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Immunisation with a Partially Effective Vaccine
Niels G Becker
National Centre for Epidemiology and Population Health
Australian National University
Ideally vaccines immunise individuals completely. In practice
some vaccinees are infected, making it very important to
determine the efficacy of a vaccine as well as what happens
when a partially effective vaccine is used in mass
vaccination.
This lectures looks at this issue, in part to demonstrate that
infectious disease models also play an important role in the
analysis of infectious disease data.
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Background
How is the efficacy of a vaccine usually measured?
nV vaccinated individuals are observed over interval [0,T].
Of these CV become cases.
nU unvaccinated individuals are observed over interval [0,T].
Of these CU become cases.
C V /nV
VE  1 
Classical measure of vaccine efficacy
C U / nU
This sort of measure is sensible for assessing interventions in
chronic diseases, but less so for infectious diseases.
It does not account for transmission, and so can seriously
mislead us.
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To illustrate this weakness consider two types of vaccine effects
on susceptibility (protection).
1. “Complete/none” vaccine effect
The vaccine gives complete protection with probability 1f
and gives
no
protection with probability f.
2. “Partial/uniform” vaccine effect
The vaccine gives the same partial protection to every vaccinee.
The probability of a contact close enough to infect an
unvaccinated individual has only probability a,
where 0 < a < 1, of infecting a vaccinated person.
These two vaccine effects were first considered by
Smith, Rodrigues and Fine (1984).
Int. J. Epidemiol. 13, 87-96.
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Example 1
(a)
Consider a community of homogeneous individuals who
mix uniformly.
For a vaccine with a “complete/none” effect it can be shown
that
VE  1

C V /nV
C U / nU
estimates 1f,
which is a sensible estimate of vaccine efficacy for this vaccine
effect.
(b)
Consider now a vaccine with a “partial/uniform” effect.
When the infection intensity acting on individuals is t for
unvaccinated individuals and a.t for a vaccinated ones, a
sensible measure of vaccine efficacy is 1  a.
Then VE estimates VE
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
1  exp( aT )
1 
1  exp(  T )
VE
The graph of

1 
1  exp( aT )
1  exp(  T )
is
VE as a function of duration of exposure (a=0.5)
0.5
VE
0.4
0.3
0.2
0.1
0.0
0
5
Lambda_T
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Therefore the value of VE depends on
• the size of the ‘epidemic’ over the observation period
• the duration of the observation period
• the vaccination coverage
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Example 2
Suppose
(a) the study consists of households of size two, with 1
vaccinated and 1 unvaccinated in each household;
(b) the vaccine reduces susceptibility a little, but reduces
the infectivity of the vaccinee a lot.
Then, for a large study
VE  1

C V /nV
C U / nU
gives a value that depends on
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•
the magnitude of the reduction in infectivity (1-b), and
•
the secondary attack rate (SAR).
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Heuristic explanation
vaccinated
unvaccinated
Can we formulate a measure of vaccine efficacy that
accounts for transmission?
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The trouble with
VE  1

