IEEE Computational Intelligence Magazine - November 2020 - 28

where K v1 ^i h and K v2 ^i h are the Gaussian kernels before and
after the deployment of intervention at time d. This high-level
model includes three hyperparameters: variances before and
after intervention: v 1 and v 2 , and the change-point time d.
Let us denote the hyperparameters h = 6v 1, v 2, d@T . After the
latest observation Itt , the posterior estimation of i is updated
by the Bayes rule:

T % p ^i t - 1 | It1: t - 1h $ p ^Itt | i t, It1: t - 1h
p ^i t |It1: t h =
(11)
p ^Itt | It1: t - 1h

This step reflects the Bayesian principle in the key updating
step in Kalman filtering [21]. Unlike the Kalman filtering
method where uncertainty is explicitly modeled through a
covariance matrix under the Gaussian assumption, we directly
use posterior probability to capture the uncertainty of estimation. The poster ior is usually intractable but can be
approximated through grid-based methods. Given a set of
hyperparameters h i, the hybrid model evidence can be calculated as [26]:

uninformative as a uniform distribution with the pre-set lower
and upper limits (e.g., the upper limit for the European countries is set to 8 in the experiment). The shape of b 0 ^ x h is
adapted from the distribution of generation time interval w ^ x h
reported by Ferretti et al. [5]. We applied the above framework
to infer the epidemic evolution in 14 European countries,
states in the US and Wuhan city, China in Section VI.
V. Evaluation of Intervention Measures

With the estimated results from the above Bayesian updating
scheme, now we can perform statistical analysis between the
evolution of the transmission dynamics and the implementation of intervention measures. The whole framework containing data reconstruction, dynamic modeling, Bayesian updating,
and statistical analysis is presented in Figure 4. In this section,
we introduce the quantification of intervention measures and
the statistical method.
A. Data Source

For the observations, we use the aggregated data of publicly
available daily confirmed cases of 14 Europe countries (Austria,
Belgium, Denmark, France, Germany, Ireland, Italy, Netherp ^It1: t | h i h = 8 p ^It1: t, i t | h i h di t (12)

lands, Norway, Portugal, Spain, Sweden, Switzerland and the
United Kingdom) and 52 states of the United States from John
Finally, the posterior estimation p ^i t |It1: t h can be averaged
Hopkins University database [1]. The data include the time
across the hyperparameter grids weighted by the hybrid model
series of confirmed cases from 22nd January to 8th June 2020
evidence. The posterior mean and confidence intervals of p t
(accessed on 9th June 2020). Six states with accumulated conand D t as well as the corresponding R t are obtained in a dy--
firmed cases less than 1,000 are excluded from the analysis. The
namic manner. The prior of R 0 at the first timestep is set
daily number of onset patients in
Wuhan is adopted from the retrospective study by Pan et al. [3].
1) Infection Data Reconstructed From Observations
The data of intervention measures
in European countries are collected
Report of
from the Oxford Coronavirus GovernReconstructed
Deconvolution
Confirmed
Daily Infection
ment Response Tracker [2], reporting
Cases
the overall stringency index S t of intervention measures during the analysis
period (accessed on 9th June 2020).
4) Statistical Analysis
This overall stringency index is calculatMitigation pt
ed based on the policy quantification of
eight intervention measures (i.e. school
Bayesian
Instantaneous Rt
Regression
closing, workplace closing, cancel public
Update
Model
events, restrictions on gatherings, close
Suppression Dt
public transport, stay-at-home requirements, restrictions on internal move2) Parameter Estimation
ment and international travel controls)
and one health measure (i.e. public info
3) Policy Quantification
campaigns) to indicate the government
Response
Report of
Policy Index
response level of intervention.
Levels
Policy Event
According to the normalized stringency index by Oxford report [2], we
categorized the dates into five response
FIGURE 4 Components of the quantification framework. The evolution of mitigation and suplevels (Level 0: S t # 20%, minimal
pression factors are estimated using the infection data reconstructed from the daily reported
response
for reference; Level 1: 20%
confirmed cases. Given the history of government responses, the impacts of intervention mea< S t # 40%, soft response; Level 2: 40%
sures are quantified by correlating the inferred epidemic parameters to response levels.

IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | NOVEMBER 2020

IEEE Computational Intelligence Magazine - November 2020

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