Dynamic SIR/SEIR-like models comprising a time-dependent transmission rate: Hamiltonian Monte Carlo approach with applications to COVID-19

A study of changes in the transmission of a disease, in particular, a new disease like COVID-19, requires very flexible models which can capture, among others, the effects of non-pharmacological and pharmacological measures, changes in population behaviour and random events. We favour data-driven approaches over a priori and ad-hoc methods and introduce a generalised family of epidemiologically informed mechanistic models, guided by Ordinary Differential Equations and embedded in a probabilistic model. The mechanistic models SIKR and SEMIKR which divide the population into disjoint compartments for individuals Susceptible to infection, Infectious (K sub-compartments), Exposed (M sub-compartments), and Removed from the pool of susceptible are enriched with a time-dependent transmission rate, parameterised using Bayesian P-splines. Such a parameterisation enables an extensive flexibility in the transmission dynamics, without resorting to ad-hoc specifications. Our probabilistic model relies on the solutions of a mechanistic model and benefits from access to the information about under-reporting of new infected cases, a crucial property when studying diseases with a large fraction of asymptomatic infections. Such a model can be differentiated efficiently, which makes Hamiltonian-based Monte Carlo sampling feasible after a careful initialisation and tuning strategy. This is particularly important in the present setting with weakly identified directions and challenging posterior geometries. Furthermore, we apply our methodology to study the transmission dynamics of COVID-19 in the Basque Country (Spain) from mid February 2020 to the end of January 2021, showing how the framework can recover plausible temporal patterns in transmission while making explicit the dependence of the results on modelling choices and convergence diagnostics.