Sequential data assimilation of the stochastic SEIR epidemic model for regional COVID-19 dynamics

Ralf Engbert; Maximilian M. Rabe; Reinhold Kliegl; Sebastian Reich

The copyright holder for this preprint . https://doi.org/10.1101/2020.04. 13.20063768 doi: medRxiv preprint for the post-intervention period (March 29-April 4). In scenario II, we replaced the post-intervention contact parameter by its pre-intervention value (March 15-21). As a results, the two scenarios predict rather different temporal developments (decline of daily new cases for scenario I, and strong increase for scenario II). Therefore, our model predictions suggest that lifting off the current interventions would clearly switch the epidemic dynamics to the exponential increase before implementation of non-pharmaceutical interventions. Such predictions can easily be scaled up to the federal state level (Bundesländer) or to the country level; a corresponding predictive model will be potentially quite robust because of explicit modeling of spatial and temporal heterogeneities, captured by a separate time-course of the contact parameter for each region. The recent simulation study Li et al. [16] used a similar approach of sequential data assimilation for dynamic epidemic models. However, the deterministic SEIR model was implemented and extended by additional noise assumptions. We proposed the usage of the stochastic SEIR model in the formulation of a master equation [10] which can be simulated exactly and numerically efficiently using Gillespie's algorithm [12] . A more complex spatiotemporal stochastic model has been considered in [4] . Furthermore, the state-parameter estimation in [16] utilises the ensemble Kalman filter directly on an augmented state space [20] . Contrary to that study, we found a direct application of the ensemble Kalman filter to the augmented state space (X, β) not suitable because of the strongly nonlinear interaction between the model states X and the contact parameter β. This led us to the two stage approach, as presented in this study, combining the ensemble Kalman filter for state estimation with a likelihood based inference of the contact parameter β. Our current study was mainly motivated by the methodological problem of a possible contribution from data assimilation to epidemics modeling based on a stochastic SEIR model. There are obvious limitations in our current modeling framework, which we did not address because of the methodological focus. Longer-term predictions (∼ months) are important, but clearly dependent critically on the estimation of undocumented infections (see Li et al. [16] ). Such hidden infections create, after recovery, an unknown reduction in the number of susceptibles that slows down epidemic dynamics; such an effect is currently not included in our current model. However, it seems compatible with our framework to extend the SEIR model by an additional class of undocumented infected individuals [16] . Another important limitation of these results comes from the simplifying assumption that there is no coupling to neighboring regions. As a consequence, regional differences in the contact parameter might in fact be due to differences in contacts between regions. Introducing couplings between regions [16] could also be integrated in our modeling framework. However, the no-coupling approximation might be realistic in the current situation of social distancing and contact ban. Finally, at the time of this work, timestamps refer to the day at which the case was reported to the RKI ("Meldedatum"), not the time of infection ("Referenzdatum"). Recently, such information has become available