Transcript Document

Preparedness for an Emerging Infection
Niels G Becker
National Centre for Epidemiology and Population Health
Australian National University
This presentation outlines the modelling and results in:
Becker NG, Glass K, Li Z, Aldis G (2005). Controlling emerging infectious
diseases like SARS. Mathematical Biosciences 193, 205-221
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We have no vaccine to control an emerging infection, like SARS.
It is necessary to resort to basic control measures such as
• Isolating cases following diagnosis
• Quarantining households with cases
• Quarantining traced contacts of cases
• Providing advice on how to reduce exposure
• Closing schools
How well do such measures work?
How can models be used to assess the measures?
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Structure of the model
A household structure is needed because we want to assess
the effect of quarantining affected households.
Two types of individual are needed because we want to assess
the effect of closing schools:
Type 1 = school attendees, and Type 2 = others
The effect of intervention is assessed by their reduction of R.
Reduce the number of types of infective that we need to keep
track of by attributing infections as follows:
Attribute to an infective A the individuals she infects in other
households AND all infections that arise in those household
outbreaks.
Do not attribute infections to A any infectives she infects
within her own household.
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Figure 1: How we’re attributing infections.
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(single type of infectiv
=
mean number of between household infections
ν=
mean number of within household infections
By attributing infections in this way the household structure
does not add new types of infective.
Our two types are:
Type 1 = school attendees, and
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The mean matrix is
1  M 11

2  M 21
Type 2 = others
2
M 12 

M 22 
The effective reproduction number is
R 
5

1
M 11  M 22  (M 11  M 22 )2  4M 12 M 21
2

For specifying details of the model it is useful to separate out
the between and within household transmission components.
Between-household transmission
ij =
mean number type-j individuals infected by an
infective of type i.
Within-household transmission
vij =
mean number type-j cases in a household outbreak
arising when a randomly selected type-i individual is
infected.
Note that M11= 11ν11 + 12ν21 , M12= 11ν12 + 12ν22
Other terms similarly.
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Therefore
 M 11

 M 21
M 12   11
  
M 22   21
12  11  12 


22  21  22 
Between-household transmission
fS = fraction of community-time spent at school
i = proportion of individuals of Type i in the community
 reflects enhanced mixing at school
 11

 21
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12 
  1  2 
  0 
  f S 

  (1  f S )
22 
 0 
  1  2 
Within-household transmission
Assume uniform mixing within households.
This gives, for example,
 12 

 n 
: ( ) 1
m h
n1 ( )n 2 ( )
n1 ( )  n 2 ( )  1 E(N 1 )
where
 = (n1,n2) is the type of household
m = mean outbreak size in a household of type  , with 1
primary infective
h = proportion of households of type 
Use Reed-Frost transmission chain model
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Aside
Reed-Frost transmission chain model
To illustrate, consider a household of size 4.
θ = probability of avoiding infection by one infective
The probability distribution for the chains of transmission is
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Chain
Size
Probability
10
1
θ3
110
2
3θ2(1-θ).θ2
1110
3
3θ2(1-θ).2θ(1-θ). θ
112
4
3θ2(1-θ).(1-θ)2
1111
4
3θ2(1-θ).2θ(1-θ).(1-θ)
120
3
3θ (1-θ)2.θ2
121
4
3θ(1-θ)2.(1-θ2)
13
4
(1-θ)3
R-F assumption
The probability
of avoiding
infection when
exposed to two
infectives is θ2
From this we find the probability distribution for the
outbreak size to be
Size
Probability
1
θ3
2
3θ4(1-θ)
3
3θ3(1-θ)2.(1+2θ)
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balance of probability
What is ν?
Similarly for other household sizes.
(End of Aside)
Now let’s assign parameter values.
Begin with the demographic quantities which we
determine from Australian census data.
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Proportion of household type (h x 1000) for Australia
Household size (n)
Number of school children
(n1)
0
2
3
4
5
236
2
317 18
3
110 42 14
0
4
51 43 61
5
0
5
13 15 22 24
1
0
6
0
1 = 0.18
Var(N1) = 0.75
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2 = 0.82
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h(0,1) = 0.0236
1
6
11
1
0
4
For example:
0
7
7
h(2,2) = 0.0061
0
E(N1) = 0.46
Var(N2) = 0.96
E(N2) = 2.12
Cov(N1,N2) = 0.05
Assigning other parameter values for a plausible scenario
RH0 = 6
θ = 0.8
 = 2.63
fS = 0.4
=3
θ = 0.2
 = 1.38
(from school hours in Australia)
(from forces of infection estimated for measles)
Next we model the effect of the interventions.
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The effect of promoting reduced exposure
Exposure is reduced by
• wearing a mask
• making fewer close contacts
• reducing hand-to-mouth contacts
• washing hands more regularly
This makes individuals less susceptible and less infectious
Model this by:
Between households: * = S I  (= 2  for simplicity)
Within households:
log(θ*) = S I log(θ) (= 2 log(θ) for simplicity)
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The effect of isolating each case at diagnosis
This reduces the infectious period
Mathematically this is the same as reduction infectivity
That is,
Between households: * =  
Within households:
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log(θ*) =  log(θ) or θ* = θ
Figure 2a
(low infection rate within households, θ = 0.8)
Figure 2b
(high infection rate within households, θ = 0.2)
Effective reproduction number for different levels of control  :
 reducing susceptibility by a factor 
 reducing infectivity by a factor 
 reducing infectious period by a proportion 
Compare graphs corresponding to the following controls:
 Lower exposure outside the household only
 Lower exposure within and outside of the household
 Isolating cases when diagnosed
 Lower exposure and isolation
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(all of the above)
Figure 2a (θ = 0.8)
Reduced suscy & infty: community and household
Reduced suscy & infty: community only
Isolating cases
All controls
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Figure 2b (θ = 0.2)
Reduced suscy & infty: community and household
Reduced suscy & infty: community only
Isolating cases
All controls
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The effect of closing schools
Recall that
 11

