IEEE Computational Intelligence Magazine - November 2020 - 25

-formulated the R t 's updating function by introducing external factors (e.g. interventions and mobility). Thus, the estimated R t curve is largely constrained by the factors that are
considered in the model.
Data Assimilation [20] lends itself naturally to this problem
since it provides a framework to enable dynamically updating
the model states and parameters when new observations
become available while also taking into account model and
observation uncertainty. Data assimilation technologies, such
as Kalman filter and variational method [21], have been widely used in signal tracking, oceanology, environment monitoring and weather forecasting where physical models and
observation data are assimilated to produce accurate predictions. Data assimilation for epidemiological modeling was first
proposed in [22] where compartment models were used as
the underlying model for assimilation. In [25] and [26], estimating time-varying parameters in the compartment models
was further investigated. To the authors' best knowledge, our
work is the first study to apply data assimilation to the renewal process-based model.

suppression to disentangle the intervention effects. As illustrated in Figure 1, we uste two complementary metrics p t and
D t to model these factors, respectively.
The suppression effects mainly shorten the infectious period
of the infected population, corresponding to the truncation of
b ^ x h along the horizontal axis. We use a time-varying parameter D t to denote the effective infectious window induced by
suppression. The mitigation effects attenuate the overall infectiousness by shielding the susceptible population, corresponding to the scaling in the vertical direction. We introduce another
time-varying parameter p t to describe this attenuation effect
induced by mitigation. Formally, we parameterize the evolution
of the infectiousness profile as:

III. Epidemic Modeling of COVID-19 Transmission

Therefore, the impact of intervention measures on R t
reduction is disentangled: mitigation factor p t attenuates the
overall infectiousness through shielding the susceptible population, and suppression factor D t shortens the infectious period

In this section, we propose a time-varying renewal process with
two complementary parameters p t and D t to model the
evolving infectiousness profile. We adopted a time-varying
renewal process for epidemic modeling. The renewal process
[8] of infectious disease transmission is:

where the unit-normalized transmission rate w ^ x h is the probability density function of generation time, i.e. the interval
between the primary infection and the secondary infection. In
the early stage without intervention, the infectiousness profile
remains time-independent as the baseline b 0 ^ x h which
describes the transmission dynamics when the susceptible population is infinite. The corresponding R is the well-known
basic reproduction number R 0. In reality, the infectiousness
profile b ^ x h will evolve with time t, therefore we introduce
b t ^ x h to address the change in its distribution caused by intervention measures.
To quantify the impacts of intervention measures to the
evolution of R t, we propose two factors: mitigation and

x
x

1 Dt
(3)
$ Dt

Dt
0

Baseline Infectiveness Profile
β(τ )

R $ w ^ x h (2)

b0 ^xh $ pt

R t = p t $ 8b 0 ^ x h dx (4)

∞

b^xh =

Accordingly, the instantaneous reproductive number R t can
be derived:

I ^ t h = 8I ^t - x h b ^ x h dx (1)

where I ^ t h is the incident infection on time t and b ^ x h is the
infectiousness profile. The infectiousness profile means that a
primary case infected x time ago (i.e. with the infection-age x )
can now generate new secondary cases at a rate of b ^ x h, describing a homogenous mixing process. b ^ x h is related to biological,
behavioral and environmental factors. We can calculate the
reproduction number R as the area under the curve of b ^ x h,
which is the overall number of secondary cases infected by a
primary case. Further, b ^ x h can be rewritten as:

bt ^xh =

R0 > 1
τ

Mitigation Effect

Suppression Effect

β(τ )

β(τ )
pt
τ

Instantaneous Infectiveness Profile
β(τ )

Rt < 1
τ
FIGURE 1 Disentangling the reduction of reproduction number into
mitigation and suppression factors.

NOVEMBER 2020 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE

IEEE Computational Intelligence Magazine - November 2020

Table of Contents for the Digital Edition of IEEE Computational Intelligence Magazine - November 2020