation occurring often at a much higher rate than dissipation. Also, studies have shown that the ratio of road supply to travel demand could explain the percentage of time lost in congestion as an aggregate measure2,22.

Unlike individual link traffic shockwaves in a two-dimensional time-space diagram, which are categorized as forward or backward moving, network traffic jams evolve in multiple directions over space. Therefore, we propose that a network’s propagation and dissipation can be characterized by two average rates, namely, the congestion propagation rate β and the congestion recovery rate μ, which together can predict the number of congested links in the network over time. These two macroscopic characteristics are critical in modeling congestion propagation and dissipation as a simple contagion process23.

Despite the complex human behavior-driven nature of traffic, we demonstrate that urban network traffic congestion follows a surprisingly similar spreading pattern as in other systems, including the spread of infectious disease in a population or diffusion of ideas in a social network, and can be described using a similar parsimonious theoretical network framework. Specifically, we model the spread of congestion in urban networks by adapting a classical epidemic model to include a propagation and dissipation mechanism dependent on time-varying travel demand and consistent with fundamentals of network traffic flow theory. We illustrate the model to be predictive, and validate the framework using empirical and simulation-based numerical experiments. The proposed model can be used for adaptive and predictive control of congestion in an urban network. By monitoring the network in real time and observing the number of congested links, the model can be applied to develop optimal control strategies with different objectives, such as minimizing the total duration of congestion, minimizing the total number of congested links, and minimizing the recovery time. Similar to other macroscopic model-based control applications24,25,26, we can employ the proposed SIR-based model to improve the traffic network performance by controlling and optimizing the network input through optimal metering of traffic flow or by increasing the network recovery rate through improved signal timing, bottleneck removal, and capacity expansion.

## Results

We use empirical data from Google that contains estimated time-dependent traffic speeds on every link in the road network across six different cities in the world, namely Chicago, London, Paris, Sydney, Melbourne, and Montreal (see Methods). We also use simulated data from a calibrated and validated mesoscopic dynamic traffic assignment model of Melbourne (see Methods)1. Using both empirical and simulation data, we demonstrate that the proposed modeling framework can successfully describe the dynamics of congestion propagation and dissipation in urban networks.

In the proposed network-theoretic framework, nodes represent the intersections (controlled and uncontrolled) and links represent the physical roads between any two intersections. For each link i in the network, we have the time-dependent speed vi(t). We represent the maximum speed on the link as $$v_i^{{\mathrm{max}}}$$. To reveal how congestion propagates and dissipates in a network, we define

$$\lambda _i(t) = \frac{{v_i(t)}}{{v_i^{{\mathrm{max}}}}},$$
(1)

where λi(t) is the ratio of link speed vi(t) over link maximum speed $$v_i^{{\mathrm{max}}}$$. We then classify each link as in either a congested si = 1 or uncongested si = 0 state (also known as free flow) using a threshold ρ as below

$$s_i(t) = \left\{ {\begin{array}{*{20}{c}} {1,} & {\lambda_i(t)\;<\;\rho }, \\ {0,} & {\lambda_i(t)\ge\rho }, \end{array}} \right.$$
(2)

where ρ is a pre-specified threshold that represents different congestion levels11. Figure 1 illustrates the identified congested network for different values of ρ at a given t using data from the simulation-based dynamic traffic assignment model of Melbourne. The size of the congested network grows as ρ increases. Alternatively, one can also construct the congested network using traffic density measurements or any other classification method.

