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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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BackgroundHow 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.

Classical measure of vaccine efficacy

UU

VV//1 VEnCnC

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

1f

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 estimates 1f,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 estimates

UU

VV//1 VEnCnC

)exp(1)exp(11

T

TaVE

5

0 5 100.0

0.1

0.2

0.3

0.4

0.5

Lambda_T

V EVE as a function of duration of exposure (a=0.5)

)exp(1)exp(11

T

TaVE

The graph of is

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 studyUU

VV//1 VEnCnC

gives a value that depends on • 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

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.

UU

VV//1 VEnCnC

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Let us begin by proposing suitable measures for the vaccine effects.

Reduction in susceptibilitySuppose 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 protectionA=1 implies no vaccine effectPr(A=1) = f, Pr(A=0) = 1- f implies complete protection apart from a fraction f of failures.

11

02468

0 5 10 15 20

Reduction in infectivityx = infectiousness function indicates how infectious an unvaccinated individual is x time units after being infected.

X

days

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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 OUTBREAKSuppose that every individual has the same exposure to infectious individuals. Then we can show

1 - cV/cU VES 1- log(1cV) / log(1cU) irrespective of the distribution of A

cU = proportion of unvaccinated participants who become casescV = 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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((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 ?

1 - cV/cU VES 1 - log(1cV) / log(1cU)

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Unvaccinated Vaccinated

Cases 96 8 Not infected 174 57 270 65

ApplicationOutbreak of mumps in school of Ashtabula County, Ontario

The estimatesCU/nU = 96/270 = .356, CV/nV = 8/65 = .123give 0.654 VES 0.702

(.118) (.111)These bounds are of practical value!

1cV/cU VES = 1 E(A) 1log(1cV)/log(1cU)

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Can we estimate VEI and VEIS?OUTBREAKS in HOUSEHOLD PAIRSP = probability of an unvaccinated susceptible individual being infected from outside the pair = 1 - Qp = probability of a susceptible individual being infected by an infected partner = 1 - qTo illustrate, assume uniform (a,b) response and hshlds size 2, with 1 vaccinated and 1 unvaccinated member.

0 cases U case V case 2 cases Q1+a Q(1-Qa)qb PQaqa balance n00 n10 n01 n11

Likelihood inference for a and b is straightforward.

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Illustration with reference to smallpox dataEpidemic of variola minor in Braganca Paulista County (Brazil), 1956A total of 338 householdsHousehold sizes from 1 to 12 (mean 4.6)809 vaccinated and 733 unvaccinated 85 and 425 were infectedVaccine response modelThree vaccine responses

c1 – f – c fProbability

(0,•)(a,b)(1,1)Response (A,B)

B, O’Neill & Britton (2003),

Biometrics.

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Transmission modelUnvaccinated 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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Posterior distribution for c

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Posterior distribution for a

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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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Collaborators1. 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 EndThe End