Mathematics
and Statistics of SARS
~ ABSTRACTS ~
Multistage SIR model
K. W. Liang, S. L. Lee, X.B. Pan and Roger C. E. Tan, Department
of Mathematics, NUS
The Susceptible-Infected-Recovered (SIR) model can be used in
simulating the natural spread and course of an epidemic, such as
SARS, in a small community. However, in practice, as more is
known of the disease and practical measures such isolation and
quarantine are successfully implemented the course of the
epidemic changes and the SIR model is modified to adapt to the
new measures. A variant of the SIR model is considered, in which
the recovered set (R) is replaced by the isolation set (L) and
the contact number and isolation factor are adaptively computed
at various stages in the course of the epidemic.
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SEIR model: SARS outside the hospitals
K.W. Liang, S. L. Lee, X.B. Pan and Roger C.E. Tan, Department
of Mathematics, NUS
The epidemic dynamics of SARS among individuals outside the
hospitals is examined using a SEIR model. The model consists of
a system of delay differential equations with parameters that
include among others, the probability of infection and
incubation period.
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System dynamics simulation of SARS propagation in Singapore
Ang Chee Chien and Ang Teck Siang, Operations Research
Laboratory, Centre for Decision Support, DSO National
Laboratories
This paper describes the system dynamics model that have been
built to model the trend of SARS propagation in Singapore. The
model is intended to be used as a "management flight
simulator" to determine the factors that are most critical
in controlling the propagation of the virus in Singapore.
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Measure if you can, simulate if you must: an
investigative attempt to understand the nature of SARS
Joseph S.Y. Lee, DSO National Laboratories, Pierce K.H.
Chow, Singapore General Hospital, Jorgen Seldrup, Clinical
Trials and Epidemiology Research Unit, and T.K. Tan, DSO
National Laboratories
Severe Acute Respiratory Syndrome (SARS) emerged only in late
2002 but the rapid transmission of the disease worldwide within
a few months compelled the medical profession and health
authorities to institute public health measures based on
rudimentary data. Cases were defined syndromically in the
absence of diagnostic tests and knowledge of the epidemiology of
SARS remains incomplete. In particular, data on sub-clinical or
asymptomatic infection though critical remained unavailable. On
6th April 2003, 70 patients exposed to a SARS virus outbreak in
two surgical wards in SGH were quarantined and subjected to
triaging and cohorting in 3 wards according to their risks and
nursing needs. We use data from this cohort over a 21 day period
to build a graph-based computer simulation model to validate
epidemiological hypotheses pertaining to infectivity,
transmission and the relevance of sub-clinical states."
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Some branching process models and the experience of SARS in
Singapore
Bruce Brown and Lewin-Koh Sock Cheng, Department of
Statistics & Applied Probability, NUS
A two-state branching process model is formulated to describe
the spread of SARS in Singapore. This is an extension of simple
branching process models which have been applied in past
epidemiology studies to the spread of infectious diseases,
especially in the early stages. The model is relatively easy to
analyze and produces readily interpretable formulas for the
total number of cases in a hypothetical future outbreak, The
analysis of the model depends on the future availability of a
detailed record of SARS transmission in Singapore.
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Phylogenetic analysis of the SARS virus and other
coronaviruses
Zhang Louxin, Department of Mathematics, NUS
The talk presents our preliminary results on the analysis of
the genomic sequences of the SARS coronaviruses and some other
viruses. The objective is to track the step-by-step transmission
of the SARS and to answer where the SARS virus came from. We
analyzed the genomic sequences of the SARS virus and other
viruses that are closely related to the SARS using VISTA. Using
phylogenetic methods, we also studied the genetic variance of
different strains that were found in Beijing, Guangdong, Hong
Kong, Singapore, Taiwan, and Toronto.
