Modeling Vehicular Traffic Flow With Taillight Effect on Deteriorated Roads
Corresponding Author
Gabriel O. Fosu
Department of Mathematics , Kwame Nkrumah University of Science and Technology , Kumasi , Ghana , knust.edu.gh
Search for more papers by this authorGideon K. Gogovi
Department of Biostatistics and Health Data Science , Lehigh University , Bethlehem , Pennsylvania , USA , lehigh.edu
Search for more papers by this authorRichard Owusu
Department of Mathematics , Kwame Nkrumah University of Science and Technology , Kumasi , Ghana , knust.edu.gh
Search for more papers by this authorKwame A. Gyamfi
Department of Mathematics , Kwame Nkrumah University of Science and Technology , Kumasi , Ghana , knust.edu.gh
Istituto Nazionale di Geofisica e Vulcanologia , L’Aquila , Italy , ingv.it
Search for more papers by this authorCorresponding Author
Gabriel O. Fosu
Department of Mathematics , Kwame Nkrumah University of Science and Technology , Kumasi , Ghana , knust.edu.gh
Search for more papers by this authorGideon K. Gogovi
Department of Biostatistics and Health Data Science , Lehigh University , Bethlehem , Pennsylvania , USA , lehigh.edu
Search for more papers by this authorRichard Owusu
Department of Mathematics , Kwame Nkrumah University of Science and Technology , Kumasi , Ghana , knust.edu.gh
Search for more papers by this authorKwame A. Gyamfi
Department of Mathematics , Kwame Nkrumah University of Science and Technology , Kumasi , Ghana , knust.edu.gh
Istituto Nazionale di Geofisica e Vulcanologia , L’Aquila , Italy , ingv.it
Search for more papers by this authorAbstract
This study addresses a critical gap in traffic flow modeling by developing a macroscopic framework that simultaneously accounts for taillight signaling effects and deteriorated road conditions—a common but understudied scenario in developing regions. Building on the observation that reduced road quality and limited visibility significantly alter driver behavior, particularly in response to preceding vehicles’ taillights, we formulate a model that captures these complex interactions neglected by classical approaches. Through systematic mathematical analysis, we first demonstrate that the vector–matrix formulation yields a strictly hyperbolic and anisotropic system, characterized by finite-speed wave propagation along distinct characteristic fields. We then establish the model’s theoretical foundations by conducting a linear stability analysis that quantitatively links traffic stability to key parameters including driver sensitivity, headway, and road condition factors. Numerical simulations validate our analytical framework, revealing how small initial density perturbations evolve into characteristic traffic wave phenomena: shock formations, rarefaction waves, and stop-and-go clusters. These results not only confirm the model’s ability to reproduce realistic traffic patterns under infrastructure constraints but also provide mechanistic insights into the emergence of congestion in poor road conditions. By integrating taillight dynamics with road quality effects, this work advances macroscopic traffic theory while offering practical tools for traffic management in low-maintenance road networks.
Conflicts of Interest
The authors declare no conflicts of interest.
Open Research
Data Availability Statement
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
References
- 1 Hossain M. A. and Tanimoto J., A Microscopic Traffic Flow Model for Sharing Information From a Vehicle to Vehicle by Considering System Time Delay Effect, Physica A: Statistical Mechanics and its Applications. (2022) 585, 126437, https://doi.org/10.1016/j.physa.2021.126437.
- 2 Yu S., Chen Y., Song L., Xuan Z., and Li Y., Modelling and Mitigating Secondary Crash Risk for Serial Tunnels on Freeway via Lighting-Related Microscopic Traffic Model With Inter-Lane Dependency, International Journal of Environmental Research and Public Health. (2023) 20, no. 4, https://doi.org/10.3390/ijerph20043066, 36833757.
- 3 Zhai C., Wu W., and Xiao Y., Cooperative Car-Following Control With Electronic Throttle and Perceived Headway Errors on Gyroidal Roads, Applied Mathematical Modelling. (2022) 108, 770–786, https://doi.org/10.1016/j.apm.2022.04.010.
