  The algorithms used for the optimal management of an ambulance fleet require
an accurate description of the spatio-temporal evolution of the emergency
events. In the last years, several authors have proposed sophisticated
statistical approaches to forecast ambulance dispatches, typically modelling
the data as a point pattern occurring on a planar region. Nevertheless,
ambulance interventions can be more appropriately modelled as a realisation of
a point process occurring on a linear network. The constrained spatial domain
raises specific challenges and unique methodological problems that cannot be
ignored when developing a proper statistical approach. Hence, this paper
proposes a spatio-temporal model to analyse ambulance dispatches focusing on
the interventions that occurred in the road network of Milan (Italy) from 2015
to 2017. We adopt a non-separable first-order intensity function with spatial
and temporal terms. The temporal dimension is estimated semi-parametrically
using a Poisson regression model, while the spatial dimension is estimated
non-parametrically using a network kernel function. A set of weights is
included in the spatial term to capture space-time interactions, inducing
non-separability in the intensity function. A series of tests show that our
approach successfully models the ambulance interventions and captures the
space-time patterns more accurately than planar or separable point process
models.
