Control of Dose Dependent Infection

1. Niels Becker National Centre for Epidemiology and Population Health Australian National University Frank Ball Mathematical Sciences, University of Nottingham The observations • 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 pose the question How does dose-dependent infection alter the performance of vaccination strategies? Control of Dose Dependent Infection

2. We will assess vaccination strategies by their effect on Rv, the reproduction number of infectives. Motivation for this: • 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 stagesWhich branching process?Trick:Attribute to an infective all cases directly infected in other households AND all cases arising in those householdsThis means we only have two types of infectives (M and S) ThenRVis largest eigenvalue of the corresponding 2 x 2 mean matrix

4. A Mild infective Severe 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

5. Between-household transmissionij= mean number type-j individuals infected by an infective of type i.Within-household transmissionvij= mean number type-j cases in a household outbreak arising when a randomly selected type-i individual is infected.The mean matrix is

6. This mean matrix can be written as the matrix product where the LHS matrix contains means for number infected between households, andthe RHS matrix contains means for number infected within householdsNext 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.

7. Preamble for Vaccine Response Model 1 Infectivity function x = infectiousness function indicates how infectious an unvaccinated individual is x time units after being infected. X BU days

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 potentialfor 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. Vaccine Response Model 1Random vaccine response (A,B ). Realisation (a,b ) Buis the area under the infectiousness function of an unvaccinated infectiveWhat does this mean with respect to between and within household transmission?Pr(A =ai, B =bi) = pii =1,2, … , k

10. Summary measures for vaccine response 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 responsesReduce 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

12. Some matrix algebra allows us to deduce that the non-zero eigenvalues are equal those of the 2 x 2 matrix Simplify transmission within households: Assume everyone gets infected, with high dose, once the infection enters the household.Then the mean matrix becomes 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. Strategies1. 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 possibleChoose the largest nk and vaccinate one individual in each such household, and continue until coverage v is reached.

15. Example 1Tecumseh household distributionPr(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 = 0.5Complete/none (CN) protective vaccine response withPr(A=0) = 0.8 and Pr(A=1) = 0.2 (VES =0.8)

17. Example 2Same, namelyTecumseh household distributionMM = 0.25, MS = 0.1, SM = 0.25, SS = 0.5EXCEPTPartial/uniform (PU) protective vaccine response with, Pr(A=0.2, B=1) = 1 (VES =0.8)

19. The strategies of vaccinating • whole households at random, • individuals at random, • 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): • It is assumed that every household member of an affected household becomes infected, so vaccination offers protection only bypreventing the initially contacted individual in a household being infected, AND • 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 randomfrom 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 2Unvaccinated person exposed to low-dose: mild infection Unvaccinated person exposed to high-dose: severe infection Vaccinee exposed to low-dose: no infectionVaccinee exposed to high-dose: mild infectionAssume equal household sizeOptimal 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 sizeResults still hold when there are vaccine failuresExample 3Equal 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 End