Mathematical Modeling of SARS Transmission in Singapore: from a Public Health Perspective

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Mathematical Modeling of SARS Transmission in
Singapore from a Public Health Perspective

• Stefan Ma1, Marc Lipsitch2
• 1Epidemiology Disease Control Division
• Ministry of Health, Singapore
• 2Department of Epidemiology
• Harvard School of Public Health, United States

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Introduction

• On 31 May 2003, Singapore was removed from the
  list of areas with recent local transmission of
  SARS.
• As of 3 June 2003, using a modification of the
  WHO case definition, a total of 206 probable
  cases of SARS have been reported in Singapore.

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• 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.

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Objectives

• To use mathematical models of SARS transmission
  to estimate the infectiousness of SARS from the
  rate of increase of cases, assess the likelihood
  of an outbreak when a case is introduced into a
  susceptible population, and
• To draw preliminary conclusions about the impact
  of control.

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TTSH Cluster
SGH Cluster
First 3 imported cases
Last onset 5 May Isolated 11 May
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Epidemiological parameter for assessing the
likelihood of an outbreak

• Reproductive number of an infection, Ro is
  defined as the expected number of secondary
  infectious cases generated by an average
  infectious case in an entirely susceptible
  population.
• However, during the course of an epidemic, R the
  effective reproductive number will be used.
• To stop an outbreak, R must be maintained below
  one.

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Problem of using R

• Since the control measures were implemented
  during the course of the epidemic, R can be
  estimated, but R0 may not be known.
• However, the R0 can be estimated (Lipsitch et al
  2003)
• R0 ? S? f(1-f)(S?)2
• where ?(t) ln(y(t))/t f denotes the ratio of
  the infectious period to the serial interval S
  denotes mean serial interval

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R0 of SARS epidemic in Singapore

• Using the Hong Kong SARS reported cases,
• Y(t) 425 cases t 41 days
• And using the Singapore SARS data,
• the mean serial interval was 8.3 days and f 0.7
• The estimated R0 was about 3 (90 credible
  interval 1.5-7.7).
• It means that a single infectious case of SARS
  will infect about 3 secondary cases in a
  population while without control measures
  implemented.

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Conceptual Model of SARS Transmission (SEIR)
kb
Susceptible Quarantined (XQ)
Susceptible (X)
rQ
kb
q
Latent Infection (E)
Latent Infection Quarantined (EQ)
p
p
Infectious, Undetected (IU)
Infectious, Quarantined (IQ)
m
w
m
w
m
Infectious, Isolated (ID)
Death due to SARS (D)
v
v
Recovered, Immune (R)
v
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SEIR can be solved by a set of ordinary
differential equations (Lipsitch et al 2003)
dX / dt - kbIUX / N0 rQXQ dXQ / dt qk(1 -
b)IUX / N0 - rQXQ dE / dt -pE kb(1 - q)IUX /
N0 . . . dD / dt m(IU ID IQ) Simple model
can be derived.
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Simplified model for the effect of quarantine

• In order to access the impact of control
  measures, such as isolation of SARS cases and
  quarantine of their asymptomatic contacts,
• Rint R(1-q)Dint / D,
• where q denotes the proportion of contacts
  quarantined Dint denotes the duration of
  infectiousness in the presence of interventions
  and R 3.
• This is a simplified model!!!

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• To be conservative, if the reduction in time from
  symptom onset to hospital admission/isolation
  assumed, D to be half after the introduction of
  intervening measures (i.e. Dint/D 0.5),
• and in order to prevent the outbreak, the
  effective reproductive number, Rint should be
  maintained below one, hence at least 60 (q
  0.6) of contacts need to be quarantined.

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Conclusion

• If no control measure is implemented, about 3
  secondary SARS cases in population will be
  infected by a single infectious case. In fact,
  the R was less than one in the first 8 weeks.
• In Singapore, there was a significant decline in
  the time from symptom onset until hospital
  admission or isolation from 9 days in the first
  week to a mean 6 days in the second week, to a
  mean less than 2 days in most weeks thereafter.

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Conclusion

• These declines could be resulted of effective
  control measures including
• Placing in home quarantine for those persons
  identified as having had contact with a SARS
  patient
• Screening of passengers at the airport and
  seaports
• Concentration of patients in a single
  SARS-designated hospital,
• Imposition of a no-visitors rule for all public
  hospitals and
• Use of a dedicated private ambulance service to
  transport all possible cases to the
  SARS-designated hospital.

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Conclusion

• Mathematical modeling is a useful and helpful
  tool for monitoring over the course of the
  epidemic as well as assessing the impact of
  control measures.

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• References
• The materials used in this presentation are
  extracted from the following papers
• MMWR. Severe acute respiratory syndrome
  Singapore, 2003. May 9, 2003/Vol. 52/No.18.
• Lipsitch M, Cohen T, Cooper B, Robins JM, Ma S,
  James L, Gopalakrishna G, Chew SK, Tan CC, Samore
  MH, Fisman D, Murray M. Transmission dynamics and
  control of severe acute respiratory syndrome.
  http //www.sciencexpress.org/23 May 2003/Page
  1/10.1126/science.1086616
• WHO SARS Update 70 Singapore removed from list
  of areas with local transmission.

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• Acknowledgments
• Thank you the medical officers and staffs of Tan
  Tock Seng Hospital, Singapore for their courage
  and dedication in caring of SRAS patients.
• Thank you the Epidemiological Unit of Tan Tock
  Seng Hospital, Singapore for data collection,
  collation and facilitation for this
  epidemiological analysis.

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Thank You