Age-related changes in the rate of disease transmission: implications for the design of vaccination programmes

Abstract

Mathematical models are developed to aid in the investigation of the implications of heterogeneity in contact with infection within a community, on the design of mass vaccination programmes for the control of childhood viral and bacterial infections in developed countries. Analyses are focused on age-dependency in the rate at which individuals acquire infection, the question of 'who acquires infection from whom', and the implications of genetic variability in susceptibility to infection. Throughout, theoretical predictions are based on parameter estimates obtained from epidemiological studies and are compared with observed temporal trends in disease incidence and age-stratified serological profiles. Analysis of case notification records and serological data suggest that the rate at which individuals acquire many common infections changes from medium to high and then to low levels in the infant, child and teenage plus adult age groups respectively. Such apparent age-dependency in attack rate acts to reduce slightly the predicted levels of herd immunity required for the eradication of infections such as measles, when compared with the predictions of models based on age-independent transmission. The action of maternally derived immunity in prohibiting vaccination in infants, and the broad span of age classes over which vaccination currently takes place in the U.K., however, argue that levels of herd immunity of between 90 and 94% would be required to eliminate measles. Problems surrounding the interpretation of apparent age-related trends in the acquisition of infection and their relevance to the design of vaccination programmes, are discussed in relation to the possible role of genetically based variation in susceptibility to infection and observations on epidemics in 'virgin' populations. Heterogeneous mixing models provide predictions of changes in serology and disease incidence under the impact of mass vaccination which well mirror observed trends in England and Wales.

The epidemiology of varicella–zoster virus infections: a mathematical model

“…The following formulation therefore describes the pattern of viral transmission in a large age-structured community of constant size N. It is compartmental in structure consisting of a set of partial differential equations [19,20], which describes the densities of individuals classified into subgroups. The host population is divided into the following categories: passively immune infants with maternally derived antibodies (M), susceptible individuals (S), incubators who are infected but not yet infectious (E), and infectious individuals (I).…”

“…The relationship between the death rate and age is discussed below. The model described so far is a standard formulation employed in the representation of directly transmitted primary micro-parasitic infections like varicella, which engender permanent subsequent immunity, and has been described and analysed in detail [19,20].…”

“…Because at a population level very few infections occur in the elderly there is often no need in mathematical models of these diseases to be concerned with the representation of processes amongst later age groups [19]. In such models all individuals can be assumed to survive until the mean life expectancy, after which they are removed from the compartments of the model.…”

“…Such a function is particularly useful in any model of varicella-zoster infections, because zoster occurs in a wide range of ages, with a predominance of cases in older people. In the models presented here, the Gompertz survival function suggested by Anderson and May [19] is used. This corresponds to an exponential increase in the death rate:…”

“…For the purpose of modelling, this force of infection is calculated for n discrete age groups and fl(a', a) is described by a matrix Ilij of transmission coefficients. This is the 'Who Acquires Infection From Whom' (WAIFW) matrix, which has been extensively described [19]. In the calculation of the n values for this matrix we use a form of the matrix: which describes a peak in transmission amongst children of school age.…”