Epidemiological effects of seasonal oscillations in birth rates

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Abstract

Seasonal oscillations in birth rates are ubiquitous in human populations. These oscillations might play an important role in infectious disease dynamics because they induce seasonal variation in the number of susceptible individuals that enter populations. We incorporate seasonality of birth rate into the standard, deterministic susceptible–infectious–recovered (SIR) and susceptible–exposed–infectious–recovered (SEIR) epidemic models and identify parameter regions in which birth seasonality can be expected to have observable epidemiological effects. The SIR and SEIR models yield similar results if the infectious period in the SIR model is compared with the “infected period” (the sum of the latent and infectious periods) in the SEIR model. For extremely transmissible pathogens, large amplitude birth seasonality can induce resonant oscillations in disease incidence, bifurcations to stable multi-year epidemic cycles, and hysteresis. Typical childhood infectious diseases are not sufficiently transmissible for their asymptotic dynamics to be likely to exhibit such behaviour. However, we show that fold and period-doubling bifurcations generically occur within regions of parameter space where transients are phase-locked onto cycles resembling the limit cycles beyond the bifurcations, and that these phase-locking regions extend to arbitrarily small amplitude of seasonality of birth rates. Consequently, significant epidemiological effects of birth seasonality may occur in practice in the form of transient dynamics that are sustained by demographic stochasticity.

Introduction

Communicable disease surveillance over the last century has produced many valuable time series that reveal the temporal and spatial epidemic patterns caused by a wide variety of pathogens (Anderson and May, 1991, Grenfell and Harwood, 1997, Earn et al., 1998, Earn et al., 2002, Grenfell et al., 2001). These time series have stimulated the development and analysis of numerous mathematical models of infectious disease transmission, which aim to identify the mechanisms that generate observed epidemic patterns and to design strategies for control and eradication (Kermack and McKendrick, 1927, Bartlett, 1960, Bailey, 1975, Anderson and May, 1991).

Until the 1970s, the mathematical theory of infectious diseases was focused on autonomous deterministic and stochastic models. London and Yorke (1973) recognized that the contact rate among individuals is not constant but varies seasonally as a result of the aggregation of children in schools. Hence, the disease transmission rate is subject to exogenous seasonal forcing, making the system non-autonomous and leading potentially to complex dynamics. Indeed, subsequent studies established that seasonal forcing of the transmission rate can lead to multiple coexisting stable cycles (Schwartz and Smith, 1983) and chaos (Olsen and Schaffer, 1990, Glendinning and Perry, 1997). Recent work has shown that a suitably parameterized, seasonally forced transmission model can successfully predict observed transitions in the temporal structure of epidemics for a variety of childhood infectious diseases, based on slow changes in vaccination levels and birth rates (Earn et al., 2000, Bauch and Earn, 2003).

In addition to the secular changes that occur over decades, it is well known that birth rates oscillate seasonally, and this has been implicated in the incidence patterns of some non-communicable diseases such as schizophrenia and diabetes (Miura, 1987). From the point of view of infectious diseases, birth rate seasonality represents an additional source of exogenous seasonal forcing. In this paper, we investigate how seasonal oscillation in birth rates influences the dynamics of the simplest standard models of childhood infectious disease transmission (the susceptible–infectious–recovered or SIR model and the susceptible–exposed–infectious–recovered or SEIR model). We begin with a brief discussion of some birth data showing seasonal oscillations, from which we estimate the magnitude of seasonal forcing of birth rates. We then review the dynamics of the basic SIR model before exploring the theoretical consequences of birth rate seasonality.

Section snippets

Data showing seasonality of birth rates

Seasonality of birth rate has been observed in almost all historical populations (Trovato and Odynak, 1993). Two distinct patterns are evident in modern populations: the American pattern with a trough in April–May and a peak in September, and the European pattern with a peak in spring–summer and a secondary peak in September (Trovato and Odynak, 1993, Doblhammer et al., 2000, Haandrikman, 2003). In spite of its proximity to the United States, Canada's pattern of births since the early 1900s has

The standard unforced SIR model

The standard SIR model, originally investigated by Kermack and McKendrick (1927), can be written asS˙=νN-βNI+μS,I˙=βNIS-(γ+μ)I,R˙=γI-μR.Here, S, I, and R denote the numbers of individuals that are susceptible, infectious, and recovered, respectively, and recovery is assumed to entail lifelong immunity. The total population size is N=S+I+R. There are two demographic parameters, the per capita birth and natural death rates, which are given by ν and μ, respectively. The epidemiological parameters

The SIR model with seasonal oscillation in birth rate

We introduce seasonality of birth rate into the SIR model by replacing every occurrence of ν in Eq. (7a), (7b), (7c) withν[1+ɛsin(2πt/tf+φ)].Here, the amplitude of forcing of the birth rate is ɛ[0,1] and period of forcing istf=1yr.The phase shift φ[0,2π) plays no dynamical role; it is included above because it is convenient to associate integer values of t with the start (1 January) of each year when comparing model time series with data.

Our principal goal is to determine whether birth rate

The SEIR model with seasonal oscillation in birth rate

When we use the SIR model (7a), (7b), (7c), we are implicitly making the approximation that the latent period (the time between initial infection and becoming infectious) is zero. For childhood diseases, the latent period is often comparable to the infectious period (Anderson and May, 1991), so assuming a zero latent period might be a poor approximation.

Non-zero latency is usually introduced by adding an “exposed” (E) compartment in which individuals are infected but not yet infectious. If the

Discussion

We have investigated the effects of seasonal oscillations in birth rate on infectious disease transmission dynamics. In Section 2 we found that in North America the amplitude of oscillations is on the order of 10% of the mean birth rate (ɛ0.1). In Section 4, analysis of the SIR epidemic model in the presence of birth seasonality revealed a number of potentially significant dynamical effects, depending on the characteristics of the disease in question. In Section 5, we found that predictions

Acknowledgment

We thank Chris Bauch, Paul Higgs, Nicholas Kevlahan, Junling Ma, James Watmough, Gail Wolkowicz, and two anonymous referees for helpful comments.

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