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.