Exploring the relationship between incidence and the average age of infection during seasonal epidemics
Introduction
Age-structured epidemic models have long been used to understand how transmission of infections might vary among different age groups. These models were originally developed to provide insight into how policies such as vaccination affect the age distribution of infection (Griffiths, 1974; Schenzle, 1984). The long-term dynamics of such models are relatively well understood. It has been shown that at equilibrium, the average age of first infection is inversely proportional to the force of infection (λ), and thus the incidence of disease (Anderson and May, 1983, Dietz, 1975). This is consistent with observations that the mean age of infection with a number of childhood diseases was lower in urban compared to rural areas (Fales, 1928), and that the average age of pediatric patients with mild and severe clinical malaria decreased with increasing transmission intensity in sub-Saharan Africa (Snow and Marsh, 1998). Interventions that reduce the incidence of infection, such as vaccination, are expected to lead to an increase in the average age of infection (Griffiths, 1973). This is a trend that has been observed for poliomyelitis in association with improved hygiene in the US (Anderson and May, 1991) and for post-vaccination measles, rubella, and pertussis (Anderson and May, 1991; Broutin et al., 2005).
However, many infections do not reach an endemic equilibrium, but rather exhibit regular epidemic cycles due to the interaction between susceptible and infectious hosts. In the absence of stochastic effects or seasonal forcing, the basic SIR (susceptible–infectious–recovered) family of models predict that these oscillations will be damped until a stable endemic equilibrium is reached (Anderson and May, 1991). Seasonal variations in climatic factors, immune system function, and/or population contact rates help sustain these oscillations (Dowell, 2001; Grassly and Fraser, 2006). These seasonal variations in epidemic parameters generally occur with a period of 1 year, but the period of the long-term oscillations in disease incidence due to the dynamical interaction of susceptible and infected hosts may be annual or multi-annual. If the natural period of oscillations is an approximate linear multiple of the 1-year seasonal forcing, then resonance can lead to large-amplitude fluctuations in the number of infections at the underlying sub-harmonic frequency (Dietz, 1976). Certain aspects of the short-term dynamics have also been studied extensively, such as the effect of age-structure on the periodicity of oscillations in incidence and the tendency of such models to exhibit chaos (Bolker and Grenfell, 1993; Earn et al., 2000, Earn et al., 1998; Olsen and Schaffer, 1990; Rohani et al., 1999). Relatively little attention, however, has been paid to short-term seasonal changes in the average age of infection predicted by such models.
Hitherto, most researchers have ignored seasonal changes in the age distribution of infection, or have assumed that such changes reflected seasonal differences in mixing patterns, primarily due to school terms (Bolker and Grenfell, 1993; Grenfell and Anderson, 1985). While the presence of seasonal or regular epidemics has been shown to have little impact on the estimation of certain parameters important in determining the long-term dynamics (Whitaker and Farrington, 2004), examining the relationship between the number of cases and the average age of first infection may help to provide additional insight into the etiology of diseases for which the involvement of an infection has been hypothesized, but for which a specific infectious agent has yet to be identified. One such disease is Kawasaki disease (KD), an acute systemic vasculitis that occurs primarily in children under 5 years of age. Two aspects of the epidemiologic evidence which suggest an infectious etiology for KD include seasonal fluctuations in incidence and the age distribution of cases (Burgner and Harnden, 2005). If one assumes that an infectious agent is responsible for KD, then by examining the correlation between the number of cases and the average age of such cases, it may be possible to predict the values of certain parameters of that infectious agent, such as the duration of infectiousness and the nature of immunity to re-infection. Comparing these predictions against measured values for candidate etiologic agents could help to narrow the range of possibilities, providing a source of evidence independent of the types of evidence usually considered in identifying etiology.
We define a basic age-structured SIR model and examine how the relationship between the number of cases and the average age of first infection varies for an immunizing infection depending on the duration of infectiousness and assumptions about population mixing. We find that the average age of first infection is greatest at or soon after the epidemic peak regardless of the duration of infectiousness when we assume homogeneous mixing. This relationship emerges as a consequence of fluctuations in the susceptible population, and can be verified mathematically under some simplifying assumptions. We go on to explore how sensitive this relationship is to assumptions about population mixing, the mechanism of seasonality, the presence of a latent period prior to onset of infectiousness, and whether we consider prevalent or incident infections to be more representative of case report data.
Section snippets
Description of the model
We first define a basic age-structured SIR model. We model a hypothetical population in which individuals are equally divided among 80 one-year age classes. Individuals are born into the first susceptible age class (S1) at a rate of ν=12.6 births per 1000 people per year, and age into the next age class at a rate μ=1 yr−1. All individuals live to 80 years old, at which point they die (or are dropped from the model), leading to a rectangular age distribution that approximates that of a developed
Sensitivity to population mixing assumptions
We explore a variety of assumptions about how individuals belonging to different age classes mix with one another, and construct “who acquires infection from whom” (WAIFW) matrices based on these assumptions (Appendix B). The simplest assumption is that individuals are equally likely to contact any other individual in the population regardless of age (homogeneous mixing), which we explored in Section 2. Alternatively, we might assume that individuals are 10-times more likely to contact other
Mechanism of seasonality
The means by which seasonal oscillations in the incidence of infectious diseases are generated is largely unknown, although there are numerous hypotheses. Climatic factors such as colder temperatures and lower relative humidity (Lowen et al., 2007), host physiological factors such as photoperiod effects on levels of vitamin D (Cannell et al., 2006), and behavioral factors such as the aggregation of children in schools (Fine and Clarkson, 1982) have all been cited as possible causes of the
Age-structured SEIR model
A slight variation on the basic age-structured SIR model includes the addition of a latent period following infection of a susceptible individual during which that person is not yet infectious to others; this is the SEIR (susceptible–exposed–infectious–recovered) model. We assume that the average duration of the latent period is one week, consistent with infections such as measles, rubella, and chickenpox (Anderson and May, 1991). The set of differential equations describing the system is as
Prevalent versus incident infections
Our primary analysis considers the relationship between the number of incident infections and the average age of such infections. Thus, our comparison to case notification data assumes that cases are reported at beginning of their infectious period or at a constant interval, e.g. 1 week, following the onset of infectiousness. Alternatively, one might assume that there is an equal probability of a case being reported at any point during the infectious period. If this is true, it would be more
Discussion
We use seasonally forced age-structured models to examine what aspects of the disease dynamics affect the relationship between the number of cases and the average age of first infection over the course of a single epidemic period. We explore a variety of parameter combinations and mixing assumptions, and find that the average age of first infection is greatest at or near the peak of the epidemic when we assume homogenous mixing regardless of the duration of infectiousness. When population
Acknowledgments
The authors thank Caroline Colijn for useful feedback in the analysis of the model, Megan Murray and Jamie Robins for constructive advice on the structure of this project, and David Burgner, Cecile Viboud, and Lone Simonsen for stimulating discussions that led to the worked described here. V.E.P. was supported by Training Grant T32 AI07535, and V.E.P. and M.L. were supported by Cooperative Agreement 5U01GM076497 (Models of Infectious Disease Agent Study) from the National Institutes of Health.
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Current address: Center for Infectious Disease Dynamics, Biology Department, 208 Mueller Laboratory, The Pennsylvania State University, University Park, PA 16802, USA.