1

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

2

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

3

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

4

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

5

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

6

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

7

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

8

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

9

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

10

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)

11

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

**

**

11

11

1

1

bcbc

bcbc and

1

c ,

1

bSSSM

MSMMt

SSt

SM

tMS

tMM

kkk pva

pva

-v

b

b

νν

νν

12

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.

13

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.

14

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

15

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)

16

17

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)

18

19

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?

20

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

21

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.

22

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.

23

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

24

Whole households Equalising strategy

n = 3 n = 4

25

Collaborator

Ball F, Becker NG (2005). Control of transmission with two types of infection. Submitted to Mathematical Biosciences.

The EndThe End