Reviewer 1's Reviews

Comments on initial submission:
“A brief summary of the paper

The authors present a detailed analysis of the 2001 Foot-and-Mouth (FMD) outbreak in the UK.

They modify and extend a model previously used by Keeling et. al (2001) in order to describe the dynamics of disease. Then, they adopt a Bayesian approach to draw inference about the model’s parameters using Markov Chain Monte Carlo (MCMC) methods.


The authors note in the introductory section of the paper, that the UK experienced a range of the severe economic and social effects of the FMD epidemic and therefore, understanding the risk factors underlying the transmission dynamics of that epidemic will potentially allow minimisation of the scale and the cost of any future outbreak. It is also mentioned that a weakness of some of the modelling studies undertaken in 2001 (during the outbreak) was the relatively ad hoc nature of the parameter estimation methods.
In the second section of the paper, entitled as “Methods”, the authors first describe the level of detail in their dataset and briefly provide some descriptives statistics. Then, they describe in detail the simplest model they used in their analysis. In addition, they refer to further extensions.
Furthermore, they provide the statistical framework they used so as to draw inference for the parameters and compare the various models of different complexity. Finally, an extensive section with the results is presented, as well as the corresponding conclusions.

The main findings of the paper are that cattle are more infectious and also more susceptible than sheep. In addition, there is evidence for some difference in the parameter estimates between and after the interventions which took place during the epidemic. Furthermore, the authors allowed for nonrandom mixing between species and inferred that like species mix with like. Of those farms which were indicated as Contiguous Premises (CPs) or Dangerous contacts (DCs) the authors estimated a very low proportion of being infectious before culling.

Comments - Suggestions

The authors provide a nice introduction explaining why understanding the evolution of the 2001 FMD outbreak is still, 6 years after the epidemic, of scientific interest. Although this section is quite short, they authors successfully give the key references regarding the work which has been done on modelling the epidemic by researchers during the outbreak. A key reference in the analysis of purely temporal incidence data using MCMC methods is also O’Neill and Roberts (1999), and should be possibly added.

In the second section, where the data are described, it is not obvious why the statement by Keeling (via personal communication) about overestimation of the sheep numbers in Wales is necessary - unless the authors have adopted his approach as well. It is not very clear in the second last paragraph of the same section which numbers of the culled farms someone should sum up in order to get the total number of the 7457 (non-IPs) farms.

Although the authors explain why the have used the specific form of the spatial kernel in the “Discussion” section, it would have been more coherent if they have placed this argument in the “Model Formulation” section. Furthermore, the use of the Euclidian distance as distance metric should have been mentioned in the same section where the kernel is defined rather than the discussion section.

The authors attempt to account for the possibility that between-species and between-farm mixing might not be at random; however,I haven’t really understood how this assortative parameter _ is connected with between-species and between-farm mixing. Possibly more explanation about the role of _ in the model is needed.

It is assumed in the paper that the transmission parameters are constant in time throughout the epidemic. Nevertheless, the authors also look at more complex models which allow the parameters to be different before and after of the two interventions which took place during the outbreak: the 23rd of February and the 31th of March. It would have been useful if the more complex models were explicitly written down (not necessarily in the main body of the paper).

The section entitled as “Statistical Inference and Model Comparison” could have been divided in different subsections in order to be easier for the reader to follow. In addition, a reference for the Metropolis Hastings algorithm should be given, eg. Metropolis et. al. (1953), Hastings (1970). The last sentence of the this section “Re-parameterizing the models by estimating an infectivity ratio (cattle/sheep) and sheep infectivity ... ” is a bit confusing as far this ratio does not appear in Table 1 and is questionable whether it is necessary or not. Moreover, a reference is missing eg. Roberts et. al. (1997), regarding the optimal acceptance rate of the Metropolis Hastings algorithm.

In the “Results” section the authors perform informal statistical model comparisons and summarize their conclusion about the parameters which govern the disease’s transmission. Apart from point estimates, they also report “credible intervals”. It should be clarified whether equal-tailed or highest posterior density intervals were used. If the latter, then the method used to obtain them should be specified.

In both Tables 1 and 2 credibility intervals for a parameter named as “Infectivity Factor” are given; nevertheless, the definition of this parameters is not given in the text. On the other hand, an “infectivity ratio” parameter is given in the text but is not shown in any of the tables; perhaps they are the same? Regarding the predictive maps which are given in the paper, a connection between the infectiousness of a specific farm j to all farms i, i.e. Equation (12), and the basic farm-specific reproductive number R0 could have been made (Figures 4 and 5).

The authors present a simple method to predict how many farms, particularly DCs or CPs were infected but according to their model not diagnosed. This is done by weighting the probability of exposure time (T) prior to culling for a proactively culled farm with the distribution of time infection-report. In general, such an idea, seems very interesting since it will allow assessment of the adopted proactive culling policy. Nevertheless, it would have been better if the method was explained in more detail. For instance, is the following quantity evaluated:

Pr(0<Exposure Time<Time from Infection to Report) ?

If yes, are these probabilities estimated for every single DC or CP? In the Conclusions section the authors claim that “ ... it is almost certain that only a small proportion of the DCs and CPs culled were infected”. If such a statement is not based on their derived results, then further sufficient justification for it is needed.

References •

• Hastings, W. K. (1970), ”Monte Carlo sampling methods using Markov chains and their applications” Biometrika, 57, 97-109
• Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. & Teller, E. (1953), ”Equations of state calculations by fast computing machines,” Journal of Chemical Physics, 21, 1087-92
• O’Neill, P. D. and Roberts, G. O. (1999). Bayesian inference for partially observed stochastic epidemics. J. Roy. Statist. Soc. Ser. A, 162:121-129
• Roberts, G. O., Gelman, A., Gilks, W. R. (1997), ”Weak Convergence and Optimal Scaling of Random Walk Metropolis Algorithms,” Annals of Applied Probability, 7, 110-120”

Comments on first revision
“The authors have made some changes to the paper which was originally submitted. They have corrected the typos, expanded some of the model equations and rephrased some sentences in order the text to become more coherent and clearer to the reader. They also explained in more detail in this revised version their method to predict how many farms of unobserved infectious status, particularly DCs or CPs, were infected. Nevertheless it is still not very transparent how the assortative parameter ρ is related between-species and between-farm mixing. It might be helpful to see how Equation 1.2 changes when the parameter ρ is inserted and this could possibly explain intuitively its usage.

Regarding the sensitivity analysis presented in the last section. By adopting a Bayesian inference and using data augmentation techniques such as those cited in the paper, the problem of not observing the infection times can be overcome. Moreover, inference whether CPs or DCs were infectious before culling can also be drawn using similar methodology. Nevertheless, this is not an easy task due to the high dimension of the data and requires indeed non−standard MCMC methods and techniques.

Nevertheless, randomising the infectious times (by adding some random Gaussian noise) shouldn’t massively affect the point estimates of the model parameters, such as relative infectivity and susceptibility.

What it might affect, is the uncertainty about these estimates. Therefore, apart from showing point estimates which are obtained from the different simulations, it might be worth showing (approximate) credible intervals as well.”

N.B. These are the general comments made by the reviewer when reviewing this paper in light of which the manuscript was revised before publication. Specific points addressed during revision of the paper are not shown.