Modelling latent variables is essential to the whole-system
model.
Bayesian state-space models are a form of Bayesian hierarchical model
that allow Bayesian networks to easily distinguish dynamic biological
processes such as changes through time from unavoidable errors due to
the imperfect detection of disease. Accordingly, approaches such as
those used by McDonald or Wells may be described as state-space methods
since they quantify the randomness and variability of ecological
processes (Table 3). Specifically, they do this by accounting for
whether a parameter is unobserved or observed, as well as any associated
sampling error . This means that Bayesian state-space models are
especially good at, for example, teasing apart demographic stochasticity
and sampling error .
The inferred latent variables and ecological findings of the Bayesian
state-space models presented in Table 3 demonstrate that disease
parameters are mainly studied on group or population levels even though
most use individual serology data to inform models. Table 3 illustrates
that the application of Bayesian state-space models within wildlife
disease epidemiology is limited, but in our searches we also encountered
problems caused by vague or inconsistent model terminologies.
Underpinning this observation are three search methods referenced within
Table 3. The first is a comprehensive Web of Science search using
combinations of the terms: “Bayesian”; “state-space”; “disease”;
“wildlife”, which only yielded four relevant studies. A further
relevant study was found using the search topic “state-space model”
when filtering by the Web of Science category “ecology”. Two
additional relevant examples were found incidentally in the absence of
either “state” or “space” as a keyword.
Three key conclusions can be drawn from the examples within Table 3.
First, that observed serological data is common to all studies:
presumably because most disease states in wildlife remain latent
following visual surveillance. Second, it is encouraging that ecological
stochasticity is modelled in the dimensions of space and time, often
within the same study. And most importantly, the conclusions drawn by
all examples found in Table 3 only regard population- or species-
levels. Based on these conclusions, we observe that state-space models
often span two levels of a hierarchy but rarely multiple latent
variables. An example of this is demonstrated within previous work on
the badger bTB system by , which used state-space models to infer the
latent variable “alive” but ignored uncertainty in diagnostic test
outcomes.
When combined with Bayesian state-space methods, Table 3 demonstrates
that serological data collected over space and time can yield powerful
conclusions about high-level disease parameters. But, state-space models
are also an obvious tool for filling in any unknown relationships
between individual disease states. For example, frequentist multi-state
modelling revealed that epidemiological and demographic parameters vary
between disease states in badgers , yet a vast number of unknown
complexities still exist within this relationship which cannot be
quantified without Bayesian methods. There is a fundamental need to
parameterise processes within disease models more rigorously by applying
Bayesian state-space theory. Although the need for good epidemiological
parameter inference has been apparent for over a decade , the potential
of state-space models has not yet been realised: they can help define
the mutable nature of disease across any level of the
“whole-system” model, inclusive of space, as well as time.