### Modeling contagion dynamics of network traffic jams

Below, we describe the dynamics of congestion propagation with a system of ordinary differential equations (ODEs), which is analogous to the well-known SIR model:

$$\frac{{{d}c(t)}}{{{d}t}} = - \mu c\left( t \right) + \beta kc\left( t \right)\left( {1 - r\left( t \right) - c\left( t \right)} \right),$$
(3)
$$\frac{{{d}r(t)}}{{{d}t}} = \mu c\left( t \right),$$
(4)
$$\frac{{{d}f(t)}}{{{d}t}} = - \beta kc\left( t \right)\left( {1 - r\left( t \right) - c\left( t \right)} \right),$$
(5)

The formulated model simultaneously describes the dynamics of congestion propagation as well as congestion dissipation or recovery in a network, given estimated parameters β and μ, which are dependent on a definite time-dependent travel demand profile as in real-world networks. Analogous representations to the well-known susceptible-infected (SI) and susceptible-infected-susceptible (SIS) model of epidemics can also be formulated to describe the spread of traffic in a network as a two-state model (see Supplementary Note 1). If a SI model is adopted, the estimated c(t) describes propagation of congestion in the network as long as demand for travel continues and congestion eventually propagates to the entire network with no recovery leading to a full gridlock, also known as “complete jam”27,28 or “collapse of the network”29. Here, gridlock is defined as a state of the system under which traffic in the entire network or a portion of the network comes to a complete standstill with zero (or minimal) flow20 (see Fig. 2a). If an SIS model is adopted, the congestion propagation dynamics are described such that congestion continues to grow but does not propagate to the entire network and remains invariant after some time leading to a partial network gridlock (see Fig. 2b). While from a theoretical perspective, both SI and SIS models could also be adopted to describe congestion propagation in a network, they fall short of realism. In a partial network gridlock, the number of congested links can grow or shrink depending on whether the gridlock is propagating or resolving itself (dissipating) as what often happens in real-world networks. Therefore, we expect that the SIR model, as a three-state model, provides a more realistic representation as will be empirically demonstrated in the next section (see Fig. 2c).

### Empirical evidence

We apply the proposed contagion-based model to empirical data collected from six different large metropolitan cities around the world (see Fig. 3a–d). We explore changes in the fraction of congested links c(t) in the selected networks. The proposed dynamics in Eq. 35 are fit to traffic data between the onset and offset of congestion (Fig. 3b) to estimate the propagation rate β and the recovery rate μ using an ordinary least-square (OLS) method with a pattern search algorithm (see Methods) for different values of ρ as in Eq. 2 and assuming an average k = 3 for the studied networks based on the distributions of the node degree.

The applied simple contagion process successfully describes the congestion spreading patterns in different cities at a macroscopic level. While the selected cities have significantly different topology and travel demand patterns, their estimated R0 values are almost the same for ρ = 0.2 and only slightly vary for ρ = 0.3 (see Fig. 3a). This suggests existence of a universal measure represented by R0 for the spread of congestion in urban network similar to what is usually observed for infectious diseases. Note that cmax and the time where cmax occurs are different between cities. For larger ρ values, the observed difference between the estimated R0 grows, which is mainly due to the loose definition of congestion when ρ is large. Therefore, we expect to see divergence between cities as ρ increases. The fraction of recovered and free flow links in the studied networks is also illustrated in Fig. 3c, d, and shown to be consistent with the expected outcomes of the theoretical SIR model.

### Revealing the underlying dynamics with simulation

We now explore changes in the fraction of congested links c(t) in the network, given the traffic data obtained from the simulation-based dynamic traffic model of Melbourne (Supplementary Note 3) and the identified congested links. See Supplementary Note 4 for a comparison between empirical and simulation-based data from the Melbourne network. Figure 4a shows the evolution of c(t) over time for different values of ρ using simulated data with a calibrated travel demand profile for the morning peak period 6:00–10:00 a.m., followed by a 4-h recovery period with zero demand. Introducing zero demand after a complete network loading is a common approach in network traffic analysis2,21, which allows the network to go under a complete recovery. Many of the interesting properties of network traffic flow can only be observed during a full recovery such as the formation of hysteresis in the network flow-density relationship.

The simulated congestion propagation and dissipation patterns follow the commonly observed spreading patterns in epidemics in which the propagation follows an initial exponential growth regime, followed by a sum of multiple exponential processes during recovery. Smaller values of ρ are used as they better reflect congestion formation compared to larger values of ρ. For example, when ρ = 0.1 is being used, the fraction of congested links in the network is almost zero for the first hour of the simulation as congestion has not yet formed in any link across the network. However, when ρ = 0.9 is used, nearly 15% of the links in the network are considered congested at the beginning of the simulation. The adopted and formulated SIR model expressed in Eq. 35 successfully describes the evolution of c(t) over time as shown in Fig. 4b. The model is applied to traffic data between the congestion onset and offset times at which the traffic jam starts propagating and almost finishes dissipating, respectively.

