IEEE Computational Intelligence Magazine - November 2020 - 26

Infection

Onset

through isolating the infected population. It is noted that the
R t can be derived from p t and D t, both of which provide
more mechanistic details about the evolution of the infectiousness profile.

Report

s(τ )

IV. Adaptive Parameter Estimation

(a)

We aim to develop a comprehensive framework to estimate parameters of renewal process models using the
Bayesian updating approach of data assimilation, especially
three key parameters: < R t, p t, D t >. The impacts of different interventions can be quantified through monitoring
the evolution of < R t, p t, D t >. This framework contains
all components of a data assimilation system: building an
observation function to map observations to model state,
modeling and Bayesian updating as shown in Figures 2
and 3. By applying the observation function, we reconstruct the number of daily infections from reports of confirmed cases, taking into account the incubation time and
report delay with a deconvolution algorithm. Then < R t,
p t, D t > is estimated through a Bayesian updating approach
of data assimilation.

0.12
0.1
s(τ )

0.08
0.06
0.04
0.02
0

0

5

10

15
Days
(b)

20

25

30

FIGURE 2 Reconstruction of daily infection from the confirmed cases
using deconvolution algorithms. The time delay between the infection and
onset and report is demonstrated (a). The estimated distribution between
infection and report is presented which is used for deconvolution (b).

Likelihood
(t = 1)

Likelihood
(t = 2)

Likelihood
(t = N )

l1:2

l1

Prior (t = 1)

l1:N

Prior (t = 2)

Prior (t = N )

...

T(.)
Posterior
(t = 1)

p1
Rt

D1

T(.)
Posterior
(t = 2)

p2

D2

...
Posterior
(t = N )

pN

DN

Time
FIGURE 3 Illustration of the Bayesian updating framework for estimating suppression and mitigation factors. We employ a two-level hierarchical
model: For each time step, the low-level model (i.e. renewal process) provides the likelihood of p t, D t (green). The posterior (orange) is calculated through the element product of the likelihood and the prior (blue) from the previous time step. To generate the prior for next time step,
we use the high-level model (i.e. the transformation T ) to induce the evolution of parameters. The high-level model is a piecewise gaussian
random walk process where the fluctuations of p t and D t differ before and after an intervention time. The instantaneous reproduction number
R t can be derived from the posterior distribution of p t and D t .

26

IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | NOVEMBER 2020



IEEE Computational Intelligence Magazine - November 2020

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