Spatial modelling

This work has led to a coherent, unified, Bayesian statistical analysis of multiple streams of epidemic surveillance data from the 2009 A/H1N1pdm outbreak in England, producing age and region stratified epidemic reconstructions (with associated uncertainty) and robust estimates of the parameters of the transmission process. We have explored two modelling approaches: the PR and the MR models. Both fit adequately well the various data sources, with highly comparable estimates for both model parameters and epidemic characteristics that are consistent with existing literature.

Each approach has its strengths and limitations.

The PR approach is found to be parsimonious, yet sufficiently flexible to capture the underlying dynamics. It is also ‘non-parametric’, in the sense that the parameters representing the epidemic growth and initial seeding of infectiousness in each region are estimated without being subject to any parametric assumption. The assumed lack of relation between these parameters means that the spread of infection between regions cannot be forecast, and significant epidemic activity has to be observed in all regions to enable estimation of the epidemic burden. This lack of predictive ability is a limitation in the use of the PR approach. However, the greater flexibility becomes an advantage when it comes to epidemic reconstruction, and this is observed in a significant improvement in the model fit of the PR approach to the 2009 pandemic data. An additional advantage is that this modelling approach does not rely on the validity of the commuter data to describe the spread of infection, nor does it rely on the assumption that individuals maintain routine commuting behaviour regardless of infection status. Despite the permitted spatial variation in epidemic growth rates, the PR model provides estimates for $R_0^{\text{init}}$ that are consistent across regions. Therefore, the spatial heterogeneity in infection is being accounted for through the initial seeding of infectiousness.

The MR model incorporates spatial heterogeneity in transmission, arising from the interaction between regional populations, through commuting flows. This gives the MR model greater power to predict the spatial spread of influenza, enabling the prediction of which will be the next region to experience widespread infection. Early in a pandemic, therefore, the MR approach is more useful in a predictive modelling setting. However, it has been seen elsewhere that long–range interactions have a declining role in the spread of a pandemic once infection is widespread in each region.⁵⁷ This is exacerbated for A/H1N1pdm influenza as school-age children, the demographic group most affected, do not contribute to commuter movements. This marginalises, to some degree, the key benefit of this approach. Additionally, the MR approach involves an increased computational burden that limits its use as a tool for timely epidemic tracking as data accumulate over time.

One variant of the MR model investigated here involved the stratification of the population within each region into commuters and non-commuters. This has the effect of assuming that each region contains a fixed subpopulation of individuals who commute daily. This formulation yields no consistent improvement in model performance, while further increasing the computational cost. Factoring in the ‘random’ movements of casual and occasional travellers, who have been quoted to potentially increase the rate of transmission between regions by 25%,⁵⁸ would involve further computational burden, and is particularly difficult to implement in an inferential setting without appropriate auxiliary information (e.g. if the census data contained information on the purpose of travel).

The MR model could be made more realistic and detailed by assuming that a proportion of those with symptomatic illness may not travel, or that asymptomatic illness is less infectious. However, consideration of such factors would only lessen the contribution of long-range transmission, leaving the conclusions unchanged.

To summarise, using a Bayesian statistical framework the PR model is found to be sufficiently flexible to provide a good fit to data, is quick to implement as it includes lower-dimension contact matrices, and the non-interacting nature of the regions means that the likelihood calculations can be easily parallelised. Reassuringly, it also provided concurring estimates for the basic reproductive number ($R_0^{\text{init}}$) across the regions, in agreement with the MR approach. However, the PR model can provide little insight into inter-region transmission and the determinants of spatial heterogeneity in the spread of infection because of its simple structure. In a situation where school-age children are the main agents of transmission and baseline transmissibility is not high, spatial models that concentrate on local transmission, like the PR model, provide a powerful and timely tool for use by public health services, helping to inform effective control and containment measures.

