Inferring the effective reproductive number from deterministic and semi-deterministic compartmental models using incidence and mobility data

The effective reproduction number (R-t) is a theoretical indicator of the course of an infectious disease that allows policymakers to evaluate whether current or previous control efforts have been successful or whether additional interventions are necessary. This metric, however, cannot be directly observed and must be inferred from available data. One approach to obtaining such estimates is fitting compartmental models to incidence data. We can envision these dynamic models as the ensemble of structures that describe the disease's natural history and individuals' behavioural patterns. In the context of the response to the COVID-19 pandemic, the assumption of a constant transmission rate is rendered unrealistic, and it is critical to identify a mathematical formulation that accounts for changes in contact patterns. In this work, we leverage existing approaches to propose three complementary formulations that yield similar estimates for R-t based on data from Ireland's first COVID-19 wave. We describe these Data Generating Processes (DGP) in terms of State-Space models. Two (DGP1 and DGP2) correspond to stochastic process models whose transmission rate is modelled as Brownian motion processes (Geometric and Cox-Ingersoll-Ross). These DGPs share a measurement model that accounts for incidence and transmission rates, where mobility data is assumed as a proxy of the transmission rate. We perform inference on these structures using Iterated Filtering and the Particle Filter. The final DGP (DGP3) is built from a pool of deterministic models that describe the transmission rate as information delays. We calibrate this pool of models to incidence reports using Hamiltonian Monte Carlo. By following this complementary approach, we assess the tradeoffs associated with each formulation and reflect on the benefits/risks of incorporating proxy data into the inference process. We anticipate this work will help evaluate the implications of choosing a particular formulation for the dynamics and observation of the time-varying transmission rate. Author summary Policymakers use the effective reproduction number (R-t) to determine whether an epidemic is growing (R-t > 1) or shrinking (R-t < 1). One can estimate this quantity by simulating compartmental models fitted to data. These models can be seen as the ensemble of two structures: one that describes the course of a disease in an individual and another one that accounts for behavioural patterns. Nevertheless, these estimates are sensitive to the assumptions embedded in the model, such as the formulation of the time-varying transmission rate. In this paper, we couple an SEIR-type structure with three complementary formulations: 1) non-negative random-walks (Geometric Brownian Motion) 2) non-negative random-walks pulled toward a long-term goal (Cox-Ingersoll-Ross) 3) Gradual approximations towards a long-term goal (exponential smoothing). We refer to each coupling as a Data Generating Process (DGP). In essence, we simulate trajectories from these DGPs to identify plausible sets of transmission rates (based on incidence and mobility data) that explain Ireland's first COVID-19 wave. Here, we assume that mobility data is a proxy measurement for the transmission rate. These DGPs yield similar average estimates for R-t, albeit with dissimilar degrees of uncertainty. Finally, we reflect on the tradeoffs of choosing each particular formulation.

Automated summary

The effective reproduction number (R-t) is an important indicator for evaluating the development of infectious diseases. This paper proposes three complementary mathematical models to estimate R-t based on data from the first wave of COVID-19 in Ireland.