A major goal of biomathematical modelling is to seek conceptual simplicity in the face of biological complexity. In general, the questions of interest should dictate the form and complexity of the model used to address them. In the present context in particular, the appropriate degree of aggregation over population inhomogeneities (such as spatial, socio-economic, genetic and age variation) may depend on the goals of the investigation (Levin et al. 1997
; May 2004
). For measles, analyses based on models of homogeneous mixing populations have proved useful in the study of seasonality (Fine & Clarkson 1982
; Bjørnstad et al. 2002
; Ferrari et al. 2008
) and effects of climate drivers (Lima 2009
). Metapopulations of weakly coupled homogeneously mixing populations are central to current understanding of spatio-temporal disease dynamics (Grenfell et al. 2001
; Xia et al. 2004
). In addition, such models can yield insight into the fundamental predictability of disease outbreaks (Stone et al. 2007
) and have been found adequate to predict the effects of variation in birth rate and some dynamical features associated with the institution of vaccination programmes (Earn et al. 2000
). Homogeneous mixing models were also the basis of early work on local extinctions and vaccination strategies (Bartlett 1957
), though age structure and infectious period distribution may have substantial roles to play in these questions (Schenzle 1984
; Keeling & Grenfell 1997
; Lloyd 2001b
; Conlan et al.
submitted). In particular, estimates of the parameter R0
are known to be highly sensitive to assumptions on age structure (Wallinga et al. 2001
). Interpretations of homogeneous mixing models must be made in the context of their limitations.
Clinical and household studies of infectious diseases yield information regarding the transmission and progression of infection on the scale of individuals or families (Hope Simpson 1948
; Bailey 1956
). Such studies are complemented by data on disease prevalence or incidence on larger spatial scales. The recent development of likelihood-based approaches capable of dealing with the full spectrum of unavoidable complexities (Cauchemez & Ferguson 2008
; Cauchemez et al. 2008
; King et al. 2008
)—unobserved variables, measurement error, process noise, nonlinearity, non-stationarity and covariates—means that we can now view a disease from the population point of view with something like the same clarity that we have for many years been able to view individual infections. Specifically, although we have long understood how to describe the population dynamics of infectious diseases using mathematical models, we have only recently gained the ability to fit these models to data using statistically sound methods. When biological quantities estimated in this way agree with those estimated from clinical or household studies, it can be interpreted as confirmation of the assumptions embodied in the model. Disagreement between the small- and large-scale points of view, however, raises interesting scientific questions. In the present study, we find broad agreement with previous studies concerning many of the model's parameters. On the other hand, we find significant departures with respect to three important quantities—R0
, the infectious period and the intensity of extra-demographic stochasticity.
The quantity R0 is central in epidemiological theory because it has interpretations in terms of so many quantities of interest, including mean age of first infection, mean susceptible fraction, exponential-phase epidemic growth rate and vaccination coverage required for eradication. It is important to realize that the conjunction of these interpretations occurs only in the context of very simple models. Simple models necessarily lack flexibility. In reality, these interpretations diverge owing to heterogeneities in age, spatial location, host genetics, etc. It is therefore unrealistic to expect that estimates of R0 (or any other single quantity) derived from fitting a simple model to one sort of data should agree with estimates derived from other data sources. Rather, the key question should be: which biological interpretations are relevant in the context of the data used to inform the model? From a statistical point of view, this corresponds to the question: to what features of the data are the parameter estimates sensitive? Our estimate of the basic reproductive ratio, R0, like those of earlier studies focusing on aggregated case-count time series, is somewhat greater than estimates based on serological surveys and age-stratified incidence data. It is possible that this reflects the sensitivity of the time-series-derived estimates of R0 to peak incidence. Specifically, peak incidence strongly influences the estimates of both R0 and reporting rate ρ, but since ρ is well identified by other features of the data (namely, the long-run cumulative incidence), we suspect that the model requires a high R0 to match the observed peak epidemic case counts. Alternatively, the high R0 estimates may reflect the sensitivity of estimated R0 to early phase epidemic growth. If this is in fact the case, our estimates of R0 may more accurately reflect contact rates among the core group of school-age children than they do those of the population at large. In this case, the model is effectively extrapolating these rates to the adult population, about which these data have little direct information since so few adult cases occur. In contrast, from this point of view, the interpretation of R0 in terms of its definition as mean number of secondary infections in a fully susceptible population is hopelessly extrapolated. Likewise, the interpretation of R0 in terms of mean age of first infection is unjustified, since it necessarily depends on the age structure of transmission, which is not part of the model. Perhaps the most direct information pertaining to mean age at first infection in these data comes from the lag between changes in birth rate and their subsequent effects on incidence. This lag is explicitly captured in our model by the delay, τ, between birth and recruitment into the susceptible pool.
