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.