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Contact Patterns and Their Implied Basic Reproductive Numbers: an Illustration for Varicella-Zoster Virus

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Contact Patterns and Their Implied Basic Reproductive Numbers: an Illustration for Varicella-Zoster Virus Epidemiol. Infect. (2009), 137, 48–57. f 2008 Cambridge University Press doi:10.1017/S0950268808000563 Printed in the United Kingdom Contact patterns and their implied basic reproductive numbers: an illustration for varicella-zoster virus T. VAN EFFELTERRE1*, Z. SHKEDY1,M.AERTS1, G. MOLENBERGHS1, 2 2,3 P. VAN DAMME AND P. BEUTELS 1 Hasselt University, Center for Statistics, Biostatistics, Diepenbeek, Belgium 2 University of Antwerp, Epidemiology and Social Medicine, Center for Evaluation of Vaccination, Antwerp, Belgium 3 National Centre for Immunization Research and Surveillance, University of Sydney, Australia (Accepted 25 February 2008; first published online 9 May 2008) SUMMARY The WAIFW matrix (Who Acquires Infection From Whom) is a central parameter in modelling the spread of infectious diseases. The calculation of the basic reproductive number (R0) depends on the assumptions made about the transmission within and between age groups through the structure of the WAIFW matrix and different structures might lead to different estimates for R0 and hence different estimates for the minimal immunization coverage needed for the elimination of the infection in the population. In this paper, we estimate R0 for varicella in Belgium. The force of infection is estimated from seroprevalence data using fractional polynomials and we show how the estimate of R0 is heavily influenced by the structure of the WAIFW matrix. INTRODUCTION and social factors affecting the transmission rate be- tween an infective of age ak and a susceptible of age a An essential assumption in modelling the spread of [1, 2]. For the discrete case with a population divided infectious diseases is that the force of infection, which into a finite number, say n, of age groups, Anderson & is the probability for a susceptible to acquire the May [3] introduced the WAIFW (Who Acquires infection, varies over time as a function of the level of Infection From Whom) matrix in which the ij th entry infectivity in the population [1]. For many infectious of the matrix, b , is the transmission coefficient from diseases, the force of infection is also known to de- ij an infective in age group j to a susceptible in age pend on age. The equation describing the dependence group i. Let I be the total number of infectious of the force of infection on age and time is given by i individuals in the ith age group at time t, i=1, …, n, ð L then the age- and time-dependent force of infection 0 0 0 l(a, t)= b(a, a )I(a, t)da: (1) can be approximated by the matrix product 0 The coefficients b(a, ak) are called the transmission l=WI: (2) coefficients and I(ak, t) is the number of infectious Here, I=(I1,…,In) is the vector in which the ith el- individuals at age ak and time t. These transmission ement is the number of infectious individuals (preva- coefficients combine epidemiological, environmental lence of infectivity) in age group i, l=(l1,…, ln)is the vector in which the ith element is the force of * Author for correspondence: Dr T. Van Effelterre, Hasselt infection specific to age group i and W is a known University, Center for Statistics, Biostatistics, Agoralaan 1, B3590 WAIFW matrix. The configuration of the WAIFW Diepenbeek, Belgium. (Email: tvaneff@yahoo.com) matrix represents a priori knowledge (or assumptions) Downloaded from https://www.cambridge.org/core. IP address: 170.106.34.90, on 02 Oct 2021 at 00:55:42, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0950268808000563 R0 and the WAIFW matrix 49 about the mixing patterns in the population. Several The basic reproductive number R0 can be computed configurations are discussed in the literature (see e.g. as the dominant eigenvalue of a matrix for which [1, 2, 4–6]). For example, for a model with five age the ijth entry is the basic reproductive number R0ij , groups the WAIFW matrix W1 in equation (3) re- specific to the transmission from an infective in age presents a mixing pattern for which individuals are group j to a susceptible in age group i. More precisely, mixing only with individuals from their own age R0ij =bi, jDNi where D is the duration of infectiousness group (assortative mixing [5]) with a specific age-de- assumed independent of age and Ni is the size of the pendent transmission coefficient while W2 represents a population in age group i. Therefore, the