Since the comparatively high estimate of R0
does not appear to be a mere artefact of time discretization, the question remains as to why the population-level data suggest a more communicable disease than do the individual-level data. Existing estimates of R0
are sensitive to assumptions about the age structure of transmission that have not yet been fully resolved (Wallinga et al. 2001
). Perhaps transmission in schools is relatively more effectual than within households. Alternatively, it may be that this discrepancy points to a need for a better description of infectious- and/or latent-period distribution, of the disease's age structure or both. In our results, the departure of the estimated latent and infectious periods from plausible values obtained from household studies grows with city size, and so the most likely explanation for this correlation is the spatial heterogeneity of transmission within towns and cities. The latter, we have shown, cannot usefully be captured via the simple device of an exponent in the transmission term. More detailed explorations of disaggregated data and/or models with explicit spatial structure have the potential to shed light on this question. Finally, the strong evidence in favour of extra-demographic stochasticity raises the question of precisely why such stochasticity aids in the explanation of the data. To what extent does this finding indicate the presence of genuine environmental stochasticity? To what extent does it indicate model misspecification? To address these questions, again, analysis based on more detailed models is called for. In the case of measles, there are sufficient data to entertain models featuring spatial inhomogeneity (Grenfell et al. 1995
; Xia et al. 2004
; Bjørnstad & Grenfell 2008
), age structure inhomogeneity (Schenzle 1984
; Bolker & Grenfell 1996
; Keeling & Grenfell 1997
) or both (Bolker & Grenfell 1995
). Although it has been demonstrated that such models can do a good job of accounting for gross features of the data, there has been less emphasis on requiring these models to account for all features of the data.
Progress on inference methodology for parameter estimation from measles time series has emphasized SEIR-type models for homogeneous populations (Ellner et al. 1998
; Bjørnstad et al. 2002
; Cauchemez & Ferguson 2008
; Keeling & Ross 2008
). The study of age structure and spatial effects for measles has placed less emphasis on inferring parameters from data (one exception is Xia et al. 2004
). This may be partly because of the additional difficulties of inference for such systems and partly because models based on homogeneous mixing do an impressive job of describing key dynamic features (Earn et al. 2000
; Grenfell et al. 2002
). Regardless of one's view of the importance of paying further attention to population inhomogeneities, there is a natural methodological question: how applicable are the techniques presented here for larger, more complex models? There is a computational price to pay for the convenience of plug-and-play statistical methods. For the results in , one application of the plug-and-play-iterated filtering inference procedure (based on 50 iterations, with a Monte Carlo size of 104
) implemented via the Pomp
package in R (R Development Core Team 2006
; King et al. 2009
) took 5 h to run on a desktop machine. Computational effort scales roughly linearly with the number of parameters plus the number of state variables, so one can see that only a modest amount of additional structure could be included without hitting computational limitations. To increase computational efficiency, however, the iterated filtering algorithm can be implemented in a non-plug-and-play mode, in which the filter is either tailored to the particular model or takes advantage of available analytic properties of the transition densities. Such extensions, which would be required for much larger models, are a topic for future research. One can seek inspiration for future possibilities from numerical climate models, for which filtering operations (the computationally intensive step in the iterated filtering algorithm) have been carried out on systems with 5 × 106
state variables (Anderson & Collins 2007
). Filtering in such high-dimensional situations requires the development and evaluation of appropriate approximations for the system under investigation.
In this article, we have carried out the first scientific investigation based on a new framework for continuous-time, discrete-state population dynamics with both demographic and extra-demographic noise (probabilistic and statistical properties of this model class were investigated by Bretó et al. (2009)
. Extra-demographic stochasticity (interpreted as noise in the rates of a discrete population Markov population model) is equivalent to the possibility of multiple individuals moving simultaneously between compartments (Bretó et al. 2009
), and, as such, may be due to social events that affect the behaviour of many individuals (e.g. sporting events) or events that change disease transmissibility, such as variations in temperature and humidity. Variability in the rates gives the model additional flexibility that can also describe model misspecification. Analogously, when carrying out linear regression, it is customary to fit a line to data while understanding that the variation of the data around the line corresponds to unknown and unmodelled deterministic effects as well as to random fluctuations. For linear regression, one typically treats both these sources of uncertainty equally, and certainly all the usual standard errors and test statistics do not discriminate between them. We maintain that the same approach can be applied to dynamic models; in other words, the distinction between model misspecification and process stochasticity should be noted, though it will not usually affect the subsequent analysis.
The term extra-demographic stochasticity
encompasses all sources of variability beyond the intrinsic demographic stochasticity that would be present in a homogeneous population. There are many circumstances in which such variability can be expected to be important. In particular, the variability of rates in our new framework offers an approach to modelling superspreading events (Lloyd-Smith et al. 2005
). These events occur when variability between individuals, environmental effects or an interaction between the two results in a highly skewed distribution for the number of secondary cases caused by an index case. Superspreading has been documented in measles, but is of greater dynamic importance in other diseases such as severe acute respiratory syndrome (Lloyd-Smith et al. 2005
). Conventional population models amenable to non-plug-and-play statistical analyses have been unable to include such effects readily. This is, therefore, one more example in which the flexibility of plug-and-play methodology holds the potential to encourage the development of scientifically appropriate models.