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. 2006 Oct 7;273(1600):2541-50.
doi: 10.1098/rspb.2006.3604.

Seasonal infectious disease epidemiology

Nicholas C Grassly  1 , Christophe Fraser
Affiliations

Seasonal infectious disease epidemiology

Nicholas C Grassly et al. Proc Biol Sci. .

Abstract

Seasonal change in the incidence of infectious diseases is a common phenomenon in both temperate and tropical climates. However, the mechanisms responsible for seasonal disease incidence, and the epidemiological consequences of seasonality, are poorly understood with rare exception. Standard epidemiological theory and concepts such as the basic reproductive number R0 no longer apply, and the implications for interventions that themselves may be periodic, such as pulse vaccination, have not been formally examined. This paper examines the causes and consequences of seasonality, and in so doing derives several new results concerning vaccination strategy and the interpretation of disease outbreak data. It begins with a brief review of published scientific studies in support of different causes of seasonality in infectious diseases of humans, identifying four principal mechanisms and their association with different routes of transmission. It then describes the consequences of seasonality for R0, disease outbreaks, endemic dynamics and persistence. Finally, a mathematical analysis of routine and pulse vaccination programmes for seasonal infections is presented. The synthesis of seasonal infectious disease epidemiology attempted by this paper highlights the need for further empirical and theoretical work.

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Figures

Figure 1
Figure 1
Estimated average seasonal transmission parameter (mean centred) for measles in the UK compared to sinusoidal forcing with σ=0.28. The monthly estimates from Soper (1929) are averages over 1905–1916 for Glasgow, and the biweekly estimates from Finkenstadt & Grenfell (2000) are averages for England and Wales over 1944–1964. School terms are indicated at the top of the figure by the thick bars.
Figure 2
Figure 2
The distribution of final outbreak size following the introduction of a single case in a stochastic SIR model in the presence (σ=0.5) and absence (σ=0) of seasonal variation in the transmission parameter for different values of the average basic reproductive number R¯0. Each curve represents the outcome from 10 000 single case importations occurring on a randomly distributed day of the year. The expected distribution based on branching process theory is shown in grey. We ignore the demographic processes of birth and death since outbreaks occur over a relatively short period. In this case, the probability of an infection occurring in a small time interval dt is β(t)S(t)I(t)dt/N(t) and that of recovery to the immune class I(t)dt/D, where S(t) and I(t) are the number of susceptible and infected individuals, respectively, at time t, the total population size N(t)=107 is constant over time, β(t) is the transmission parameter and D=2 weeks is the mean infectious period. The asterisked line shows the increase in larger outbreaks seen for a shorter duration of infection (one week) when R¯0=0.9 and σ=0.5.
Figure 3
Figure 3
Endemic dynamics of deterministic and stochastic SIR models of (a) non-seasonal and (b) seasonal infections, with the introduction of routine vaccination of 60% of the population at birth after 20 years. (a) The non-seasonal deterministic model shows damped oscillation in incidence following perturbation of the system by vaccination, with the period T≃2πAD predicted by stability analysis of the endemic equilibrium. In the stochastic case, these oscillations are sustained both pre- and post-vaccination, although with different periods reflecting the change in the mean age at infection A following vaccination. (b) The change in the natural period of the system following vaccination in the seasonal model results in a transition from annual to triennial epidemics in the deterministic case. In the stochastic model post-vaccination, more complex dynamics are observed, with switching between the attractors with annual and triennial periodicity driven by the stochasticity. These simulations are for an infection with R¯0=8, an infection period of two weeks, population size N=106 and rate of exit from the population μ=0.1. Vaccination moves individuals from the susceptible to the recovered compartment. The seasonal model assumes a sinusoidally varying transmission parameter with σ=0.1. In the stochastic model, we assume a small rate of import of infection of 10−5 yr−1 N−1 to avoid extinction of infection.
Figure 4
Figure 4
Pulse vaccination and seasonal infectious disease dynamics. (a) A contour plot showing the percentage reduction in R¯v for annual pulse vaccination three months prior to peak transmission (δ=0.25), compared to six months (δ=0.5), for a seasonal infection with sinusoidal variation in transmission with strength σ and vaccination coverage (1−q). (b) Numerical simulations of a stochastic, sinusoidal seasonal SIR model showing the fraction of populations where the infection goes extinct and the cumulative cases of disease for an infection with robustly annual epidemics and pulse vaccination at six months prior to peak transmission (δ=0.5; triangles, solid line) and at three months (δ=0.25; diamonds, dashed line). Each point represents 100 simulations for 100 years post-vaccination, with model parameters σ=0.6, μ=0.16, N=105 and D=6 weeks. Pulse vaccination is assumed to successfully immunize 40% of susceptible children (q=0.6). (c) Prevalence of infection over time for a seasonal stochastic SIR model where the introduction of pulse vaccination at time 10 years results in a switch from biennial to annual epidemics. The model parameters are R¯0=6, σ=0.3, N=106, δ=0.25 years and D=2 weeks.
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