Modeling the SARS epidemic in Hong Kong

Dr. Liu Hongjie, Prof. Wong Tze Wai Department of Community & Family Medicine

The Chinese University of Hong Kong

Dr. James DerrickDepartment of Anaesthesia & Intensive Care

Prince of Wales Hospital

May 13, 2003

Modeling the SARS epidemic in Hong Kong

• We aim to construct a model of the SARS epidemic in community (i.e., we have excluded the outbreaks among health care workers or the “common source” outbreak in Amoy Garden.

• The model only applies to a person-to-person mode of transmission of SARS.

Objectives of the Media Release:

1. Explain the natural course of an epidemic – the relationship between the population that is infectious (including patients and infected individuals who will become patients), susceptible population and immune (recovered) population;

2. Show how the natural progression of an epidemic is affected by the effectiveness and timeliness of public health measures, by introducing our mathematical model;

3. Using the assumptions and limitations of our model, discuss the current situation in terms of the trend of the epidemic and the likelihood of its resurgence;

Dynamics of disease and of infectiousness at the individual level

Times (days)

Clinical onset

Incubation period

Time ofinfection

Resolution

Relapse

Symptomatic periodSusceptible

• immune• carrier• dead• recovered

Dynamics of disease

Onset ofinfectiousness

Latent period

End ofinfectiousness

infectious periodSusceptible

Dynamics of infectiousness

Dynamics of infectiousness at the population level

Susceptible St

InfectiousIt

Recovered / immuneRt

SIR

Basic Reproductive Number (R0)

• The average number of individuals directly infected by an infectious case during his/her entire infectious period

• In a population

if R0 > 1 : epidemic

if R0 = 1 : endemic stage

if R0 < 1 : sucessful control of infection

• If population is completely susceptible

measles : R0 = 15-20

smallpox : R0 = 3 – 5

SARS: ???

Basic (R0) reproductive number

R0 =

Number of contactper day

Transmissionprobabilityper contact

Duration of

infectiousnessx x D =

• Average number of contacts made by an infective individual during the infectious period: D e.g. 2 persons per day X 5 days = 10 persons

• Number of new infections produced by one infective during his infectious period:

No. of contacts during D ( D ) X transmission probability per contact ( )

e.g. 10 persons X 0.2 = 2 infected cases

Basic (R0) reproductive number

R0 =

Number of contactper day

Transmissionprobabilityper contact

Duration of

infectiousnessx x D =

Preventive measures targeting reducing any parts of the components will halt SARS epidemic

SIR Model

Susceptible St

InfectiousIt

Recovered / immuneRt

SIR

St: Proportion of population (n) that is susceptible at time t

It: Proportion of n that is currently infected and infectious at time t

Rt : Proportion of n that is recovered / immune

SIR mode is used to predict the three proportions at different scenarios.

Estimate of the 3 proportions changing over time t

Susceptible St

InfectiousIt

Recovered / immuneRt

SIR

Time derivatives of 3 proportion

dX/dt, where X could be S, I or R

At any time t during the epidemic, the 3 equations will be:

dS/dt = - S I

dI/dt = SI – I/D

  dR/dt = I/D

A close look at dS/dt = - S I

Susceptible St Infectious It Recovered / immune Rt SIR

In population, there are 6 different types of possible contacts

Susceptible meets susceptible (S - S, no transmission)

Susceptible meets infectious (S - I, transmission)

Susceptible meets resistant (immune) (S - R, no transmission)

Infectious meets infectious (I - I, no transmission)

Infectious meets resistant (I - R, no transmission)

Resistant meets resistant (R - R, no transmission)

Assumptions of this model

1. the average household size is 3 (census data in 2001);2. the interval between onset of disease and admission

to hospitals is 5 days (based on the paper by Peiris et al. Coronavirus as a possible cause of SARS. Lancet online, 8 April, 2003); 

3. Once SARS patients are hospitalized, they are not able to disseminate the infection back to the community;

4. Patients are infectious one day before the onset of their illness till hospitalized.

Guideline for estimate R0

In Hong Kong, the mean life expectancy is about 80 years.

The average age at the SARS infection is about 40, thus

R0 80 / 40 2

The value of R0 is used to estimate the parameters in modeling the

natural history of the SARS epidemic in Hong Kong.

(Ref. Anderson and May. Infectious Diseases of Humans: Dynamics and Control, 1991)

R0 Mean life expectancy / Average age at infection

Estimate of parameters----- Natural history

• Duration of infectivity (day): 6 days 1 day before onset of symptoms 5 day-delay in seeking treatment (Peiris’s paper)

: No. of contacted person: 14 Household (HH): 2 (Average household size is 3 according HK censes in 2001) Social contacts (SC): 12 No. of contacted persons / day: 14/6 = 2.33

: Risk of transmission per contact HH: 0.25 SC: 0.1 Weighted average : 0.149995

• Two infectious cases enter the susceptible population

Proportions of S, I and R Natural history of SARS epidemic

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

10 10 20 30 40 50 60 70 80 90 10

0

110

120

130

140

150

160

170

180

190

200

210

Days of epidemic

Prop

ortio

n

S

I

R

Susceptible

Recovered/Immune

Infectious

Proportions of infectious population (Control started on different dates)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 12 24 36 48 60 72 84 96 108

120

132

144

156

168

180

192

204

216

228

240

252

264

276

288

300

312

324

336

348

Days of epidemic

Prop

ortio

n

Ro= 1.44

Ro= 2.01

Natural

Day 10

Day 40

Day 30

Day 20

Proportions of infectious population at different Ro

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 12 24 36 48 60 72 84 96 108

120

132

144

156

168

180

192

204

216

228

240

252

264

276

288

300

312

324

336

348

Days of epidemic

Prop

ortio

n

Ro = 2.01

Ro =

1.44Ro =

1.39

Ro = 1.3 Ro =

1.17

Control started on day 10

0

20

40

60

80

100

120

140

160

180

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

105

110

115

120

125

130

135

140

Days of epidemic

Infe

ctio

us c

ases

Predicted number of SARS infectious cases

Control Stage 1 starting from on day 20: D = 5, Ro=

1.44

Control Stage 2 starting from on day 30: D = 3, Ro= 0.84

Proportions of S, I and R on log scale (Control at two stages)

0.00000001

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

Days of epidemic

Prop

ortio

n on

log

scal

e

S

I

R

Susceptible

Recovered/Immune

Infectious

Computer Assisted SARS Modeling

Main Messages to bring across

1. If the epidemic is allowed “to run its natural course”, in other words, to die down by itself, up to several million people will fall victim to SARS. Sufficient herd immunity that will protect the community from further epidemics will only be achieved at the expense of this magnitude of community infection;

2. An epidemic will die down only when the basic reproductive number, Ro (number of people infected by a patient) is less than one. This can be achieved only in two ways:- when herd immunity is high enough (natural course of events), or when effective public health measures limit the spread of the epidemic;

Main Messages to bring across:

3. At present, with all the public health measures in place, it appears our public health measures are capable to reduce the number (Ro) to <1;

4. To effectively control the epidemic, efforts must be sustained keep Ro to <1. Otherwise, the epidemic can start again at any time, because the proportion of immune individuals in our population (herd immunity) is far too low to offer any “natural protection”.