 21
12 
  1  2 
  0 
  f S 

  (1  f S )
22 
 0 
  1  2 
Relatively more transmission occurs at school when  > 1
We assess the effect of closing schools by comparing R for the
case of fS = 0 relative to fS = 0.4
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Figure 2c
(low infection rate within households, θ = 0.8)
Figure 2d
(high infection rate within households, θ = 0.2)
The effect of closing schools
Effective reproduction number for different levels of control  :
 reducing susceptibility by a factor 
 reducing infectivity by a factor 
 reducing infectious period by a proportion 
Compare graphs corresponding to the following controls:
 Lower exposure outside the household only
 Lower exposure within and outside of the household
 Isolating cases when diagnosed
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 Lower exposure and isolation
Figure 2c (θ = 0.8)
Reduced suscy & infty: community and household
Reduced suscy & infty: community only
Isolating cases
All controls
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Figure 2d (θ=0.2)
Reduced suscy & infty: community and household
Reduced suscy & infty: community only
Isolating cases
All controls
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The effect of quarantining affected households
Quarantining is a more stringent control
• It isolates potentially infected individuals, without waiting for their
diagnosis
• Usually some quarantined individuals are not infected
When the time from infection until isolation is larger than twice
the latent period, then quarantining households can prevent
secondary household cases from infecting individuals from
other households.
Primary case latent
Primary case
infected
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If primary case
isolated here
Earliest time a
secondary case
can be infected
If primary case
isolated here
The effect of quarantining traced contacts
Household members are considered to be traced contacts.
As well, a fraction of contacts outside the community is identified
and quarantined.
Calculations require assumptions about the duration from
infection until (i) infectious, (ii) diagnosed and (iii) recovered.
Calculating the exact reduction in R is much more difficult.
We make a simplifying assumption.
For traced primary cases who were infected assume that they
were infected as soon as their “source” became infectious.
This leads to a conservative estimate of the reduction in R
achieved.
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Choice of parameters
(motivated by SARS)
• The incubation period is 6.5 days
• The latent period is 6.5 days.
• The infectious period is 9 days.
Figure 3a
(low infection rate within households, θ = 0.8)
Figure 3b
(high infection rate within households, θ = 0.2)
Compare graphs corresponding to the following control measures:
• Isolating cases only (for reference)
• Quarantining entire household at first diagnosis
• Quarantining household and 50% of other contacts
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Black arrows indicate where R = 1
Figure 3a (θ=0.8)
Quarantining households
Isolating cases only
Quarantining
households and 50%
of traced contacts
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Figure 3b (θ=0.2)
Isolating cases only
Quarantining
households
Quarantining
households and 50%
of traced contacts
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Future work:
Application of these ideas to help preparedness for pandemic influenza
• Contribution of antivirals towards containment of an outbreak initiated
by one imported case
• How much are antivirals likely to delay the start of a major outbreak?
• Use of antivirals in maintaining the health care service / other essential
services.
• How many courses of antivirals are needed?
• How do we estimate the efficacy of antivirals?
The End
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