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Modelling is everywhere

The spread of Q fever in a herd

Pertinent models can be developed even when significant doubt exists about the values of parameters. One example is the spread of Q fever among herds of dairy cattle. The model shows that contagion within a herd is influenced by the date of infection and the number of animals with antibodies.

By Evelyne Lhoste - Pascale Mollier, translated by Emma Morton-Saliou
Updated on 05/24/2013
Published on 04/15/2013

Q fever is a common illness among ruminants, from which the illness spreads most easily to humans. Q fever effects the fertility of those infected. It is caused by the Coxiella burnetii bacterium, which is mostly spread through the air, making it difficult to detect and prevent. Little quantitative knowledge exists on Q fever.

Contagion in a herd varies

Veterinary epidemiologists and biomathematicians at INRA (Angers-Nantes and Jouy-en-Josas) worked together to design a unique model which evaluates the level of infectiousness among different herds according to the date and extent of the infection. For the first time, scientists were able to estimate variations in the bacterial load in the herd’s environment.
The study showed that human health risk varied between infected herds which in appearance are similar. Differences in how the illness spread were noted in relation, it is thought, to the earlier or later appearance of the epidemic prior to observations. Furthermore, resistance in infected animals and their possession of antibodies plays a central role in the spread of the illness and extends the period of bacterial shedding.

235 cows studied one by one

Researchers studied a total of 235 cows in five infected herds at different times. A sample was taken once a week for four weeks from most of the cows, followed by PCR analyses and ELISA tests to establish the presence of pathogenic bacteria and antibodies. The data was used to ascertain – with a certain margin of error – when a healthy cow becomes ill, and when a sick cow has recovered.

Much uncertainty

The model had to take into account several inaccuracies initially linked to margins of error in the diagnostic tests: a positive result does not necessarily mean a cow is infected.
In addition, 30% of data was missing, since certain samples could not be taken (of the 1,175 samples expected, only 821 were obtained).
The absence of a significant chunk of data made statistical inference especially difficult, and was the main reason that scientists used a Bayesian approach. When the data was modelled, parameters of the model considered to be random variables had to be modelled as well. Defining their probability distributions best summarizes current knowledge before the data is used. This amounts to establishing the potential orders of magnitude of each parameter. Here, statistical inference involves the modification of the probabilities associated with the model’s parameters after assessing the information available in the data. These new probabilities, which represent updated current knowledge, cannot be calculated in analytical terms – it can only be understood in numerical terms with the help of iterative computer-based algorithms.

A complex probabilistic model...

Q fever modelled using a probabilistic mathematical model. © INRA. © INRA
Q fever modelled using a probabilistic mathematical model. © INRA © INRA

While it can be formally written out by hand, the model itself is coded in computer language which can be taken up by the programme that performs the calculations. The model incorporates 35 different parameters.

To start with, data is interpreted using a probabilistic mathematical model presented in graph form above. Each of the four boxes represents a cow’s state of health: S (healthy but susceptible to the illness), I (infected, shedding bacteria in the environment, no antibodies), I+ (infected, shedding bacteria in the environment, with antibodies) and R (resistant and not shedding). A cow’s state of health can only evolve in one of the directions indicated by a large arrow. Each arrow is linked to a probability, a key parameter of the spreading of the epidemic. These parameters are not the only factors guiding this system – bacterial load in the surrounding air is essential, and depends for the most part on the number of cows infected (I- and I+).

After that, the model is simultaneously applied to all the cows in the study. Also included are pertinent intra-herd interaction and the primary uncertainties in the observations (e.g. a positive result does not necessarily mean a cow is infected ). The structure of these relationships is illustrated in the directed acyclic graph (DAG) below. Each ellipsis represents either an item of data (observation, diagnostic test result) or a hidden parameter or variable (dynamic characteristic of the environment) in the model. Arrows indicate a direct relationship between them. Three rectangles virtually multiply the ellipses and arrows by the number of objects they represent: number of herds (h), number of cows per herd (i) and length of time observed (t), respectively.

A directed acyclic graph (DAG) of the model of the spread of Q fever © INRA. © INRA
A directed acyclic graph (DAG) of the model of the spread of Q fever © INRA © INRA

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Applied Mathematics and Informatics
Associated Centre(s):
Angers-Nantes, Jouy-en-Josas

References

Courcoul, A., Vergu, E., Denis, J.-B. and Beaudeau, F. 2010. Spread of Q fever within dairy cattle herds : key parameters inferred using a Bayesian approach. Proceedings of the Royal Society, B.2857-2865

This study was conducted over an 18-month period as part of the doctoral thesis of Aurélie Courcoul. She is now a research veterinarian at the ANSES (France’s National agency on food safety, environment, and workplace security). She is specialized in animal epidemiology and is currently working on bovine tuberculosis.