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Open Access

Epidemiological consequences of household-based antiviral prophylaxis for pandemic influenza

Andrew J. Black, Thomas House, M. J. Keeling, J. V. Ross
Published 6 February 2013.DOI: 10.1098/rsif.2012.1019
Andrew J. Black
Stochastic Modelling Group, School of Mathematical Sciences, The University of Adelaide, Adelaide, South Australia 5005, Australia
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Thomas House
Warwick Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
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M. J. Keeling
Warwick Mathematics Institute, University of Warwick, Coventry CV4 7AL, UKSchool of Life Sciences, University of Warwick, Coventry CV4 7AL, UK
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J. V. Ross
Stochastic Modelling Group, School of Mathematical Sciences, The University of Adelaide, Adelaide, South Australia 5005, Australia
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    • Epidemiological consequences of household-based antiviral prophylaxis for pandemic in uenza: Supplementary material

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    Figure 1.

    The basic household model used in this paper. There are three levels to this model: (a) the individual level, (b) the within-household level where there is strong mixing, and (c) the population level in which there is weaker mixing and a distribution of household sizes.

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    Figure 2.

    General behaviour of antivirals. Household basic reproduction number, R*, and doubling time early in the pandemic, Td, as a function of the mean delay to antiviral allocation for the SEEIIR model. The mean delay is taken as the time from the first infectious case to when antivirals take effect. This is composed of notification and further allocation delay. (a,c) SEEIIR with constant and exponential delay. (b,d) SEEIIR with notification (σ = 1). Dashed and solid lines are from the constant and exponential delay models, respectively, for three values of σ, where 1/σ is the average exposed period. (b) The coloured lines show the model results with notification for σ = 1. Black dashed and solid lines are constant and exponential delay models, respectively. The coloured curves end at minimum delay possible for a given value of the notification rate, rn. (c,d) The doubling time, using the same colour scheme. Other parameters: k = 3, βk = 2, α = 1, γ = 1, τ = 0.8 and ρ = 0.8. All rates are given in terms of days–1. (a,c) Blue lines, σ = 50; green lines, σ = 1; red lines, σ = 0.5. (b,d) Blue lines, rn = 10; green lines, rn = 1; red lines, rn = 0.2.

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    Figure 3.

    The reduction in transmission needed to reduce the household reproductive ratio, R* < 1, for different household size distributions. (a–c) Three household size distributions for (a) UK 2001, (b) Indonesia 2003 and (c) Sudan 1990. The values of R* for an uncontrolled epidemic are 2.3, 3.4 and 4.7 for the distributions (a–c), respectively. (d) The minimum value of the antiviral efficacy, τ, and the maximum mean delay to reduce R* = 1 for the three distributions (a–c). The dashed lines are for the model assuming constant delay and solid lines are for the exponential delay (green, UK; blue, Indonesia; red, Sudan). Other parameters: βk = 2, α = 1, γ = 1, σ = 1 and ρ = 0.8. All rates are given in terms of days–1.

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    Figure 4.

    Pandemic influenza parameter estimates. (a(i–vii)) Kernel density plots for the within-household transmission rate, βk, from 10 000 random samples from the posterior distributions given in [24]; (c) 2000 (randomly selected from the 10 000) random samples for the posterior distribution for the infectious and latent period parameters, γ and σ, estimated by fitting to the serial interval data (b); points are coloured according to their likelihood value as per the scale on the right. (d,e) The posteriors for the reduction in transmission, τ, (for both zanamivir (blue) and oseltamivir (green)) and the reduction in susceptibility, ρ.

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    Figure 5.

    Pandemic model results for the exponentially delayed model. (a,b) Kernel density plots for the household reproductive ratio, R*, as a function of the mean delay to allocation for the two types of antivirals oseltamivir and zanamivir. Solid and dashed white curves in (a) and (b) mark the mean and the 95% credible intervals of these distributions, respectively. (c,d) The doubling time, Td, as a function of the mean delay for oseltamivir and zanamivir, respectively. Black lines show the mean, and dashed red and blue lines show the 50% and 95% credible intervals, respectively.

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    Figure 6.

    Pandemic model results for the constant delay model. (a,b) Kernel density plots for the household reproductive ratio, R*, as a function of the mean delay to allocation for the two types of antivirals oseltamivir and zanamivir. Solid and dashed white curves in (a) and (b) mark the mean and the 95% credible intervals of these distributions, respectively. (c,d) The doubling time, Td, as a function of the mean delay for oseltamivir and zanamivir, respectively. Black lines show the mean, and dashed red and blue lines show the 50% and 95% credible intervals, respectively.

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    Figure 7.

    Additional public-health considerations. (a) Trade-offs giving a percentage reduction in the household reproductive number R*. Here we show the percentage reduction in R* for different combinations of antiviral efficacy and mean delay. We assume the exponentially distributed delay model and pandemic influenza mean parameters, ρ = τ (blue line, τ = 0.5; green line, τ = 0.65; red line, τ = 0.8; light blue line, τ = 0.95. (b) Posterior estimates for R* with a delay distribution (c) taken from [29]. Kernel density plots are shown for R* assuming no interventions (black curve) and a distribution for the mean delay with zanamivir (solid blue curve) and oseltamivir (dashed green curve).

Tables

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  • Table 1.

    The transitions and associated rates defining the stochastic SEEIIR model for the within-household dynamics; k is the size of the household. Only the states that change in a given transition are shown, all others remain constant. The parameters τ and ρ control the reduction in transmission and susceptibility when antivirals are administered to all members the household, hence τ = ρ = 0 for the uncontrolled epidemic.

    eventtransitionrate
    internal infection(S, E1) → (S − 1, E1 + 1)βk(1 − τ)(1 − ρ)S(I1 + I2)/(k − 1)
    latent progression(E1, E2) → (E1 − 1, E2 + 1)2σE1
    start shedding(E2, I1) → (E2 − 1, I1 + 1)2σE2
    infection progression(I1, I2) → (I1 − 1, I2 + 1)2γI1
    recoveryI2 → I2 − 12γI2
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6 April 2013
Volume 10, issue 81
Journal of The Royal Society Interface: 10 (81)
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Keywords

A(H1N1)pdm
Bayesian
doubling time
Markov chain
R*
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Epidemiological consequences of household-based antiviral prophylaxis for pandemic influenza
Andrew J. Black, Thomas House, M. J. Keeling, J. V. Ross
J. R. Soc. Interface 2013 10 20121019; DOI: 10.1098/rsif.2012.1019. Published 6 February 2013
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Epidemiological consequences of household-based antiviral prophylaxis for pandemic influenza

Andrew J. Black, Thomas House, M. J. Keeling, J. V. Ross
J. R. Soc. Interface 2013 10 20121019; DOI: 10.1098/rsif.2012.1019. Published 6 February 2013

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