IEEE Computational Intelligence Magazine - November 2020 - 24

and regions (e.g. the United States, Hong Kong) alert us to
monitor the epidemic evolution carefully while intervention
measures are being relaxed.
Mathematical models play a key role in understanding
and responding to the emerging COVID-19 pandemic [3]-
[5]. Compartmental models (e.g. SIR, SIER) and time-sinceinfection models (i.e. renewal process-based models) are the
two well-known approaches describing the underlying transmission dynamics [6], [7]. The compartmental models
describe the transmission among sub-populations while the
renewal process-based approach starts from the inter-individual transmission. Despite different nomenclatures and applications, each model contains parameters characterizing the
epidemic dynamics. One of the most well-known parameters
is the reproduction number R, which represents the average
number of secondary cases that would be induced by an
infected primary case [8]. This key parameter is related to the
final epidemic size of an infectious disease [9]. Intervention
measures aim to maintain the reproduction number under
one so that the epidemic can be contained along with time.
Thus, the estimation of time-varying R will reflect the
impacts of an intervention.
The basic reproduction number R 0 is the reproduction
number at the beginning of an epidemic outbreak, when the
susceptible population is approximately infinite and without
intervention measures. When various intervention measures are
being introduced, the instantaneous reproduction number R t
(also called effective reproduction number) is of greater interest.
To gain insights into epidemic evolution, most existing studies
such as [3], [10] focus on estimating time-varying instantaneous
reproduction number R t.
However, the nowcasting of R t from reported data is
not an easy task. Several approaches have been proposed to
estimate R t with different advantages [11]-[13], but the
timeliness and accuracy are still of concern. Nowcasting
results are affected by different factors, such as assumptions
of the epidemic models, statistical inference methods and
uncertainty of data resources. Inappropriate interpretation
or imprecise estimation of R t are criticized for providing
misleading information [14]. For example, the nowcasting
from reported confirmed cases will fall behind the nowcasting from onset data because there is a delay from symptom onset to case report. We hypothesize that more detailed
characteristics of the time-varying infectiousness profile
could be estimated from the publicly available reports (e.g.,
death data, confirmed data, onset data and laboratory data)
and help to better understand and evaluate the efficiency
of interventions.
In this study, we propose a comprehensive Bayesian updating scheme for reliable and timely estimation of parameters in
epidemic models. The transmission dynamics are modeled as a
concise renewal process with time-varying parameters. To
monitor the evolving impacts, more fine-grained modeling of
the transmission dynamics is required. Instead of the wellknown R t , we introduce two complementary parameters: the

IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | NOVEMBER 2020

mitigation factor ( p t ) captures the effect of shielding susceptible population (e.g. through social distancing), and the suppression factor ( D t ) captures the effect of isolating the
infected population (e.g. through quarantine) to stop virus
transmission. We propose a novel method to estimate these
parameters by taking the data assimilation approach with
Bayesian updating methods. We use daily reports of confirmed cases as the observation. A deconvolution method is
used to build an observation function to estimate the infection cases by taking into account the incubation time and
report delay. The evolution of the time-varying infectiousness
profile (i.e. p t and D t ) is estimated from the adjusted epidemic curve through a Bayesian approach of data assimilation.
Such a fine-grained infectiousness profile enables us to quantify the impacts of various intervention measures in a comprehensive way.
The paper is structured as follows: We introduce the related
work in Section II. In III, we present the overview of a timevarying renewal process model where the two parameters p t
and D t are proposed. In IV, we present in detail the Bayesian
updating scheme for estimating the dynamic parameters. In V,
we develop a statistical analysis method of assessing the intervention impacts based on the estimated results and the report
of intervention policies. In VI, as applications of our approach,
we investigate the impacts of intervention measures in European countries, the United States and Wuhan to illustrate the importance of this development.
II. Related Work

At the beginning of the COVID-19 outbreak in Wuhan,
China, compartmental models (e.g. SIR, SEIR model) have
been used to investigate the epidemic dynamics [15]-[17],
where the basic reproductive number was estimated from
the models with static parameters. With the spread of
COVID-19 worldwide, renewal process-based models (i.e.
time-since-infection model) are also widely used in the
study of COVID-19. The R package 'EpiEstim' [11], [12] is
the most widely used in estimating the time-varying R t
with a sliding window. In [10], 'EpiEstim' was applied to
infer R t via the discrete renewal process for policy impact
assessment. Similar work has been done in [3] to infer R t
using 'EpiEstim' from laboratory-confirmed cases in Wuhan
and hence evaluated the impact of non-pharmaceutical public health interventions. The work in [18] has pointed out
that the infection data is usually not available and death data
were used as observation for R t estimation. Instead of simply applying 'EpiEstim' to reported data, they estimated R t
by employing the renewal equation as a latent process to
model infections and connecting the infections to death
data via a generative mechanism. However, the estimated R t
is in a piecewise form and the number of changing points
was assumed to be determined by the imposed interventions. [19] estimates R t from the death data as well
while linking the disease transmissibility to mobility using
the renewal equation. In general, [18] and [19] explicitly

IEEE Computational Intelligence Magazine - November 2020

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