Transcript R 0 - The Chinese University of Hong Kong

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 Derrick
Department 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-toperson 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
Dynamics of disease
Time of
infection
Susceptible
Susceptible
Clinical
onset
Incubation period
Latent period
Resolution
Symptomatic period
• immune
• carrier
Relapse
• dead
• recovered
infectious period
Onset of
infectiousness
Dynamics of infectiousness
End of
infectiousness
Times (days)
Dynamics of infectiousness at the population level
Susceptible
St
Infectious
It
Recovered / immune
Rt
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
contact
per day
x
Transmission
probability
per contact
x
Duration
of
infectiousness
= 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
contact
per day
x
Transmission
probability
per contact
x
Duration
of
infectiousness
= D
Preventive measures targeting reducing any parts
of the components will halt SARS epidemic
SIR Model
Susceptible
St
Infectious
It
Recovered / immune
Rt
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
Infectious
It
Recovered / immune
Rt
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 = SI – I/D
dR/dt = I/D
SIR
A close look at dS/dt = -    S  I
Susceptible St
Infectious It
Recovered / immune Rt
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)
SIR
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
R0  Mean life expectancy / Average age at infection
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)
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
1
0.9
Susceptible
0.8
S
I
Recovered/Immune
0.6
R
0.5
0.4
0.3
0.2
0.1
Infectious
Days of epidemic
210
200
190
180
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
0
Proportion
0.7
Proportions of infectious population (Control
started on different dates)
0.2
Natural
0.18
0.16
Ro= 2.01
0.12
0.1
Day 20
0.08
Day 30
Day 10
0.06
Day 40
0.04
Ro= 1.44
0.02
Days of epidemic
348
336
324
312
300
288
276
264
252
240
228
216
204
192
180
168
156
144
132
120
108
96
84
72
60
48
36
24
12
0
0
Proportion
0.14
Proportions of infectious population at different Ro
0.2
Ro = 2.01
0.18
0.16
0.12
0.1
0.08
Ro =
1.44
0.06
Ro =
1.39
0.04
Ro = 1.3
0.02
Control started
on day 10
Ro =
1.17
Days of epidemic
348
336
324
312
300
288
276
264
252
240
228
216
204
192
180
168
156
144
132
120
108
96
84
72
60
48
36
24
12
0
0
Proportion
0.14
Predicted number of SARS infectious cases
180
Control Stage 1 starting from on day 20:
 = 0.144,  = 2, D = 5, Ro= 1.44
160
Control Stage 2 starting from on day 30:
 = 0.14,  = 2, D = 3, Ro= 0.84
120
100
80
60
40
20
Days of epidemic
140
135
130
125
120
115
110
105
100
95
90
85
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0
0
Infectious cases
140
Proportions of S, I and R on log scale
(Control at two stages)
95
10
0
10
5
11
0
11
5
12
0
12
5
13
0
13
5
14
0
85
90
75
80
65
70
55
60
45
50
35
40
25
30
15
20
5
10
0
1
0.1
Susceptible
S
Proportion on log scale
0.01
I
0.001
Recovered/Immune
R
0.0001
0.00001
Infectious
0.000001
0.0000001
0.00000001
Days of epidemic
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”.