C V /nV
C U / nU
is that it estimates a different quantity in different settings.
What we need is a measure (a parameter) with the same
interpretation in every setting.
The cost is that we then need to find an appropriate estimate
of this parameter for different vaccine trials and different
observational settings.
In fact we need more than one measure, because we are
interested in the reduction in susceptibility, the reduction in
infectivity and the reduction in community transmission.
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Let us begin by proposing suitable measures for the vaccine
effects.
Reduction in susceptibility
Suppose that vaccination reduces an individual's probability of
disease transmission, per close contact, by a random factor A.
That is, a force of infection (t) acting on an unvaccinated
individual at time t reduces to A.(t) for a vaccinated individual.
A=0 implies complete protection
A=1 implies no vaccine effect
Pr(A=1) = f, Pr(A=0) = 1- f implies complete protection apart
from a fraction f of failures.
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Reduction in infectivity
x = infectiousness function indicates how infectious an
unvaccinated individual is x time units after being infected.
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X
4
2
0
0
11
5
10
days
15
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The effect of the vaccine on infectiousness, in the event that
a vaccinee is infected, might be a shorter duration of illness,
shorter infectious period, a lower rate of shedding pathogen,
etc. than they would have if not vaccinated.
From public health point of view there is interest in the
reduction in transmission.
The potential for an infective to infect others is the area
under x
• BU when infective unvaccinated
• BV when the infective is vaccinated.
Relative infection potential B =BV/BU is random
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We call (A,B) the vaccine response (or vaccine effect)
We allow A and B have any probability distribution
(Expect them to lie between 0 and 1, and be correlated)
Summary measures of vaccine effects
1. Define
VES = 1  E(A)
(protective vaccine efficacy)
• For the “complete/none” response
Pr(A = 1) = f ,
Pr(A = 0) = 1  f
giving VES = 1 - f, which is sensible.
• For the “partial/uniform” response
Pr(A = a) = 1
giving VES = 1 - a, which is also sensible.
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2. Define
VEI = 1  E(AB) / E(A)
• When Pr(A=a)=1, then VEI =1-E(B)
• E(A) = 0 when Pr(A=0), but this is not of concern as
no vaccinee becomes infected
• VEI is more difficult to estimate than VES
3. Define
VEIS = 1  E(AB)
To demonstrate that this is an important parameter,
consider a community of homogeneous individuals who
mix uniformly. Vaccinate a fraction v of them.
Then
RV = [1  v + v.E(AB)] R0
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Consider first VES= 1 - E(A)
A SINGLE OUTBREAK
Suppose that every individual has the same exposure to
infectious individuals. Then we can show
1 - cV/cU 
VES
 1- log(1cV) / log(1cU)
irrespective of the distribution of A
cU = proportion of unvaccinated participants who become cases
cV = proportion of vaccinated participants who become cases
The bounds are parameters, written in a way that makes
estimates obvious.
[ B & Utev (2002) Biometrical Journal, 44, 29-42.]
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1 - cV/cU 
VES  1 - log(1cV) / log(1cU)
(i) LH bound is attained for ‘complete/no protection’ response.
(ii) RH bound is attained for ‘partial and uniform protection’
response.
(iii) The above two responses are extremes in the sense that,
with a common mean E(A) , the CN response has the most
variation for A and the PU response has the least variation.
(iv) Estimate cU by CU/nU and cV by CV/nV.
(v) We have standard errors.
(vi) VES is not identifiable when only CU and CV are observed.
(vii) These bounds are useful when cU and cV are small ?
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1cV/cU  VES = 1 E(A)  1log(1cV)/log(1cU)
Application
Outbreak of mumps in school of Ashtabula County, Ontario
Unvaccinated Vaccinated
Cases
96
8
Not
infected
174
57
270
65
The estimates
CU/nU = 96/270 = .356,
give
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CV/nV = 8/65 = .123
0.654  VES  0.702
(.118)
(.111)
These bounds are of practical value!
Can we estimate VEI and VEIS?
OUTBREAKS in HOUSEHOLD PAIRS
P = probability of an unvaccinated susceptible individual being
infected from outside the pair = 1 - Q
p = probability of a susceptible individual being
infected by an infected partner = 1 - q
To illustrate, assume uniform (a,b) response and hshlds size 2,
with 1 vaccinated and 1 unvaccinated member.
0 cases
Q1+a
n00
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U case
Q(1-Qa)qb
n10
V case
PQaqa
n01
2 cases
balance
n11
Likelihood inference for a and b is straightforward.
Illustration with reference to smallpox data
Epidemic of variola minor in Braganca Paulista County
(Brazil), 1956
A total of 338 households
Household sizes from 1 to 12 (mean 4.6)
809 vaccinated and 733 unvaccinated
85
and
B, O’Neill &
Britton (2003),
Biometrics.
425 were infected
Vaccine response model
Three vaccine responses
Response (A,B)
Probability
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(1,1)
(a,b)
(0,•)
f
1–f–c
c
Transmission model
Unvaccinated are homogeneous.
3 types of vaccine responses, BUT immunes can be ignored;
and vaccine failures are like the unvaccinated, so we need to
deal with 2 types only.
Numerical computation of
u,v(i,j) = Pr [ (i,j) out of (u,v) are infected ]
is manageable.
Numerical computation of the likelihood function is
manageable.
Inferences:
Bayesian inferences via MCMC methods
Uniform (0,1) priors for a, b, c, f, q, Q.
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a = partial reduction in susceptibility
c = Pr(complete protection) = Pr(A=0)
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The data are not
compatible with
small c and large a
No points here
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If c is large, then the data are
compatible with most values of a
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If a is small, then the
data are compatible
with most values of c
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It is difficult to distinguish between
(a  0 c  0) and (a  1 c  .8)
(both indicate low susceptibility)
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Conclusions
• Precise estimation of VES =
1  E(A) is possible.
• Estimation of VEIS = 1  E(AB) is surprisingly good.
• Estimation of the reduction in infectivity, per se, was not
particularly effective. It may be more effective from data on
smaller households.
Further work
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Collaborators
1.
Becker NG, Starczak DN (1998). The effect of random vaccine response on
the vaccination coverage required to prevent epidemics. Mathematical
Biosciences 154, 117-135.
2.
Becker NG, Utev S (2002). Protective vaccine efficacy when vaccine response
is random. Biometrical Journal 44, 29-42.
3.
Becker NG, Britton T, O’Neill PD (2003). Estimating vaccine effects on
transmission of infection from household data. Biometrics 59, 467-475.
4.
Becker NG, Britton T (2004). Estimating vaccine efficacy from small outbreaks.
Biometrika 91, 363-382.
5.
Becker NG, Lefevre C, Utev S (2005). Estimating protective vaccine efficacy
from large trials with recruitment. Journal of Statistical Planning and Inference
(in press).
6.
Becker NG, Britton T, O’Neill PD (2005). Estimating vaccine effects from
studies of outbreaks in household pairs. Statistics in Medicine (in press).
The End
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