- 4 Xiao Y., Wu W., Zhai C., Zhai M., and Zhang J., Analysis of Empirical Lane-Changing Rate Effect on Multi-Lane Traffic on Curved Roads, Chinese Journal of Physics. (2025) 95, 260–274, https://doi.org/10.1016/j.cjph.2025.01.041.
- 5 Zhai C., Wu W., and Xiao Y., The Jamming Transition of Multi-Lane Lattice Hydrodynamic Model With Passing Effect, Chaos, Solitons & Fractals. (2023) 171, 113515, https://doi.org/10.1016/j.chaos.2023.113515.
- 6 Zhai C., Zhang R., Peng T., Zhong C., and Xu H., Heterogeneous Lattice Hydrodynamic Model and Jamming Transition Mixed With Connected Vehicles and Human-Driven Vehicles, Physica A: Statistical Mechanics and its Applications. (2023) 623, 128903, https://doi.org/10.1016/j.physa.2023.128903.
- 7 Zhai C., Wu W., Zhang J., and Xiao Y., Congested Traffic Patterns of Mixed Lattice Hydrodynamic Model Combining the Perceptual Range Differences With Passing Effect, Chinese Journal of Physics. (2024) 92, 1174–1187, https://doi.org/10.1016/j.cjph.2024.10.022.
- 8 Chen S., Piao L., Zang X., Luo Q., Li J., Yang J., and Rong J., Analyzing Differences of Highway Lane-Changing Behavior Using Vehicle Trajectory Data, Physica A: Statistical Mechanics and its Applications. (2023) 624, 128980, https://doi.org/10.1016/j.physa.2023.128980.
- 9 Li J., Ling M., Zang X., Luo Q., Yang J., Chen S., and Guo X., Quantifying Risks of Lane-Changing Behavior in Highways With Vehicle Trajectory Data Under Different Driving Environments, International Journal of Modern Physics C. (2024) 35, no. 11, 2450141, https://doi.org/10.1142/S0129183124501419.
- 10
Fosu G. O.,
Oduro F. T., and
Caligaris C., Multilane Analysis of a Viscous Second-Order Macroscopic Traffic Flow Model, SN Partial Differential Equations and Applications. (2021) 2, no. 1, 1–17, https://doi.org/10.1007/s42985-020-00054-8.
10.1007/s42985-020-00054-8 Google Scholar
- 11 Jafaripournimchahi A., Cai Y., Wang H., Sun L., Tang Y., and Babadi A. A., A Viscous Continuum Traffic Flow Model Based on the Cooperative Car-Following Behaviour of Connected and Autonomous Vehicles, IET Intelligent Transport Systems. (2023) 17, no. 5, 973–991, https://doi.org/10.1049/itr2.12320.
- 12 Khan Z. H., Gulliver T. A., Imran W., Khattak K. S., Altamimi A. B., and Qazi A., A Macroscopic Traffic Model Based on Relaxation Time, Alexandria Engineering Journal. (2022) 61, no. 1, 585–596, https://doi.org/10.1016/j.aej.2021.06.042.
- 13 Mohan R. and Ramadurai G., Multi-Class Traffic Flow Model Based on Three Dimensional Flow–Concentration Surface, Physica A: Statistical Mechanics and its Applications. (2021) 577, 126060, https://doi.org/10.1016/j.physa.2021.126060.
- 14 Ngoduy D., Noise-Induced Instability of a Class of Stochastic Higher Order Continuum Traffic Models, Transportation Research Part B: Methodological. (2021) 150, 260–278, https://doi.org/10.1016/j.trb.2021.06.013.
- 15
Qaiser T.,
Altamimi A. B.,
Khan F. A.,
Alsaffar M.,
Alreshidi A.,
Khattak K. S.,
Khan Z. H., and
Khan W., A New Macroscopic Traffic Flow Characterization Incorporating Traffic Emissions, Applied Sciences. (2023) 13, no. 9, https://doi.org/10.3390/app13095545.
