CHARACTERIZING THE DYNAMICS OF RUBELLA RELATIVE TO MEASLES: THE ROLE OF STOCHASTICITY

Wednesday, 10th of December 2014

CHARACTERIZING THE DYNAMICS OF RUBELLA RELATIVE TO MEASLES: THE ROLE OF STOCHASTICITY

Ganna Rozhnova, C. Jessica E. Metcalf, Bryan T. Grenfell

Abstract and introduction are below; full text, with figures and equations, is at http://rsif.royalsocietypublishing.org/content/10/88/20130643.long

Rubella is a completely immunizing and mild infection in children. Understanding its behaviour is of considerable public health importance because of congenital rubella syndrome, which results from infection with rubella during early pregnancy and may entail a variety of birth defects. The recurrent dynamics of rubella are relatively poorly resolved, and appear to show considerable diversity globally. Here, we investigate the behaviour of a stochastic seasonally forced susceptible–infected–recovered model to characterize the determinants of these dynamics and illustrate patterns by comparison with measles. We perform a systematic analysis of spectra of stochastic fluctuations around stable attractors of the corresponding deterministic model and compare them with spectra from full stochastic simulations in large populations. This approach allows us to quantify the effects of demographic stochasticity and to give a coherent picture of measles and rubella dynamics, explaining essential differences in the recurrent patterns exhibited by these diseases. We discuss the implications of our findings in the context of vaccination and changing birth rates as well as the persistence of these two childhood infections.

1. Introduction

Rubella is a completely immunizing, directly transmitted viral infection, generally presenting as a mild and potentially even asymptomatic childhood disease [1]. As a result, rubella tends to be underreported, and its recurrent dynamics are fairly poorly characterized. Nevertheless, because infection during early pregnancy may cause spontaneous abortion or congenital rubella syndrome (CRS), which may entail a variety of birth defects [2], understanding the dynamics of rubella is of considerable public health importance. Dynamical features of rubella may alter the CRS burden via their effects on the average age of infection. Episodic dynamics may increase the average age of infection, as the intervals between larger outbreaks provide the opportunity for individuals to age into later age classes [3,4]. Likewise, local extinction dynamics can allow individuals to remain susceptible as they age into their childbearing years [5,6], resulting in the potential for a considerable CRS burden once rubella is reintroduced.

Empirically, rubella seems to be linked to either (i) annual dynamics, as in Mexico [7], Peru [5] or parts of Africa [8,9]; (ii) spiky dynamics, as in Canada [10]; and (iii) some hint at multi-annual regularity, as in Japan [11], England and Wales [12], and various European countries [13]. In figure 1, we show three time series that represent the range of observed rubella dynamics. Spectral analyses of time series are particularly useful for understanding temporal patterns exhibited by different data [14,15]. The characteristic feature of rubella spectra is an annual peak at 1 year and a multi-annual peak at 5–6 years exhibited by all data in figure 1. Rubella also seems to experience regular fade-outs [7], which is of great epidemiological importance, particularly in the context of increasing global control efforts. The propensity for stochastic extinction is characterized by the critical community size (CCS), below which the infection tends to go extinct in epidemic troughs. Analyses of dynamics in Mexico and Peru suggest a CCS of over 10⁶ for rubella [5,7].

Time series of the case reports of rubella and the corresponding spectrum in (a,d) the Hidalgo district, Mexico; (b,e) Japan and (c,f) the province Ontario, Canada. To resolve low-frequency periodicities, these time series include short intervals of vaccination (years 1998–2001 for Mexico and 1989–1992 for Japan). Before the spectrum was taken, each series was normalized, setting the mean to zero and the variance to unity. The smooth spectrum (thick black lines) was obtained from the raw spectrum (thin grey lines) using two passes of a three-point moving average of the spectral ordinates. The dashed black lines are 90% confidence limits on the smooth spectrum. The confidence intervals represent the uncertainly in the observations. The method of computation of the spectra and confidence limits is described in detail in [14, ch. 4].

