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Control of Dose Dependent Infection
Niels Becker National Centre for Epidemiology and Population Health Australian National University
Frank Ball Mathematical Sciences, University of Nottingham
1. How effective a vaccination strategy is depends on how we distribute vaccines across households.
2. The ‘size’ of the exposure is likely to be higher within households.3. Severity of illness for measles, varicella, etc, seems to depend on
the size of exposure• primary households cases tend to be less ill than subsequent
cases
How does dose-dependent infection alter the performance of vaccination strategies?
pose the question
The observations
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We will assess vaccination strategies by their effect on Rv,
the reproduction number of infectives.
Motivation for this:1. Epidemics are prevented when the vaccination
coverage v is such that Rv < 1.
2. Incidence is generally less when R is smaller
Here we dichotomise severity of illness:Suppose there are two types of infection, namely mild (M) and severe (S).
Then we have a ‘next generation matrix’, or mean matrix.Rv is the largest eigenvalue of this matrix
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Consider a branching process approximation for the population of infectives during the early stages
Which branching process?
Trick: Attribute to an infective all cases directly infected in other households AND all cases arising in those households
This means we only have two types of infectives (M and S)
Then RV is largest eigenvalue of the corresponding 2 x 2 mean matrix
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A
Mild
infective
Not infected
2 mild and 3 severe cases are attributed to
infective A
In fact, direct contacts with A resulted in 2 mild
and 2 severe infections
Severe
infective
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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.
The mean matrix is
SSSSMSSMSMSSMMSM
SSMSMSMMSMMSMMMM
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where
the LHS matrix contains means for number infected between households, and
the RHS matrix contains means for number infected within households
Next we need to determine how the matrix elements change when part of the community is vaccinated.
For this we use two alternative models for vaccine response.
SSSM
MSMM
SSSM
MSMM
This mean matrix can be written as the matrix product
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0
2
4
6
8
0 5 10 15 20
Preamble for Vaccine Response Model 1Infectivity function
x = infectiousness function indicates how infectious an unvaccinated individual is x time units after being infected.
X
days
BU
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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.
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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Unvaccinated
Vaccinated
Prob of infection t dt a t dt
Mean number infected
BU b BU
Vaccine Response Model 1
Random vaccine response (A,B ). Realisation (a,b )
Bu is the area under the infectiousness function of an unvaccinated infective
What does this mean with respect to between and within household transmission?
Pr(A =ai, B =bi) = pi i = 1,2, … , k
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Summary measures for vaccine response 1
1. Define VES = 1 E(A)
2. Define VEI = 1 E(AB) / E(A)
When Pr(A=a)=1, then VEI =1 E(B)
3. Define VEIS = 1 E(AB)
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The vaccinated community has 2k+1 types of infective,namely ‘unvaccinated’, k Mild and k Severe corresponding to different vaccine responses
Reduce the size of the mean matrix to by taking one vaccine response to be complete immunity
FURTHER the proportional effect of vaccination on the elements of the matrix simplifies things.
Specifically, with partial vaccination the mean matrix becomes
**
**
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11
1
1
bcbc
bcbc and
1
c ,
1
bSSSM
MSMMt
SSt
SM
tMS
tMM
kkk pva
pva
-v
b
b
νν
νν
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Some matrix algebra allows us to deduce that the non-zero eigenvalues are equal those of the 2 x 2 matrix
bcbc
bcbc**
**
SSt
SMt
MSt
MMt
SSSM
MSMM
νν
νν
Simplify transmission within households:
Assume everyone gets infected, with high dose, once the infection enters the household.
Then the mean matrix becomes
D
CDC
SSSM
MSMM
0
where C = 1-v+vE(AB) and D = v Cov(A,B) +f/E(N),
and only f = E{[N -V+V E(A)][N -V+V E(B)]
depends on the vaccination strategy.
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Strategies
1. Vaccinate all members of randomly selected
households (Rhh,v)
2. Vaccinate individuals at random (Rind,v)
3. Vaccinate the same fraction of members in every
household (Rfrac,v)
Result:
Rhh,v ≥ Rind,v ≥ Rfrac,v for fixed v.
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Optimal strategy
To minimise Rv make Var(N V) as small as possible
Choose the largest nk and vaccinate one individual
in each such household, and continue until coverage v
is reached.
)]1(E)1(E/[)1(E)1(E2let and
1)(E and 1)(E Suppose
BABA
BA
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Example 1
Tecumseh household distribution
Pr(N=1) = 133/567, Pr(N=2) = 189/567, Pr(N=3) = 108/567,
Pr(N=4) = 106/567 and Pr(N=5) = 31/567
MM = 0.25, MS = 0.1, SM = 0.25, SS = 0.5 Complete/none (CN) protective vaccine response with
Pr(A=0) = 0.8 and Pr(A=1) = 0.2 (VES =0.8)
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Example 2
Same, namely
Tecumseh household distribution
MM = 0.25, MS = 0.1, SM = 0.25, SS = 0.5
EXCEPT
Partial/uniform (PU) protective vaccine response with,
Pr(A=0.2, B=1) = 1 (VES =0.8)
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The strategies of vaccinating
i. whole households at random,
ii. individuals at random,
iii. the same fraction of members in every
household
are equally effective.
Why is it so?
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Explanation (of the equal effectiveness of the 3 strategies):
i. It is assumed that every household member of an affected household becomes infected, so vaccination offers protection only by preventing the initially contacted individual in a household being infected, AND
ii. E(Nv) = E(N) for all three strategies
where
N = size of the household of an individual chosen at random from the population
Nv = size of the household of an individual chosen at random from the vaccinated individuals of the population
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Explanation
(of why the critical vaccination strategy for a PU vaccine response is
larger than that for a CN response with the same VE):
With a PU response, each time a vaccinee is exposed to a contact, s/he becomes infected independently with probability 1 ε
With a CN response, the first time a vaccinee is exposed to a contact, s/he becomes infected with probability 1 ε, but if they avoid infection at their first exposure then they necessarily avoid infection at all subsequent exposures.
Thus for fixed ε, the CN response results in greater reduction in the transmission of disease.
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Vaccine Response Model 2
Unvaccinated person exposed to low-dose: mild
infection
Unvaccinated person exposed to high-dose: severe
infection
Vaccinee exposed to low-dose: no infection
Vaccinee exposed to high-dose: mild infection
Assume equal household size
Optimal strategy can be either the equalising strategy
or vaccinating whole households, depending on model
parameters.
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When det[ij] ≠ 0 the optimal strategy may also
depend on the distribution of the household size
Results still hold when there are vaccine failures
Example 3
Equal household size (a) n=3, (b) n=4.
MM = 1.3, MS = 0, SM = 0, SS = 0.3
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Whole households Equalising strategy
n = 3 n = 4
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Collaborator
Ball F, Becker NG (2005). Control of transmission with two types of infection. Submitted to Mathematical Biosciences.
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