To further reveal the congestion spreading patterns, we also conduct a simulation with 1 h of peak demand loading followed by several hours of recovery with zero demand, consistent with an analysis previously reported by Olmos et al.2. Here we focus on a target group of simulated vehicles that enter the network within the peak hour (8:00–9:00 a.m.), specified as the hour immediately before the peak point on the demand profile. See Fig. 5a in which congestion propagation in the network can be approximated by an initial growth followed by a decay. Figure 5b, c illustrates the changes in the estimated parameters β, μ, and R0 for a range of ρ values. It is counter-intuitive that when ρ increases, the rates of congestion propagation and dissipation both decrease exponentially. However, R0 increases when ρ increases as expected. In fact, propagation and dissipation rates here must be interpreted relatively. Therefore, their individual values may not provide an absolute indication of the extent that congestion is spreading. Instead, it is the ratio of β over μ, or more accurately R0, that has a physical meaning. This is analogous to queuing theory in which the size of a queue depends on the difference between the arrival and departure curves rather than the individual arrival and departure rates. Here, R0 can also be seen as a representation of the rate in which the network shockwave evolves. Further, R0 follows an interesting linear relationship with ρ, as shown in Fig. 5c.

### Connection with travel demand

Here, we examine the impact of changing demand on the propagation and dissipation dynamics of congestion. For the same network with 1 h of loading followed by 4-h recovery, we have conducted multiple simulation runs with a scaling factor of ζ = 0.75, 1.0, 1.25, 1.5, 1.75, and 2.0 to decrease or increase demand while keeping the origin–destination demand patterns the same. We then solve the proposed ODE model and estimate the model parameters β and μ as described in Supplementary Note 5 and Methods section. When demand increases, the fraction of congested links also increases for the same network, and as a result, complete recovery of congestion takes longer while the start of the recovery phase remains the same. This is reflected in the reduction of μ in response to the increase in demand for any given ρ (see Fig. 7a for the evolution of c(t) over time for various demand levels). Counter-intuitively, increasing demand also results in reduction of β for a given ρ (see Fig. 7b, c). This should not be interpreted as a slower propagation of congestion. In fact, β is not independent of μ. The two macroscopic characteristics vary interdependently, so c(t) reaches its peak value at t = 75 min according to the demand profile. However, R0 increases when demand increases for any given ρ with a clear indication that the size of the network shockwave will grow larger and it takes longer to recover. R0 has a linear relationship with ρ in which the slope also follows an almost linear relationship with demand (see Fig. 7d, e). Results suggest that there is also a three-dimensional relationship between R0, ρ, and demand as illustrated in Fig. 7f, in which for smaller values of ρ, the relationship between R0, and demand is almost linear. The relationship with demand is critical for applying the proposed model to travel demand management and traffic control in real world. The proposed model can be used for perimeter control of a sub-network within a larger network, in which the relationship with demand guides which R0 should be used given a fixed ρ value, and vice versa. When R0 is known, the proposed model can then predict when the congestion will peak and when it fully recovers. Identifying the time of the congestion peak and recovery duration can be used to identify the time-optimal traffic control strategies (see Supplementary Note 6). What is missing here is the observed relationship with demand from empirical data. Travel demand is difficult to observe in real world. Therefore, our analysis here remains limited to the simulation environment. Passive mobile phone use data could potentially be used to obtain travel demand over time and relate with congestion propagation and dissipation in the network, which remains an interesting direction for future research.

## Methods

### Traffic simulation model

Simulation of the Melbourne network is conducted in AIMSUN, a commercially available traffic simulation software. The model is a mesoscopic simulation-based dynamic traffic assignment, which has been calibrated and validated for the morning peak period 6:00–10:00 a.m. A large number of input (demand and supply) parameters need to be calibrated before the simulation outcomes are used. Details of the calibration and validation process can be found in ref. 1 and Supplementary Note 3.