10.3390/app13095545 Google Scholar
- 16 Qiao D., Dai B., Lin Z., Guo M., Zhang X., Zhang P., and Cheng F., Study on Vehicle Fuel Consumption and Exhaust Emissions Based on a New Viscous Macroscopic Traffic Flow Model, Journal of Transportation Engineering, Part A: Systems. (2023) 149, no. 2, 04022137, https://doi.org/10.1061/JTEPBS.TEENG-7506.
- 17 Wang Z. and Zhu W. X., Modeling and Stability Analysis of Traffic Flow Considering Electronic Throttle Dynamics on a Curved Road With Slope, Physica A: Statistical Mechanics and its Applications. (2022) 597, 127225, https://doi.org/10.1016/j.physa.2022.127225.
- 18 Zhai C., Wu W., and Xiao Y., Non-Lane-Discipline-Based Continuum Model Considering the Effect of Lateral Gaps and Electronic Throttle Dynamics, Chinese Journal of Physics. (2023) 83, 253–269, https://doi.org/10.1016/j.cjph.2023.03.013.
- 19 Zhai C. and Wu W., A Continuum Model With Traffic Interruption Probability and Electronic Throttle Opening Angle Effect Under Connected Vehicle Environment, European Physical Journal B. (2020) 93, no. 3, 1–12, https://doi.org/10.1140/epjb/e2020-100492-6.
- 20 Zhai C. and Wu W. T., An Extended Continuum Model With Consideration of the Self-Anticipative Effect, Modern Physics Letters B. (2018) 32, no. 31, 1850382, https://doi.org/10.1142/S0217984918503827, 2-s2.0-85054433728.
- 21 Zhang J., Tang T. Q., and Yu S. W., An Improved Car-Following Model Accounting for the Preceding Car’s Taillight, Physica A: Statistical Mechanics and Its Applications. (2018) 492, 1831–1837, https://doi.org/10.1016/j.physa.2017.11.100, 2-s2.0-85035355646.
- 22 Zhai C. and Wu W., A Continuous Traffic Flow Model Considering Predictive Headway Variation and Preceding Vehicle’s Taillight Effect, Physica A: Statistical Mechanics and its Applications. (2021) 584, 126364, https://doi.org/10.1016/j.physa.2021.126364.
- 23 Zhai C. and Wu W., A New Continuum Model With Driver’s Continuous Sensory Memory and Preceding Vehicle’s Taillight, Communications in Theoretical Physics. (2020) 72, no. 10, 105004, https://doi.org/10.1088/1572-9494/aba24c.
- 24 Li Z. and Ma C., Stability Analysis of the New Traffic Flow Lattice Model Considering Taillight Effect and Speed Deviation, Journal of Advanced Transportation. (2022) 2022, no. 1, 11, 1716827, https://doi.org/10.1155/2022/1716827.
- 25 Fosu G. O., Opong J. M., Owusu B. E., and Naandam S. M., Modeling Road Surface Potholes Within the Macroscopic Flow Framework, Mathematics in Applied Sciences and Engineering. (2022) 3, no. 2, 106–118, https://doi.org/10.5206/mase/14625.
- 26 Khan Z. H., Altamimi A. B., Imran W., Alsaffar M., Khattak K. S., and Alfaisal F. F., Macroscopic Traffic Modelling on the Impact of Road Surface Potholes: Development and Numerical Solution, IEEE Access. (2024) 12, 81718–81735, https://doi.org/10.1109/ACCESS.2024.3411303.
- 27 Delitala M. and Tosin A., Mathematical Modeling of Vehicular Traffic: A Discrete Kinetic Theory Approach, Mathematical Models and Methods in Applied Sciences. (2007) 17, no. 6, 901–932, https://doi.org/10.1142/S0218202507002157, 2-s2.0-34249789588.
- 28 Bellouquid A. and Delitala M., Asymptotic Limits of a Discrete Kinetic Theory Model of Vehicular Traffic, Applied Mathematics Letters. (2011) 24, no. 5, 672–678, https://doi.org/10.1016/j.aml.2010.12.004, 2-s2.0-79551486464.
