1. 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

2. 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.

3. 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.

4. 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.

5. TTSH Cluster
SGH Cluster
First 3
imported
cases
Last onset: 5 May
Isolated: 11 May

8. 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.

9. 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

10. 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: ).
It means that a single infectious case of SARS will infect about 3 secondary cases in a population while without control measures implemented.

11. 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 w m m
Infectious,
Isolated (ID) v
Death due to SARS (D) v
Recovered,
Immune (R) v

12. 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.

13. 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!!!

15. 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.

16. 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.

17. 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.

18. 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.

19. References
The materials used in this presentation are extracted from the following papers:
MMWR. Severe acute respiratory syndrome – Singapore, 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: // May 2003/Page 1/ /science
WHO SARS Update 70 – Singapore removed from list of areas with local transmission.

20. 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.

21. Thank You