Efficient estimation and prediction

We have proposed addressing the substantive problem of real-time tracking of an emergent and realistic epidemic, assimilating multiple sources of information through the development of a suitable SMC algorithm. When incoming data are stable, this process can be automated using standard algorithms in line with approaches already in the literature.^12,27 However, in the presence of interventions or any other event that may artificially interrupt the epidemic’s trajectory or even result in a shock to the epidemic system, it is necessary to adapt the algorithm appropriately. How the algorithm is adapted will depend on the scale of the disruption to the surveillance data. The end result is a semi-automated SMC algorithm that can be tailored to the nature of the shock to limit the required computation time.

This hybrid SMC can be seen to greatly outperform MCMC when it comes to successively iterating analyses, as will be required in a pandemic scenario. Throughout, we have compared the divergence between SMC posteriors from posteriors generated by the ‘gold standard’ MCMC. However, this may be an unfair comparison, as the MCMC algorithm is based on ‘plain vanilla’ Metropolis updates, and could benefit from an in-depth tuning process itself. More sophisticated MCMC algorithms could be used, for example differential geometric MCMC or parallelised MCMC.^59,60 These could assist with improving MCMC run times. On the other hand, as MCMC steps are the main computational overhead of the SMC algorithm, any development of the MCMC algorithm may also lead to similar improvement of the SMC algorithm. It is also worth adding that the benefits of SMC for real-time analysis have been demonstrated. For a single, one-off, analysis aimed at reconstructing the epidemic dynamics, the SMC algorithm would be implemented differently and may not hold any significant advantage over MCMC.

Finally, the analyses carried out in this work have neglected the first 50 days of the epidemic, concentrating on a period when there is substantial transmission in the population and appropriate data are becoming available. As a result, a deterministic system can adequately describe the future evolution of the pandemic. Stochastic effects are significant and need to be incorporated into the model if monitoring is needed in the earlier stages. A prescription exists for what is known as ‘particle learning’ in the presence of ‘shocks’ in such a setting.⁶¹ Alternatively, to improve the robustness of the inferences, the piecewise linear quantities describing population reporting behaviour could be described by linked stochastic noise processes. This has the potential to reduce the sensitivity of estimates to the presence of change points that are not, for whatever reason, foreseeable.

Over the course of this project, the state of the art of statistical computing in epidemic models has advanced in many directions, motivated by influenza and also by recent Ebola outbreaks.^{11,27,62–64} Each approach, however, uses direct observations of cases or estimates of cases to fit models. It is believed that our approach to tackling a realistic, messy suite of epidemic data is both novel and critically important.

Pandemic data

The capacity to provide real-time estimation and prediction of an epidemic is not only dependent on the existence of models and software. Crucial to this ability is the richness of the available public health surveillance data and its timely availability. The UK is well served in terms of the depth and completeness of its influenza surveillance mechanisms, and the timely availability of data can be almost guaranteed whenever it arises as a result of routine collection and reporting. This is not quite the case for the serological information, however, which requires suitable tests to be developed, and samples to be collected and analysed. The role of serological data is shown in figure 10 of Birrell et al.,⁴⁷ in which epidemic projections have been sequentially made using only noisy primary care consultation data in the absence of serology data. A reliable picture of the epidemic is not available until the epidemic is almost over. This poses some key questions: are serological samples going to be available in a timely manner, in sufficient quantity and quality, and in the right format? The availability of this data is a potentially limiting factor.

Routine influenza surveillance data may be reliable in terms of its timely provision. However, such data may become unreliable as infection becomes more widespread and the demands placed on health-care services increase. Hospital beds and GP appointments are finite, and the health-care system may be operating at capacity for a period. Services such as NPFS are designed to alleviate this burden in primary care, but, in particular, we have to entertain the possibility that the proportion of cases that lead to hospitalisation might artificially diminish. The model will permit time variation in p^h to account for these density-dependent effects, but how to diagnose and characterise this decline are still open problems. As will be discussed in Chapter 6, section Alternative influenza-like illness surveillance, this motivates an exploration of the relationship between primary care ILI surveillance and community ILI surveillance. The aim is to find alternative data sources whose interpretation and collection is robust to high levels of influenza activity and can, therefore, constitute a valuable addition to the array of data already under consideration.