- 29 Li C. Y., Tang T. Q., Huang H. J., and Shang H. Y., A New Car-Following Model With Consideration of Driving Resistance, Chinese Physics Letters. (2011) 28, no. 3, 038902, https://doi.org/10.1088/0256-307X/28/3/038902, 2-s2.0-79952519202.
- 30 Tang T. Q., Li P., Wu Y. H., and Huang H. J., A Macro Model for Traffic Flow With Consideration of Static Bottleneck, Communications in Theoretical Physics. (2012) 58, no. 2, 300–306, https://doi.org/10.1088/0253-6102/58/2/23, 2-s2.0-84864910645.
- 31 Tang T. Q., Caccetta L., Wu Y. H., Huang H. J., and Yang X. B., A Macro Model for Traffic Flow on Road Networks With Varying Road Conditions, Journal of Advanced Transportation. (2014) 48, no. 4, 304–317, https://doi.org/10.1002/atr.215, 2-s2.0-84902472521.
- 32
Golovnin O., Data-Driven Profiling of Traffic Flow With Varying Road Conditions, CEUR Workshop Proceedings, 2019, Institute of Physics Publishing (IOP), 149–157.
10.18287/1613-0073-2019-2416-149-157 Google Scholar
- 33
Arjapure S. and
Kalbande D., Review on Analysis Techniques for Road Pothole Detection, Soft Computing: Theories and Applications: Proceedings of SoCTA 2018, 2020, Springer, 1189–1197.
10.1007/978-981-15-0751-9_109 Google Scholar
- 34 Bansal K., Mittal K., Ahuja G., Singh A., and Gill S. S., DeepBus: Machine Learning Based Real Time Pothole Detection System for Smart Transportation Using IoT, Internet Technology Letters. (2020) 3, no. 3, e156, https://doi.org/10.1002/itl2.156.
- 35 Dharneeshkar J., Aniruthan S., Karthika R., and Parameswaran L., Deep Learning Based Detection of Potholes in Indian Roads Using YOLO, Proceedings of the 2020 International Conference on Inventive Computation Technologies (ICICT), 2020, IEEE, 381–385, https://doi.org/10.1109/ICICT48043.2020.9112424.
- 36 Asad M. H., Khaliq S., Yousaf M. H., Ullah M. O., and Ahmad A., Pothole Detection Using Deep Learning: A Real-Time and AI-on-the-Edge Perspective, Advances in Civil Engineering. (2022) 2022, no. 1, 9221211, https://doi.org/10.1155/2022/9221211.
- 37
Audu-war S. O.,
Anigbogu S. O.,
Anigbogu K. S.,
Anigbogu G. N., and
Asogwa D. C., Pothole Detection Using Image Surveillance System: A Review, World Journal of Advanced Engineering Technology and Sciences. (2023) 9, no. 2, 214–222, https://doi.org/10.30574/wjaets.2023.9.2.0210.
10.30574/wjaets.2023.9.2.0210 Google Scholar
- 38 Bučko B., Lieskovská E., Zábovská K., and Zábovský M., Computer Vision Based Pothole Detection Under Challenging Conditions, Sensors. (2022) 22, no. 22, https://doi.org/10.3390/s22228878, 36433474.
- 39 Chougule S. and Barhatte A., Smart Pothole Detection System Using Deep Learning Algorithms, International Journal of Intelligent Transportation Systems Research. (2023) 21, no. 3, 483–492, https://doi.org/10.1007/s13177-023-00363-3.
- 40
Jakubec M.,
Lieskovská E.,
Buˇcko B., and
Zábovská K., Comparison of CNN-Based Models for Pothole Detection in Real-World Adverse Conditions: Overview and Evaluation, Applied Sciences. (2023) 13, no. 9, https://doi.org/10.3390/app13095810.
