Mathematical Modeling of SARS Transmission in Singapore: from a Public Health PerspectiveStefan Ma1, Marc Lipsitch2

1Epidemiology & Disease Control Division

Ministry of Health, Singapore

2Department of Epidemiology

Harvard School of Public Health, United States

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.

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.

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.

Epidemic Curve of SARS Cases in Singaporeas at 31 May 2003 (n=206)

0

2

4

6

8

10

12

14

25-F

eb

4-M

ar

11-M

ar

18-M

ar

25-M

ar

1-Apr

8-Apr

15-A

pr

22-A

pr

29-A

pr

6-M

ay

13-M

ay

20-M

ay

27-M

ay

Date of onset

Nu

mb

er

of

ca

se

s

Last onset: 5 MayIsolated: 11 May

TTSH Cluster

SGH ClusterFirst 3imported

cases

Percentage of direct secondary casesfrom probable cases of SARS

0%

20%

40%

60%

80%

100%

0 1 2 3 4 >4

Number of persons infected by an individual probable SARS patient

Weekly average time from onset of symptomsto isolation of SARS cases

0

1

2

3

4

5

6

7

8

9

10

25/0

2-03

/03

04/0

3-10

/03

11/0

3-17

/03

18/0

3-24

/03

25/0

3-31

/03

01/0

4-07

/04

08/0

4-14

/04

15/0

4-21

/04

22/0

4-28

/04

Week of onset

Tim

e f

rom

on

se

t to

is

ola

tio

n (

da

ys

)

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.

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

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.

Conceptual Model of SARS Transmission (SEIR)

Susceptible(X)

Susceptible Quarantined (XQ)

LatentInfection (E)

Latent InfectionQuarantined (EQ)

Infectious, Undetected (IU)

Infectious,Quarantined (IQ)

Infectious,Isolated (ID)

Death due to SARS (D)

Recovered,Immune (R)

vv

v

m

mm

rQ

p p

w

qkb

kb

w

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.

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!!!

0.51.01.52.0R_int=2.50.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Proportion of contacts quarantined (q)

Pro

po

rtio

n r

ed

uct

ion

inin

fect

iou

s p

erio

d (

1-D

_in

t/D)

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.

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.

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.

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.

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.

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.

Thank You