We introduce a power-law distance-dependent biased random walk model with a tuning parameter (σ) in which finite mean first-passage times are realizable if σ is less than a critical value σc. We perform numerical simulations in one dimension to obtain σc ≈ 1.14. A three-dimensional variant of this model is argued to be related to the phenomenon of chemotaxis. Diffusiophoretic theory supplemented with coarse-grained simulations for colloidal chemotaxis establish the connection with the specific value of σ = 2 as a consequence of in-built solvent diffusion.

Care pathways in hospitals around the world reported significant disruption during the recent COVID-19 pandemic but measuring the actual impact is more problematic. Process mining can be useful for hospital management to measure the conformance of real-life care to what might be considered normal operations. In this study, we aim to demonstrate that process mining can be used to investigate process changes associated with complex disruptive events. We studied perturbations to accident and emergency (A &E) and maternity pathways in a UK public hospital during the COVID-19 pandemic. Co-incidentally the hospital had implemented a Command Centre approach for patient-flow management affording an opportunity to study both the planned improvement and the disruption due to the pandemic. Our study proposes and demonstrates a method for measuring and investigating the impact of such planned and unplanned disruptions affecting hospital care pathways. We found that during the pandemic, both A &E and maternity pathways had measurable reductions in the mean length of stay and a measurable drop in the percentage of pathways conforming to normative models. There were no distinctive patterns of monthly mean values of length of stay nor conformance throughout the phases of the installation of the hospital’s new Command Centre approach. Due to a deficit in the available A &E data, the findings for A &E pathways could not be interpreted.