10.3390/app13095810 Google Scholar
- 41 Ma N., Fan J., Wang W., Wu J., Jiang Y., Xie L., and Fan R., Computer Vision for Road Imaging and Pothole Detection: A State-of-the-Art Review of Systems and Algorithms, Transportation safety and Environment. (2022) 4, no. 4, tdac026, https://doi.org/10.1093/tse/tdac026.
- 42 Pandey A. K., Iqbal R., Maniak T., Karyotis C., Akuma S., and Palade V., Convolution Neural Networks for Pothole Detection of Critical Road Infrastructure, Computers and Electrical Engineering. (2022) 99, 107725, https://doi.org/10.1016/j.compeleceng.2022.107725.
- 43 Singh P., Kamal A. E., Bansal A., and Kumar S., Road Pothole Detection From Smartphone Sensor Data Using Improved LSTM, Multimedia Tools and Applications. (2024) 83, no. 9, 26009–26030, https://doi.org/10.1007/s11042-023-16177-0.
- 44 Bexelius S., An Extended Model for Car-Following, Transportation Research. (1968) 2, no. 1, 13–21, https://doi.org/10.1016/0041-1647(68)90004-X, 2-s2.0-0041503216.
- 45 Gipps P. G., A Behavioural Car-Following Model for Computer Simulation, Transportation Research Part B: Methodological. (1981) 15, no. 2, 105–111, https://doi.org/10.1016/0191-2615(81)90037-0, 2-s2.0-0019558539.
- 46
Dafermos C. M., Hyperbolic Conservation Laws in Continuum Physics, 2005, Springer.
10.1007/3-540-29089-3 Google Scholar
- 47
Bressan A., Hyperbolic Systems of Conservation Laws, Oxford Lecture Series in Mathematics and its Applications. (1999) 12, no. 1, https://doi.org/10.5209/rev_REMA.1999.v12.n1.17204.
10.5209/rev_REMA.1999.v12.n1.17204 Google Scholar
- 48 Toro E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 2013, Springer Science & Business Media.
- 49 Helbing D. and Johansson A., On the Controversy Around Daganzo’s Requiem for and Aw-Rascle’s Resurrection of Second-Order Traffic Flow Models, European Physical Journal B. (2009) 69, no. 4, 549–562, https://doi.org/10.1140/epjb/e2009-00182-7, 2-s2.0-67650224266.
- 50 Fosu G. O. and Oduro F. T., Two Dimensional Anisotropic Macroscopic Second-Order Traffic Flow Model, Journal of Applied Mathematics and Computational Mechanics. (2020) 19, no. 2, 59–71, https://doi.org/10.17512/jamcm.2020.2.05.
- 51
Del Castillo J. M. and
Benitez F., On the Functional Form of the Speed-Density Relationship—I: General Theory, Transportation Research Part B: Methodological. (1995) 29, no. 5, 373–389, https://doi.org/10.1016/0191-2615(95)00008-2, 2-s2.0-0029391007.
10.1016/0191-2615(95)00008-2 Google Scholar
- 52
Fosu G. O.,
Akweittey E.,
Opong J. M., and
Otoo M. E., Vehicular Traffic Models for Speed-Density-Flow Relationship, Journal of Mathematical Modeling. (2020) 8, no. 3, 241–255, https://doi.org/10.22124/jmm.2020.15409.1370.
10.22124/jmm.2020.15409.1370 Google Scholar
- 53
Herrmann M. and
Kerner B. S., Local Cluster Effect in Different Traffic Flow Models, Physica A: Statistical Mechanics and Its Applications. (1998) 255, no. 1-2, 163–188, https://doi.org/10.1016/S0378-4371(98)00102-2, 2-s2.0-0032098844.
10.1016/S0378-4371(98)00102-2 Google Scholar
- 54 Kerner B. S. and Konhäuser P., Structure and Parameters of Clusters in Traffic Flow, Physical Review E. (1994) 50, no. 1, 54–83, https://doi.org/10.1103/PhysRevE.50.54, 2-s2.0-0002562202, 9961944.