Google traffic data on 27 June 2018 are acquired and used, which consist of speeds in the unit of km/h for each link in the network with a unique identifier. The time frame for congestion modeling varies from city to city, which is determined based on the profile of the fraction of congested links, so as to get a fitted model between the onset and offset times of congestion. Nevertheless, the total time window remains the same as in our simulation (i.e., 6:00–10:00 a.m.). Links with missing data for the entire day are discarded in our analysis, but the number of such links is negligible and does not affect the results. A direction for future research is to extend the analysis to multiple days to study the day-to-day spreading dynamics of network traffic.

Google traffic data record the speeds of road sections at continuous time intervals. The data were estimated using floating GPS points of mobile users once they used Google services such as Google Maps. The sampling interval of the data is 10 min, which provides an appropriate temporal resolution for our analysis. Cities that are selected include Melbourne, Sydney, London, Paris, Chicago, and Montreal, which all have a complex urban transport network with a dense population, thereby ensuring the availability of GPS data in large volumes. Note that the reliability of estimations is directly proportional to the amount of data. The calculation of $$v_i^{{\mathrm{max}}}$$ takes into account the maximum speed observed on link i in a cycle of one day (i.e., 24 h). Using this measure along with vi(t), the ratio λi(t) is calculated for the entire network. To be consistent, only the morning peak of each city is considered in our analysis, which all lies in the time period between 6:00 and 10:00 a.m., although being subject to some differences from city to city.

### Parameter estimation with global pattern search

To estimate the parameters of the proposed model, we follow a similar approach presented in ref. 30. The estimation is performed using an OLS method with a pattern search algorithm. The estimation is formulated as a minimization problem in which the pattern search seeks to find the model parameters that minimizes the root mean-squared error (RMSE) calculated as the deviation between the observed and modeled c(t) over the study time period. We have applied the global pattern search algorithm as a derivative-free global optimization method in which the modeled curves for c(t) are fit to the traffic data with the objective function to minimize the RMSE. The pattern search algorithm falls under the general category of global optimization methods in which an initial mesh is first specified in the solution space given an initial guessed solution. The algorithm computes the objective function value at each mesh point until it finds a point whose value is smaller than the objective function value at the initial solution point. The algorithm then updates the mesh size and re-computes the objective function value at each mesh point. This will iteratively continue until one of the stopping criteria is met, such as the mesh size getting smaller than a mesh tolerance threshold or if the maximum number of iterations is reached. For details about the pattern search algorithm, see ref. 31. Also see Supplementary Table 1 in Supplementary Note 5 for the estimated model parameters and associated RMSE for different values of ρ.

## Data availability

For contractual and privacy reasons, we cannot make the empirical data from Google available. However, all data from the simulation-based dynamic traffic assignment model that are needed to replicate the findings reported in the paper and Supplementary Information are available at https://github.com/meeadsaberi/trafficspreading.

## Code availability

The simulation-based dynamic traffic assignment model of Melbourne is also available as an open access model, which can be downloaded via https://github.com/meeadsaberi/dynamel.

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## Acknowledgements

We are thankful for the fruitful discussions with Dr. David Rey at the University of New South Wales and Dr. Mohsen Ramezani from the University of Sydney.

## Author information

Authors

### Contributions

M.S., H.H., L.G., and M.G. designed the study. M.S., H.H., M.A., and S.A.H. implemented the method. M.A., Z.G., and S.S. conducted the simulations. D.J.N. and V.D. supported with empirical data acquisition and analysis. M.S., H.H., M.A., S.A.H., Z.G., L.G., S.T.W., and M.G. analyzed the results and wrote the manuscript.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work.

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Saberi, M., Hamedmoghadam, H., Ashfaq, M. et al. A simple contagion process describes spreading of traffic jams in urban networks. Nat Commun 11, 1616 (2020). https://doi.org/10.1038/s41467-020-15353-2

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