Sequential Data Assimilation of the Stochastic SEIR Epidemic Model for Regional COVID-19 Dynamics

Abstract

Newly emerging pandemics like COVID-19 call for predictive models to implement precisely tuned responses to limit their deep impact on society. Standard epidemic models provide a theoretically well-founded dynamical description of disease incidence. For COVID-19 with infectiousness peaking before and at symptom onset, the SEIR model explains the hidden build-up of exposed individuals which creates challenges for containment strategies. However, spatial heterogeneity raises questions about the adequacy of modeling epidemic outbreaks on the level of a whole country. Here, we show that by applying sequential data assimilation to the stochastic SEIR epidemic model, we can capture the dynamic behavior of outbreaks on a regional level. Regional modeling, with relatively low numbers of infected and demographic noise, accounts for both spatial heterogeneity and stochasticity. Based on adapted models, short-term predictions can be achieved. Thus, with the help of these sequential data assimilation methods, more realistic epidemic models are within reach.

Keywords: COVID-19; Ensemble Kalman filter; Sequential data assimilation; Stochastic epidemic model.

Fig. 1

The SEIR model. The population is divided into four compartments that represent susceptible, exposed, infectious, and recovered individuals. The contact parameter $\beta$ is critical for disease transmission, and 1/a and 1/g are the average durations of exposed and infectious periods, respectively. Unlike in the standard model, the birth and death processes are neglected in short-term simulations discussed throughout the current study

Fig. 2

(Color figure online)Parameter recovery analysis. a Simulated data with $b=0.6$. b Negative log-likelihood $L_{\mathrm{cum}}(\beta )$ indicates a minimum at about the true parameter value

Fig. 3

(Color figure online)Contact parameter estimates for real data. a Data for Köln; date refers to report of case at RKI. b Negative log-likelihood $L_{\mathrm{cum}}(\beta )$ for Köln give a minimum at ${\beta }_{\ast }\approx 0.7$

Fig. 4

(Color figure online) Analysis of best fit time-dependent contact parameters ${\beta }_{\ast }(t_k)$; date refers to report of case at RKI. a For two regions (LK Köln and LK Münster), the cumulative numbers show a strong increase after different disease onset times. b Semi-logarithmic scaling suggests approximate exponential growth in early as well as later regimes. c The time-dependent contact parameter ${\beta }_{\ast }(t_k)$ indicates a small decrease over time due to social distancing interventions (black: average for 320 regions; red, blue: contact parameter for the examples above; gray shading: standard deviation across regions. d Scatter plot of the time averaged contact parameter ${\beta }_{\mathrm{pre}}$ before intervention and ${\beta }_{\mathrm{post}}$ after intervention. Note that the critical value for disease containment is ${\beta }_{\mathrm{crit}}=1/3$ per day in our model (red lines)

Fig. 5

(Color figure online) Simulations of the stochastic SEIR model for two example regions. Simulations I indicate an ensemble of 100 runs of the model with initial conditions from the first epidemic day with number of cases greater than or equal to 30 (gray: ensemble of trajectories; blue: observations). Simulations II start at March 26, using an ensemble size of 100 after data assimilation (gray: ensemble of trajectories; red: observations). a Cumulative cases of infected individuals over time for LK Köln. b Daily reported new cases for Köln. c Cumulative cases for LK Münster. d Daily new cases for Münster. Date refers to report of case at RKI

Fig. 6

(Color figure online) Model predictions for COVID-19 after data assimilation in comparison to data (black lines). In scenario I (green area), an assimilated ensemble of internal model states starts the forecast with contact parameter ${\beta }_{\mathrm{post}}$ (continuation of social distancing interventions). In scenario II (red area), the equivalent forecast is generated with contact parameter ${\beta }_{\mathrm{pre}}$ (termination of interventions). a Predictions for cumulative case numbers in Heinsberg. b Predictions of daily new cases in Heinsberg. c Predictions of cumulative cases for Warendorf. d Predictions of daily new cases for Warendorf. Date refers to report of case at RKI