Accounting for uncertainty in model-based prevalence estimation: paratuberculosis control in dairy herds

Abstract

Background: A common approach to the application of epidemiological models is to determine a single (point estimate) parameterisation using the information available in the literature. However, in many cases there is considerable uncertainty about parameter values, reflecting both the incomplete nature of current knowledge and natural variation, for example between farms. Furthermore model outcomes may be highly sensitive to different parameter values. Paratuberculosis is an infection for which many of the key parameter values are poorly understood and highly variable, and for such infections there is a need to develop and apply statistical techniques which make maximal use of available data.

Results: A technique based on Latin hypercube sampling combined with a novel reweighting method was developed which enables parameter uncertainty and variability to be incorporated into a model-based framework for estimation of prevalence. The method was evaluated by applying it to a simulation of paratuberculosis in dairy herds which combines a continuous time stochastic algorithm with model features such as within herd variability in disease development and shedding, which have not been previously explored in paratuberculosis models. Generated sample parameter combinations were assigned a weight, determined by quantifying the model's resultant ability to reproduce prevalence data. Once these weights are generated the model can be used to evaluate other scenarios such as control options. To illustrate the utility of this approach these reweighted model outputs were used to compare standard test and cull control strategies both individually and in combination with simple husbandry practices that aim to reduce infection rates.

Conclusions: The technique developed has been shown to be applicable to a complex model incorporating realistic control options. For models where parameters are not well known or subject to significant variability, the reweighting scheme allowed estimated distributions of parameter values to be combined with additional sources of information, such as that available from prevalence distributions, resulting in outputs which implicitly handle variation and uncertainty. This methodology allows for more robust predictions from modelling approaches by allowing for parameter uncertainty and combining different sources of information, and is thus expected to be useful in application to a large number of disease systems.

Figure 1

(a) Distribution$\stackrel{\mathbf{~}}{\mathit{q}}{\mathbf{(}\mathit{P}}_{\mathit{obs}\mathbf{)}}$of observed within-herd prevalence,P_obs, as observed using the simulated ELISA test and after conditioning onP_obs>0. This replicates how the prevalence data used in this paper, from [15], were collected. Also shown (dashed vertical lines) are the quartile boundaries of that data (denoted in the text as Q₀to Q₄), and the first, second (median) and third quartile boundaries of $\tilde{q}\left(P_{\mathrm{obs}}\right)$, showing the fit of these to Q₁to Q₃. (b) and (c) Distribution of observed prevalence for two individual parameter sets. (b) is from a set of parameters which is given a low weight by the reweighting algorithm as it does not represent the data well, while (c) is given a higher weight.

Figure 2

(a) and (b) Density plots showing the distributionq(P)of simulated true within-herd prevalence,P, conditioned onP>0, as a function of time. Along a vertical line (at a fixed time) the density corresponds to the height of the distribution for different prevalences (similar to Figure 1(a) but for true rather than ELISA observed prevalence). (a) shows the distribution before applying the reweighting algorithm, (b) shows the same distribution with reweighting. Also shown are the first, second (median) and third quartile boundaries of these distributions as a function of time, with the median shown by the solid white line and the outer quartiles by the dotted lines. The initial condition has no infected individuals but the environmental contamination equivalent to a single high shedding animal. It can be seen that the distributions approach equilibrium. (c) shows the time dependence of the corresponding herd level prevalence (i.e. the proportion of farms with infected animals) comparing the value before (circles) and after (triangles) the reweighting. (d) The distribution p(P) for both non-weighted (circles) and weighted models (triangles), at a single timepoint where the system is deemed to be in equilibrium.

Figure 3

Evolution of herd level prevalence with control. Time evolution of the distribution p(P) of the underlying (rather than ELISA observed) prevalence, with the introduction of an ELISA test and cull strategy at t=600 months.

Figure 4

Effect of control on herd level prevalence. The effect of a range of control strategies on the herd level prevalence, 1−p(0) (i.e. the proportion of farms with infected animals) over time. The control strategies are in place throughout the period shown on the x-axis. The strategies shown are: no control (∘), annual ELISA test and cull positive animals (△), infection management by reduction of both calf exposure and infection by clinical animals ( + ), ELISA and infection management together (×) and annual faecal culture test and cull positive animals combined with infection management (◇).

Figure 5

Comparison of control options. Distributions p(P) of the within herd prevalence P, conditioned on P>0, for three different control strategies and showing different timepoints, before the control commences and 5, 10 and 25 years after it starts. In all cases the mode of the distribution moves from higher to lower prevalence, showing that farms have fewer infected animals the longer the control strategy has been in place. (a) Infection management by reduction of both calf exposure and infection by clinical animals. (b) Annual ELISA test and cull positive animals. (c) ELISA and infection management together. The combination of ELISA and management can be seen to have a significant long term effect on prevalence.