Asian Research Journal of Mathematics

Reliable Origin–Destination (OD) demand information is fundamental to transport planning, fleet scheduling, infrastructure investment, and policy formulation. However, in many developing cities, particularly those dominated by informal paratransit systems, comprehensive passenger-movement data are often unavailable, making conventional demand estimation difficult. This study developed and validated an integrated Gravity–Poisson framework for the synthetic estimation of passenger demand along the Nairobi CBD–Ongata Rongai–Kiserian public transport corridor, a major commuter route characterised by severe data limitations and highly directional travel patterns.

The framework combines a doubly constrained gravity model for OD trip distribution with a Poisson stochastic process for modelling passenger arrivals. Spatial interactions were estimated using demographic and geospatial proxy variables derived from population distributions and network distances, while model calibration was achieved through parameter optimisation and the Iterative Proportional Fitting Procedure (IPFP). Theoretical properties of the framework, including the existence and uniqueness of balanced OD solutions, positivity of passenger flows, trip conservation, convergence of the balancing algorithm, distance elasticity, and parameter identifiability, were formally established. Model performance was evaluated using mean absolute error (MAE), root mean square error (RMSE), sensitivity analysis, and benchmarking against XGBoost and Long Short-Term Memory (LSTM) machine-learning models.

The results revealed a highly concentrated morning commuter flow towards the Nairobi CBD, with approximately 97% of corridor demand converging at the Railways terminal. Calibration yielded a distance-decay parameter of β = 0.1000, indicating relatively weak sensitivity of commuter demand to travel distance. The balanced OD matrix satisfied all theoretical conservation and positivity conditions, while the Poisson arrival model indicated that rainfall, holidays, demonstrations, and network disruptions reduced passenger arrivals. Model calibration substantially improved predictive accuracy, and the sensitivity analysis supported the robustness of the framework under varying operational conditions.

The study demonstrates that reliable and operationally meaningful transport-demand information can be generated even in environments where conventional OD survey data are unavailable. Beyond providing a practical decision-support tool for public transport planning and fleet allocation, the framework contributes to the theoretical foundations of synthetic demand estimation by integrating spatial interaction theory and stochastic arrival modelling within a mathematically rigorous and computationally validated framework. The approach offers a scalable methodology for transport-demand analysis and mobility planning in rapidly urbanising, data-constrained regions.