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Searching for unusual clusters of palindromes and close
inversions in the SARS genome
Choi Kwok Pui, Departments of Mathematics and of Statistics
and Applied Probability, NUS
A close inversion in DNA/RNA genome refers to a string of
nucleotide bases and its inverted complementary sequence
occurring in close proximity of each other. Close inversions in
an RNA genome, such as SARS, play an important role in the
secondary structure of the genome for possible formation of
hairpins. Palindromes are special cases of close inversions in
that the string of nucleotide bases is adjacent to its inverted
complementary sequence. They are possibly involved in many
biological functions. In this talk, we will present different
scoring schemes to identify regions in the SARS virus genome
which contain statistically significant clusters of palindromes
and close inversions. These regions potentially contain promoter
elements and transcription factor binding sites. This talk is
based on an ongoing project of Louis H Y Chen, David Chew and
Ming-Ying Leung.
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Statistical modeling of SARS epidemic propagation via
branching processes
V. Kamalesh, V. Kuralmani, Goh Li Ping, Qian Long, Fu Xiuju
and Terence Hung, Software & Computing Programme, Institute of
High Performance Computing
The study of branching processes originated with a
mathematical puzzle posed by Sir Francis Galton, the noted
cousin of Charles Darwin, in the Educational Times of 1 April
1873. Branching process may be viewed as a mathematical
representation of the evolution of a population wherein the
reproduction and death are subject to the laws of chance. A good
number of examples have been brought under this broad coverage
like the propagation of human and animal species and genes,
nuclear chain reaction and electronic cascade phenomena.
Bienayme-Galton-Watson branching process can be thought of as
stochastic model of an evolving population of particles or
individuals. It starts at time 0 with Z(0) particles, each of
which splits into a random number of offspring that constitute
the first generation, and so on. The number of
"offspring" produced by a single "parent"
particle at any time is independent of the history of the
process, and of other particles existing at the present.
Branching processes can be adopted as models for the spread of
epidemic diseases.
First, the characteristic of interest is transmitted to some
members of a group from a source at an initial point in time.
Then the individuals who have acquired the characteristic spread
it, according to a probability distribution, to the members of
other groups. The new "generation" spreads the same
characteristic again and the process continues over and over
until either it dies out or the entire population gets the
characteristic. Let Z(0) be the number of individuals who first
have acquired the particular characteristic, i.e., the zeroth
generation of ancestors. All the individuals who have acquired
the characteristic from each of the ancestors Z(0) will then
form the first generation, and each of them will convey the
characteristic to others, and so on.
Since the offspring mean of a branching process indicates
almost sure extinction or possible explosion of a population,
there is considerable interest in knowing the value of this
criticality parameter. In this paper we aim to calculate the
extinction probability of the SARS epidemic. This will further
enable us to predict the duration of SARS epidemic under the
prevailing conditions and also the effectiveness of the
quarantine order.
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SARS outbreak in Singapore - a brief description
Chia Kee Seng, Department of Community, Occupational &
Family Medicine, NUS and Young Truong, Department of Statistics
& Applied Probability, NUS
In this presentation, we will examine the role of diagnosis
and isolation as a control mechanism. This is carried out by
using a model of susceptible, exposed, infective, diagnosed, and
recovered classes of people to extract average properties and
rate constants for Singapore population. The properties of this
model and the control measure of the epidemic will be discussed.
Future research directions will also be described.
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Mathematical modeling of SARS transmission in Singapore:
from a public health perspective
Stefan Ma, Epidemiology and Disease Control Division, MOH and
Lipsitch M, Department of Epidemiology, Harvard School of Public
Health, USA
As of 31 May 2003, using a modification of the WHO case
definition, a total of 206 probable cases of SARS have been
reported in Singapore. On 31 May 2003, Singapore was removed
from the list of areas with recent local transmission of SARS.
Some questions, for example, will the current public health
measures, such as isolation of SARS cases and quarantine of
their asymptomatic contacts, be enough to bring SARS under
control?, have been asked by public health workers at the
beginning of the outbreak. However, the questions of this kind
can be quantitatively assessed via mathematical modeling. In
Singapore, there was a significant decline in the time from
symptom onset until hospital admission or isolation starting
from symptom onset in the first week of outbreak. In addition,
it also showed a rapidly decline in the secondary cases infected
by each index cases from 7 for index cases with symptom onset in
the first week of the outbreak to a mean below one in most weeks
thereafter. These declines could be resulted of effective
control measures including isolation of SARS cases and
quarantine of their asymptomatic contacts.
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