Why is Bayesian hierarchical modelling essential to a whole-system approach?
Within a whole-system model we need to analyse multiple scales of information, therefore modelling using hierarchical Bayesian methods is important. Bayesian hierarchical models can help describe Bayesian networks in which the data can be grouped into levels, and where parameters may be inferred for any grouped level . Since information may be incorporated across multiple observational scales, notes them to be especially useful for parameterising demographic infection processes, such as disease progression, mortality, and transmission.
Bayesian hierarchical models usefully combine both statistical and ecological hierarchies, so we first summarise the difference. A statistical hierarchy forms when a large and complex model is composed of simpler sub-models, where the parameters it infers may exist only at single levels of the statistical hierarchy, may apply to multiple levels, or may be aggregated at one level to inform a process at a higher or lower level . Therefore, a statistical hierarchy describes the uncertainty within nested parameters, and in a Bayesian framework, this nesting forms a hierarchical model.
The statistical hierarchy within a Bayesian hierarchical model can represent levels of an ecological hierarchy, depending on which disease processes or states are being measured. For instance, genetic susceptibility can be considered a within-individual or within-host trait, disease progression a within-host process, and disease prevalence a population-level parameter. In hierarchical models, the individual-level of a hierarchical model is defined as the smallest measurable unit of that system : for example genes, such as those coding for disease susceptibility, could be viewed as the individual unit; alternatively, very detailed models might consider individual pathogens; coarser models might start with the individual host and model within-host infection processes in the abstract. When Bayesian methods are used to infer latent and unmeasurable states, this abstraction allows truly binary or categorical states—such as dead or alive; infected or uninfected—to be inferred as probabilities, which better reflect their lack of direct measurement . Expanding on Figure 1, infection status or “being infected” is an example of a host-specific latent variable, which may inform processes at all levels of an ecological model. Likewise, actuarial senescence—a decline in fitness with advancing age—is an example of a demographic process that could reasonably exist on multiple levels of an ecological hierarchy .
Both infection status and senescence have obvious implications for disease transmission since individual state is a key driver of physiology, ageing , and ultimately time-to-death. Yet utilising information about a latent state or process can be tricky. For example, senescence can be quantified as an—albeit very uncertain—individual-level process, and equally, as a demographic parameter; yet it is unmeasurable except at the population level. Further, the biology of senescence is complex: for example, in marmot individuals, senescence is more likely to occur in favourable conditions . Moreover, senescence theory itself remains disputed: the traditional viewpoint of senescence is that fitness declines with ageing , but this is challenged by other perspectives, for example that individual state dictates senescence in a heterogeneous fashion . Importantly, there are many ways to introduce complex latent variables such as senescence into whole-system models. What matters are reasonable and justifiable a priori assumptions regarding the proxy data available.
To date, Bayesian hierarchical methods have been applied only sparingly to wildlife disease problems. Here we can only cite eight examples (Table 2) following a comprehensive Web of Science search using combinations of the terms: “Bayesian”; “hierarchical”; “model”; “wildlife”; “animal”; “disease”; “infection”; “system”. These citations do not provide an exhaustive list of all papers that use Bayesian hierarchical methods to model disease but serve to present its limited application. Bayesian hierarchical models seem to have been applied to the field of wildlife disease epidemics in a rather ad hoc manner, in terms of methodology and terminology as well as across disease system, level of ecological hierarchy, or process to be inferred. The potential and power of Bayesian hierarchical methods is described above, yet they have rarely been used to explore individual, group and population hierarchies within the same model.
Table 2 indicates that the current applications of Bayesian hierarchical models are useful for generalising large within-population and or landscape-scale processes, and that they are broadly applicable across disease systems. Yet, studies across broad ecological levels and scales are rare. Table 2 also suggests that studies purely investigating disease spatially, or spatially and temporally, are less common than those that have a temporal investigation alone. Equally, as demonstrated by our bTB case study, Bayesian hierarchical analyses of evolving longitudinal datasets are also rare, but likely integral to the discovery of fine-scale ecological interactions pertinent to understanding disease processes.