Modelling and Stabilising Traffic Congestion via a Macroscopic SI‐Based PDE Model

Understanding and stabilising congestion dynamics in vehicular traffic systems remains a major challenge due to nonlinear propagation and dissipation effects. This article develops a macroscopic framework for modelling traffic congestion dynamics through an epidemiological analogy based on the susceptible‐infected (SI) paradigm. In this formulation, free‐flowing vehicles are interpreted as susceptible states, congested vehicles as infected states, and congestion propagation is represented as a contagion‐like process. A spatially extended system of nonlinear partial differential equations (PDEs) is formulated to incorporate density‐dependent advection governed by Greenshields’ law, road‐capacity saturation effects and spatiotemporally varying dissipation representing traffic control interventions. Equilibrium states are derived and analysed using both spectral and Lyapunov stability techniques. Spectral stability is characterised through dispersion‐relation analysis, while a Lyapunov functional establishes sufficient conditions for nonlinear decay of congestion under sufficiently small perturbations and adequate dissipation. Numerical simulations support the analytical results by illustrating the suppression or amplification of congestion waves under different control regimes. The analysis yields a stability condition of the form γ ( x , t ) > β ( x ) S _e , providing a mathematical basis for interpreting adaptive interventions such as ramp metering, signal timing adjustments and localised traffic regulation. The proposed framework offers an alternative mathematical approach for analysing congestion propagation, stability transitions and traffic control in spatially varying traffic systems.

MSC2020 Classification : 35B35, 37L15, 90B20