estimator mixing pattern similar to W1, also accounting for an for R0 depends on the configuration of the WAIFW additional mixing of individuals with individuals of matrix. Farrington et al. [4] showed that different other age groups with a ‘background’ transmission configurations of the WAIFW matrix can lead to quite coefficient: different estimates for R0. For example, Farrington 0 1 0 1 b 0000 b b b b b B 1 C B 1 5 5 5 5 C B 0 b2 000C B b5 b2 b5 b5 b5 C B C B C W1=B 00b 00C, W2=B b b b b b C: (3) @ 3 A @ 5 5 3 5 5 A 000b4 0 b5 b5 b5 b4 b5 0000b5 b5 b5 b5 b5 b5 . Note that both matrices have five unknown para- et al. [4] estimated R0 for mumps to be equal to 25 5, . meters and both are symmetric. For each of these 8 0 and 3 3 for the configuration of W2, W3 and W4, contact structures, the number of parameters is equal respectively: 0 1 0 1 b b b b b b b b b b B 1 1 4 4 5 C B 1 1 3 4 5 C B b1 b2 b4 b4 b5 C B b1 b2 b3 b4 b5 C B C B C W3=B b b b b b C, W4=B b b b b b C: (6) @ 4 4 3 4 5 A @ 3 3 3 4 5 A b4 b4 b4 b4 b5 b4 b4 b4 b4 b5 b5 b5 b5 b5 b5 b5 b5 b5 b5 b5 to the number of age groups. This is a condition to Hence, the uncertainty related to the WAIFW matrix have a solution [3]. Suppose that the population is is coming from two different sources: (1) the uncer- ^ ^ ^ divided into n age groups and let l=(l1,…,ln) be the tainty about the unknown transmission coefficients estimated vector of age-specific force of infection in bi and (2) the uncertainty about the configuration of each age group. If the structure of the WAIFW matrix the WAIFW matrix. Furthermore, Wallinga et al. [6] is known and consists of n unknown parameters, the showed that the basic reproductive number for WAIFW matrix can be estimated using the equality measles ranges between 770.38 when assortative mix- 0 1 0 1 ing pattern is assumed and 1.43 when infant mixing is l^ Y1 B 1 C B C B : C B : C assumed, i.e. infants are assumed to be the source of B C ND B C B : C= WB : C, (4) all infection [6]. @ A L @ A : : In this paper, we present an investigation of the ^ ln Yn estimation of the basic reproductive number, R0, and with N the total population size, D the mean duration pc, the minimal proportion of the population that of infectiousness, L the life-expectancy at birth, and needs to be vaccinated to eliminate the infection, for Xj varicella in Belgium for which, currently, there is no x’jx1 x’j ^ Yj=e xe and ’j= li(aixaix1): (5) vaccination programme. Following the approach of i=1 Greenhalgh & Dietz [5] we show that, depending on Here, aixaix1 is the width of the ith age group. our assumption about the contact patterns, R0 ranges . Hence, as long as the WAIFW matrix has a between 3 12 and 68 57, and pc ranges between 67 9% known configuration with n unknown parameters, the and 98.5%. parameter vector b=(b1,…,bn) is identifiable. Note This paper is organized as follow. In the next section, that we expect that bio0, i=1, 2, … , n. we present six possible configurations for the WAIFW Downloaded from https://www.cambridge.org/core. IP address: 170.106.34.90, on 02 Oct 2021 at 00:55:42, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0950268808000563 50 T. Van Effelterre and others matrix for varicella and discuss the estimation of the coefficient is assumed equal to the transmission coef- age-dependent force of infection from serological data ficients in the oldest age group (b6). Note that the using fractional polynomials. In the ‘Estimation of transmission coefficient in the oldest age group and the transmission coefficients’ section, the WAIFW the ‘background’ transmission coefficient between matrices are estimated using the integrated force of different age groups are both expected to be smaller infection in the relevant age groups. Parametric and than the transmission coefficients in younger age non-parametric bootstrap is used to calculate confi- groups. The second configuration, WV2, assumes that dence intervals for the transmission coefficients. The the main route of transmission for a directly trans- estimation of R0 and pc is discussed in the ‘Estimation mitted viral infection like varicella is in kindergarten of R0 from the WAIFW matrices’ section. children or in the classroom. This is expressed by a unique coefficient b2 for the (presumed high) trans- mission between infectious and susceptible hosts in Estimation of R0 and the WAIFW matrix for varicella based on serological data the range 2–5 years and two other specific coefficients (b1 and b3) for transmission amongst other hosts aged Age-dependent transmission coefficients <12 years.
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