Highway congestion costs European road networks an estimated €100 billion annually. Highway traffic is characterised by congestion shock waves, arising when density exceeds a critical threshold density. These shock waves cause a heterogeneous density distribution across the network, reducing its overall efficiency. Crucially, this heterogeneity is path-dependent: the network operates less efficiently during recovery from congestion than during its build-up, a phenomenon known as hysteresis. Current network-level traffic models cannot reproduce this, as they rely on a memoryless speed–density relation.

This thesis extends the classical bathtub model — a network-level traffic model driven by average density — with an explicit, path-dependent congestion variable representing the spatial extent of congestion in the network. To calibrate the model, loop detector data from 124 weekday mornings on the A10 ring road in Amsterdam (January–June 2018) were used to construct empirical congestion measures. Three candidate congestion measures were compared — density spread, an unweighted congestion fraction, and a density–weighted variant — on their ability to characterise the network congestion state. The density–weighted measure was selected based on superior fit to observed network speeds (R2 = 0.960). Empirical analysis established a critical density threshold ρcrit ≈ 17.2 veh km−1 above which congestion onset and recovery follow asymmetric paths, captured by a dynamical system with separate build-up and recovery rates. Calibration confirmed that congestion builds approximately 31% faster than it dissipates. Forward simulation of the calibrated model reproduces the hysteresis loops observed on the A10.

The model offers a tractable basis for traffic management applications such as ramp metering, variable speed limits, and congestion pricing, enabling prediction of the asymmetric dynamics of morning rush congestion. A current limitation is that inflow is treated as exogenous, producing an unrealistically sharp gridlock sensitivity. Incorporating observed inflow data is identified as the most important direction for future work.

All organisms are built out of cellular tissue. Being able to recognise abnormalities in these tissues could be useful in recognizing cancerous cells. In this thesis we construct a mathematical model for cellular tissue based on its spatial structure. We consider cells as elements of the network. Touching cells are considered connected. Cells grow at different growth speeds. We determine the point in time when this network is fully connected, meaning there is a path between every pair of cells through touching cells. This point indicates the start of the last phase of the cellular growth, where friction restricts cell movement. We first use a Poisson point process to generate the locations of the cells. To make the model more similar to cellular tissue, we introduce determinantal point processes which have short-ranged repulsion, meaning points repel each other and thus spread. We compare the repulsion of Poisson and Determinantal point processes with real cellular tissue. We conclude that determinantal point processes have significantly higher repulsion than Poisson point processes. We also conclude that the repulsion in cellular tissue is higher than both point processes. Using simulations, we show that the model with determinantal point processes reaches connectivity significantly earlier than the Poisson model. We conclude that the analytically derived connectivity time point from the Poisson model can be used as an upper bound for the determinantal model.