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

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

Estimation of the time-dependence of the contact parameter is done via the model's best fit. An approximative instantaneous negative log-likelihood L(t k , β) of the contact parameter β at observation time t k is obtained from the ensemble Kalman filter (see Model inference based on sequential data assimilation). Thus, by determining the minimum of L(t k , β) with respect to β at time t k we estimate the time-dependence of the best fit β * (t k ) (Fig. 2c) . The black line reports the average time dependence for all 320 regions included in the analysis; standard deviations are indicated by the grey area. Results for the two example regions are given by their corresponding colors. The non-pharmaceutical interventions in the spread of COVID-19 were implemented at slightly varying points in time across Germany. In the majority of regions, closings of schools and other educational institutions started on March 16th, while large-scale contact bans was implemented on March 22th. Since these social distancing measures will have an impact on the contact parameter, we expected to observe a related drop in the contact parameter over time. Before we present a corresponding analysis, it should be made clear that any of these measures cannot produce an immediate effect on the observed cases of infected individuals because of the latency period. To use a reliable estimate of the contact parameter, the related interval should be as long as possible, since sequential data assimilation will need several data points to adapt the model to the data. Therefore, we selected the average value of β * (t k ) over the three days from March 17th to March 19th as a pre-intervention value. The average over March 31st to April 2nd is taken as an estimate of the post-intervention value. To analyze the effect across regions, we computed average values β pre (March [17] [18] [19] and β post (March 31-April 2) of the relevant β * (t k ) for all regions. A scatter plot indicated a clear reduction of the numerical value of the contact parameter from β pre to β post (Fig. 2d) . The reduction is statistically significant (Wilcoxon test, p < 0.01).