Measles provides a natural comparison for rubella, as it is another viral childhood disease with a very similar life cycle (particularly, direct transmission). In addition, it is perhaps the most extensively studied of the childhood infections, and its dynamics are very well understood [3,16–27]. Before the start of vaccination in England and Wales, both biennial dynamics (e.g. in London) and annual dynamics (e.g. in Liverpool) were observed [18,20,24,26]. The underlying driver of this variability has been identified as differences in birth rate, combined with annual seasonality in transmission driven by school term times [18,19,26,28]. In sub-Saharan Africa, chaotic dynamics have been shown to result from a very high birth rate, combined with extreme seasonal forcing [21]. Both highly irregular dynamics [16,20] (e.g. following vaccination in England and Wales) and triennial dynamics [29] (e.g. in Baltimore between 1928 and 1935) have also been reported. The spectral analyses of measles data exhibiting the described dynamics can be found now in the standard textbook [12]. The CCS of measles is rather smaller than that of rubella, estimated by Bjørnstad et al. [18] to be between 3 × 10⁵ and 5 × 10⁵ for England and Wales.

The two key ingredients underlying models of childhood diseases such as rubella and measles are (i) seasonality in transmission owing to schooling patterns and (ii) demographic stochasticity arising from the discrete nature of population [26,28,30,31]. Although various approaches have been used to understand the dynamics of rubella [10,32–34], most of the analyses have been essentially deterministic. Keeling et al. [32] considered a term-time forced susceptible–infected–recovered (SIR) model and compared its dynamics with rubella data in Copenhagen. From this, they concluded that the dynamics of rubella may result from switching between two cyclic attractors (annual and multi-annual limit cycles) of the deterministic model. Although the deterministic analysis they present is comprehensive, there is only a limited amount of evidence to suggest that the switching will occur in contexts that include demographic stochasticity. In particular, in this study [32], stochasticity was introduced into the model as multiplicative noise of arbitrary amplitude instead of using, for instance, standard stochastic simulations based on the Gillespie algorithm (for unforced models) [35] and its extensions (for seasonally forced models) [36]. Such simulations produce exact realizations of the stochastic process, whose full dynamics is given by the solution of the master equation as described in §2.2. For large populations, the master equation is approximated by the deterministic model with additive noise [37,38].

Bauch & Earn [10] studied a term-time forced susceptible–exposed–infected–recovered model and showed that frequencies obtained from the linear perturbation analysis of the deterministic model are in good agreement with positions of the peaks in spectra of data records of various childhood infections. The application of this approach to rubella data for Canada predicted two distinct peaks at 1 and 5.1 years, close to what we see in figure 1f. With the exception of [10], where stochastic simulations for Canada parameter values were also performed, there has been no work on rubella using a fully stochastic approach dealing with demographic stochasticity.

Here, we use this approach to characterize different rubella dynamics and illustrate patterns by comparison with measles. To this end, we perform the theoretical analysis of spectra of stochastic fluctuations around stable attractors of a seasonally forced deterministic SIR model and compare them with spectra obtained from full stochastic simulations based on a modification [36] of the algorithm by Gillespie [35]. The mathematical techniques used in this study have been developed for ecological and epidemiological models [37,39,40] and applied to model temporal patterns of measles and pertussis [38,41,42]. The picture that emerges to explain rubella dynamics is close to that proposed in reference [10] but goes beyond it, because our analysis allows us to obtain the full structure of a spectrum (as opposed to the deterministic analysis of Bauch & Earn [10] where only frequencies of the spectral peaks could be predicted). By introducing key spectral statistics (described below), we systematically investigate how the dominant period, amplitude and coherence of stochastic fluctuations change across a broad range of epidemiological parameters. We then discuss the implications of our analysis in the context of changing birth rates and vaccination levels, as well as their implications for the persistence of measles and rubella.